Algebra I Reference

Unit 1 25 topics

Equations & Inequalities

Solving for an unknown, modeling with it, and describing ranges of answers.

1.1 Linear Equations

1.1.1
Solving Linear Equations With Unknown Coefficients
Treat the other letters as numbers and isolate the variable you want. Divide by its coefficient only when that coefficient can’t be zero.
e.g. Solve $ax - 3 = 7$ for $x$ : $x = \tfrac{10}{a}$
$\displaystyle ax + b = c$
$\displaystyle \Rightarrow\; x = \frac{c - b}{a} \quad (a \ne 0)$
1.1.2
Solving Linear Equations With Unknown Coefficients by Factoring
When the variable appears in more than one term, gather those terms, factor the variable out, then divide.
e.g. $kx + 2x = 9 \Rightarrow x = \tfrac{9}{k + 2}$
$\displaystyle ax + bx = c$
$\displaystyle \Rightarrow\; x(a + b) = c$
$\displaystyle \Rightarrow\; x = \frac{c}{a + b}$
1.1.3
Solving Linear Inequalities With Unknown Parameters
Same steps as an equation. If you divide by a parameter, its sign decides whether the inequality flips.
$\displaystyle ax > b \;\Rightarrow\; \begin{cases} x > \frac{b}{a} & a > 0 \\[2pt] x < \frac{b}{a} & a < 0 \end{cases}$
1.1.4
Solving Many-Variable Equations
To solve a formula for one variable, undo the operations around it until it stands alone.
e.g. $P = 2l + 2w \Rightarrow w = \tfrac{P - 2l}{2}$
$\displaystyle A = \tfrac{1}{2}bh \;\Rightarrow\; h = \frac{2A}{b}$

1.2 Modeling With Linear Equations

1.2.1
Interpreting Linear Expressions
In $mx + b$ , $b$ is the fixed starting amount and $m$ is the amount added for each unit of $x$ .
e.g. $25 + 0.15m$ : a $25 fee plus 15¢ per mile
$\displaystyle \text{total} = \underbrace{m}_{\text{per unit}}\,x + \underbrace{b}_{\text{fixed}}$
1.2.2
Modeling With Linear Equations
Name the unknown, write the equation the words describe, solve, and check that the answer makes sense.
e.g. $30 to join plus $10 a month, $110 spent → $30 + 10m = 110$ , $m = 8$ months
1.2.3
Consecutive Integer Problems
Call the first integer $n$ . Consecutive integers are $n, n+1, n+2$ . Consecutive even or odd integers are $n, n+2, n+4$ .
e.g. Three consecutive integers sum to 48 → $3n + 3 = 48$ , $n = 15$
$\displaystyle n + (n + 1) + (n + 2) = 3n + 3$
1.2.4
Speed, Distance, Time Problems
Write $d = st$ for each traveler, then connect them: same distance, same time, or distances that add up.
e.g. Cars leave together at 50 and 60 mph in opposite directions → $110t = 330$ when 330 mi apart, $t = 3$ h
$\displaystyle d = s\,t$
1.2.5
Further Speed, Distance, Time Problems
A chaser closes a gap at the difference of the speeds. Travelers heading toward each other close it at the sum.
$\displaystyle t = \frac{\text{gap}}{s_1 - s_2} \;\text{(catching up)}$
$\displaystyle t = \frac{\text{gap}}{s_1 + s_2} \;\text{(approaching)}$
1.2.6
Modeling Work Problems
Each worker does a fraction of the job per hour. Working together, the rates add.
e.g. Pipes fill a tank in 3 h and 6 h → $\tfrac13 + \tfrac16 = \tfrac12$ → together 2 h
$\displaystyle \frac{1}{a} + \frac{1}{b} = \frac{1}{t}$
1.2.7
Modeling Mixture Problems
Track the pure ingredient: concentration times volume, added across the parts.
e.g. 10 L at 20% plus $x$ L at 50% makes 30% → $2 + 0.5x = 0.3(10 + x)$ , $x = 5$
$\displaystyle c_1V_1 + c_2V_2 = c\,(V_1 + V_2)$

1.3 Linear Inequalities

Closed dot and square bracket: the endpoint is included. Open dot and parenthesis: it is not. Infinity always takes a parenthesis.

1.3.1
Solving Compound Inequalities
For $a < bx + c < d$ , do each operation to all three parts. Dividing by a negative flips both signs.
$\displaystyle -1 < 2x + 3 < 9$
$\displaystyle \Rightarrow\; -4 < 2x < 6$
$\displaystyle \Rightarrow\; -2 < x < 3$
1.3.2
Interval Notation
A square bracket includes the endpoint; a parenthesis excludes it.
$\displaystyle [a, b]:\ a \le x \le b$
$\displaystyle (a, b):\ a < x < b$
$\displaystyle [a, b):\ a \le x < b$
1.3.3
Unbounded Intervals
Infinity always takes a parenthesis, because it is not a number you can reach.
$\displaystyle x \ge 2 \;\to\; [2, \infty)$
$\displaystyle x < 5 \;\to\; (-\infty, 5)$
$\displaystyle \text{all reals} \;\to\; (-\infty, \infty)$
1.3.4
Unions of Intervals
$\cup$ means “or”: everything in either interval.
$\displaystyle (-\infty, 1) \cup [4, \infty)$
1.3.5
Intersections of Intervals
$\cap$ means “and”: only the part both intervals share.
$\displaystyle [0, 5] \cap (3, 8] = (3, 5]$
1.3.6
Compound OR Inequalities
Solve each part on its own. The answer is the union.
$\displaystyle x < -2 \;\text{or}\; x \ge 1$
$\displaystyle \to\; (-\infty, -2) \cup [1, \infty)$
1.3.7
Compound AND Inequalities
Solve each part. The answer is the overlap, and if nothing overlaps there is no solution.
$\displaystyle x > -1 \;\text{and}\; x \le 4 \;\to\; (-1, 4]$
1.3.8
Introduction to Modeling With Inequalities
Translate the words: at least ≥, at most ≤, no more than ≤, more than >, fewer than <.
e.g. “At least 18 years old” → $a \ge 18$
1.3.9
Modeling With One-Step Inequalities
Write it, solve it, then round in the direction the situation demands.
e.g. Bags hold 12 lb, 50 lb of sand → $12b \ge 50$ , $b \ge 4.17$ → 5 bags
1.3.10
Modeling With Two-Step Inequalities
A fixed amount plus a rate, compared with a limit.
e.g. $40 budget, $10 entry plus $6 per ride → $10 + 6r \le 40$ , $r \le 5$
$\displaystyle b + rx \le L$
1.3.11
Modeling With Compound Inequalities
A quantity that must stay inside a range gives a three-part inequality.
e.g. Keep milk between 35°F and 40°F → $35 \le T \le 40$

1.4 Nonlinear Equations

1.4.1
Solving Equations Using the Square Root Method
Isolate the square, then take the square root of both sides. Keep the negative root.
e.g. $(x - 3)^2 = 16 \Rightarrow x - 3 = \pm 4 \Rightarrow x = 7$ or $-1$
$\displaystyle x^2 = k \;\Rightarrow\; x = \pm\sqrt{k} \quad (k \ge 0)$
1.4.2
Solving Equations With Odd Exponents Using the Nth Root Method
An odd power has exactly one real root, and it can be negative.
e.g. $x^3 = -64 \Rightarrow x = -4$
$\displaystyle x^n = k,\ n \text{ odd} \;\Rightarrow\; x = \sqrt[n]{k}$
1.4.3
Solving Equations With Even Exponents Using the Nth Root Method
An even power has two real roots when $k > 0$ , one when $k = 0$ , and none when $k < 0$ .
e.g. $x^4 = 81 \Rightarrow x = \pm 3$
$\displaystyle x^n = k,\ n \text{ even},\ k > 0$
$\displaystyle \Rightarrow\; x = \pm\sqrt[n]{k}$

Unit 2 33 topics

Two-Variable Equations & Inequalities

Lines, the situations they model, and where they cross.

2.5 Graphs of Linear Equations

Pick the form by what you know: slope and intercept, slope and a point, or a quick way to both intercepts.

2.5.1
Equations of Lines in Slope-Intercept Form
$m$ is the slope and $b$ is the $y$ -intercept.
e.g. $y = -2x + 5$ : slope $-2$ , crosses the $y$ -axis at $(0, 5)$
$\displaystyle y = mx + b$
$\displaystyle m = \frac{y_2 - y_1}{x_2 - x_1}$
2.5.2
Properties of Lines Given in Slope-Intercept Form
Positive $m$ rises, negative $m$ falls, zero is flat. Larger $\lvert m \rvert$ means steeper. The $x$ -intercept is where $y = 0$ .
$\displaystyle x\text{-intercept} = -\frac{b}{m}$
2.5.3
Equations of Lines in Point-Slope Form
Use it when you know the slope and one point on the line.
e.g. Slope 3 through $(2, -1)$ : $y + 1 = 3(x - 2)$
$\displaystyle y - y_1 = m(x - x_1)$
2.5.4
Equations of Lines in Standard Form
Integer coefficients with $x$ and $y$ on the same side. The quickest form for finding both intercepts.
e.g. $2x + 3y = 12$ : intercepts $(6, 0)$ and $(0, 4)$
$\displaystyle Ax + By = C$
2.5.5
Properties of Lines Given in Standard Form
Read the slope and intercepts straight from the coefficients.
$\displaystyle m = -\frac{A}{B}$
$\displaystyle x\text{-int} = \frac{C}{A} \qquad y\text{-int} = \frac{C}{B}$
2.5.6
Parallel Lines in the Coordinate Plane
Parallel lines have equal slopes and different intercepts. Perpendicular slopes multiply to $-1$ .
e.g. Parallel to $y = 2x + 1$ through $(0, -3)$ : $y = 2x - 3$
$\displaystyle m_1 = m_2$
$\displaystyle m_1 m_2 = -1 \;\text{(perpendicular)}$

2.6 Modeling With Two-Variable Linear Equations

On a distance–time graph the slope is the speed. On a speed–time graph the slope is the acceleration, and the shaded area is the distance traveled.

2.6.1
Modeling With Linear Equations in Two Variables
Find the starting value (the intercept) and the rate (the slope), then write $y = mx + b$ .
e.g. Plant 4 cm tall growing 1.5 cm a week: $h = 1.5w + 4$
2.6.2
Modeling Totals Using Two-Variable Linear Equations
Two kinds of item with a fixed total: price times count for each, added.
e.g. Adult tickets $12, child $8, $200 total: $12a + 8c = 200$
$\displaystyle ax + by = T$
2.6.3
Analyzing and Interpreting Graphs of Linear Equations
The slope is the rate of change, in $y$ -units per $x$ -unit. The $y$ -intercept is the value when $x = 0$ . The $x$ -intercept is where the quantity hits zero.
2.6.4
Further Modeling With Linear Equations in Two Variables
From two data points, compute the slope, then use point-slope form.
e.g. $50 for 2 h, $80 for 4 h → $m = 15$ , $C = 15h + 20$
$\displaystyle m = \frac{\Delta y}{\Delta x}$
2.6.5
Distance-Time Graphs
The slope is the speed. A flat stretch means stopped; a steeper line means faster; a falling line means heading back.
$\displaystyle \text{speed} = \text{slope} = \frac{\Delta d}{\Delta t}$
2.6.6
Speed-Time Graphs
The slope is the acceleration. A flat stretch means constant speed, not stopped.
$\displaystyle \text{acceleration} = \frac{\Delta v}{\Delta t}$
2.6.7
Calculating Distance From a Speed-Time Graph
Distance traveled is the area under the graph. Split it into rectangles and triangles.
e.g. 0 to 10 m/s over 4 s → $\tfrac12 \cdot 4 \cdot 10 = 20$ m
$\displaystyle d = \text{area under the } v\text{–}t \text{ graph}$

2.7 Systems of Equations

A system’s solution is where the lines meet. When elimination cancels both variables, the leftover statement tells you which case you have.

2.7.1
Introduction to the Elimination Method
Add or subtract the equations so that one variable cancels.
e.g. So $x = 7$ and $y = 3$
$\displaystyle \begin{array}{r} x + y = 10 \\ x - y = 4 \\ \hline 2x = 14 \end{array}$
2.7.2
Solving Systems of Linear Equations Using Elimination: One Transformation
Multiply one equation so a variable’s coefficients become opposites, then add.
e.g. $2x + y = 7$ , $x - 3y = 7$ : triple the first, add → $7x = 28$ , $x = 4$ , $y = -1$
2.7.3
Solving Systems of Linear Equations Using Elimination: Two Transformations
Multiply each equation by the other’s coefficient so one variable matches.
e.g. $3x + 2y = 16$ , $2x - 5y = -2$ : ×2 and ×3 → $19y = 38$ , $y = 2$ , $x = 4$
2.7.4
Systems of Linear Equations With Fractional Coefficients
Clear the fractions first: multiply each equation by its LCD.
$\displaystyle \tfrac{x}{2} + \tfrac{y}{3} = 4 \;\xrightarrow{\times 6}\; 3x + 2y = 24$
2.7.5
Systems of Linear Equations With Decimal Coefficients
Multiply by 10 or 100 to turn the decimals into integers.
$\displaystyle 0.2x + 0.5y = 1.6 \;\xrightarrow{\times 10}\; 2x + 5y = 16$
2.7.6
Systems of Equations With No Solutions and Infinitely Many Solutions
Both variables cancel. A false statement like $0 = 5$ means no solution (parallel lines). A true one like $0 = 0$ means infinitely many (the same line).
2.7.7
Consistency and Dependency in Linear Systems
Consistent: at least one solution. Inconsistent: none. Dependent: infinitely many. Independent: exactly one.
e.g. One solution is consistent and independent; the same line twice is consistent and dependent
2.7.8
Calculating the Intersection of Two Lines
Set the two expressions for $y$ equal, solve for $x$ , then find $y$ .
e.g. $y = 2x + 1$ and $y = -x + 7$ → $3x = 6$ → they meet at $(2, 5)$
$\displaystyle m_1x + b_1 = m_2x + b_2$
2.7.9
Modeling Using Systems of Linear Equations
Two unknowns need two equations. Usually one counts the items and one counts their value.
e.g. 150 tickets, adults $10, children $6, $1,140 → $a + c = 150$ , $10a + 6c = 1140$ → $a = 60$
2.7.10
Modeling Coin Problems Using Systems of Linear Equations
One equation for how many coins, one for what they’re worth. Work in cents.
e.g. → $d = 14$ dimes, $n = 16$ nickels
$\displaystyle n + d = 30$
$\displaystyle 5n + 10d = 220$
2.7.11
Solving Systems of Nonlinear Equations Using Graphs
Graph both. Each crossing point is a solution. A line and a parabola cross 0, 1, or 2 times.
2.7.12
Approximating Solutions to Systems of Linear Equations
Read the crossing off a graph, then check it by substituting. A graph reading is an estimate.
2.7.13
Approximating Solutions to Systems of Nonlinear Equations
Use a graph or a table to find where the two sides are closest, then narrow in.

2.8 Two-Variable Linear Inequalities

Draw the boundary line, decide dashed or solid, then use a test point to pick the side to shade.

2.8.1
Graphing Strict Two-Variable Linear Inequalities
For < or >, draw the boundary dashed, then shade the side that makes the inequality true. Test $(0, 0)$ if it isn’t on the line.
$\displaystyle y > mx + b$
$\displaystyle \to\; \text{dashed, shade above}$
2.8.2
Graphing Non-Strict Two-Variable Linear Inequalities
For ≤ or ≥, draw the boundary solid, because points on the line count.
$\displaystyle y \le mx + b$
$\displaystyle \to\; \text{solid, shade below}$
2.8.3
Further Graphing of Two-Variable Linear Inequalities
Rearrange into $y = \ldots$ first. Dividing by a negative flips the sign, which flips the shading.
$\displaystyle 2x - 3y > 6 \;\Rightarrow\; y < \tfrac{2}{3}x - 2$
2.8.4
Systems of Linear Inequalities
Graph each inequality. The solutions are where all the shadings overlap.
2.8.5
Modeling With Two-Variable Linear Inequalities
A budget or capacity limit becomes $ax + by \le C$ . Only sensible points count, often whole and nonnegative.
e.g. $60 for $5 books and $12 games: $5b + 12g \le 60$
2.8.6
Modeling With Systems of Linear Inequalities
Each constraint is one inequality. The workable choices lie in the overlap.
2.8.7
Modeling and Solving Systems of Linear Inequalities Using Technology
Enter each inequality in a graphing tool, read the overlap, and check its corner points.

Unit 3 13 topics

Units

Converting measurements, and quantities that vary together.

3.9 Working With Units

A conversion factor equals 1, so multiplying by it changes the units without changing the amount. The struck-through units cancel.

3.9.1
Unit Conversions Using Base Units of Mass
Metric prefixes are powers of ten: kilo is 1,000, centi is $\tfrac{1}{100}$ , milli is $\tfrac{1}{1000}$ . US: 16 oz = 1 lb, 2,000 lb = 1 ton.
e.g. $3.4\text{ kg} = 3400\text{ g}$
$\displaystyle 1\text{ kg} = 1000\text{ g}$
$\displaystyle 1\text{ g} = 1000\text{ mg}$
3.9.2
Two-Step Unit Conversions
Multiply by conversion fractions arranged so the unwanted units cancel.
$\displaystyle \frac{60\text{ mi}}{1\text{ h}} \cdot \frac{5280\text{ ft}}{1\text{ mi}} \cdot \frac{1\text{ h}}{3600\text{ s}} = 88\ \frac{\text{ft}}{\text{s}}$
3.9.3
Unit Conversions Using Base Units of Length
1 km = 1,000 m and 1 m = 100 cm = 1,000 mm. US: 12 in = 1 ft, 3 ft = 1 yd, 5,280 ft = 1 mi.
e.g. $2.5\text{ km} = 2500\text{ m}$
3.9.4
Unit Conversions Using Units of Time
60 s = 1 min, 60 min = 1 h, 24 h = 1 day, 7 days = 1 week.
$\displaystyle 1\text{ h} = 3600\text{ s}$
3.9.5
Converting Units of Area to Smaller Units
Square the length factor. Going to a smaller unit, multiply.
$\displaystyle 1\text{ m}^2 = 100^2\text{ cm}^2 = 10{,}000\text{ cm}^2$
3.9.6
Converting Units of Area to Larger Units
Square the length factor and divide.
$\displaystyle 5000\text{ cm}^2 = \frac{5000}{10{,}000}\text{ m}^2 = 0.5\text{ m}^2$
3.9.7
Determining Units in Formulas
Units follow the same algebra as the numbers: they multiply and divide.
$\displaystyle \text{m} \times \text{m} = \text{m}^2$
$\displaystyle \frac{\text{m}}{\text{s}} \div \text{s} = \frac{\text{m}}{\text{s}^2}$
3.9.8
Selecting Units for Rates of Change
A rate’s unit is the $y$ -unit per $x$ -unit. Choose a scale that gives sensible numbers.
e.g. Fingernail growth in mm per week, not km per year
$\displaystyle \text{rate unit} = \frac{y\text{-unit}}{x\text{-unit}}$
3.9.9
Converting Between Mixed Units
Convert everything to the smallest unit, compute, then convert back.
e.g. 2 h 45 min $= 165$ min; 200 min $=$ 3 h 20 min
3.9.10
Degrees of Accuracy
A measurement rounded to some unit can be off by up to half that unit either way.
$\displaystyle 7\text{ cm (to the nearest cm)}$
$\displaystyle \Rightarrow\; 6.5 \le x < 7.5$

3.10 Direct and Inverse Variation

Direct variation is a line through the origin, with y ÷ x constant. Inverse variation is a curve, with x × y constant.

3.10.1
Inverse Variation
When one quantity doubles, the other halves. Their product stays constant.
$\displaystyle y = \frac{k}{x} \quad\Leftrightarrow\quad xy = k$
3.10.2
Modeling With Direct Variation
$y$ varies directly with $x$ when $y = kx$ . Find $k$ from one pair of values, then use it.
e.g. $y = 12$ when $x = 3$ → $k = 4$ → at $x = 5$ , $y = 20$
$\displaystyle y = kx,\quad k = \frac{y}{x}$
3.10.3
Modeling With Inverse Variation
Find $k = xy$ from one pair, then $y = \tfrac{k}{x}$ .
e.g. 4 workers take 6 days → $k = 24$ → 8 workers take 3 days
$\displaystyle k = xy$

Unit 4 12 topics

Radical Expressions

Roots with variables in them, and equations that contain roots.

4.11 Radical Expressions

A root is a fractional power. Even roots need an absolute value when the sign of x is unknown; odd roots do not.

4.11.1
The Square Root of a Perfect Square With Algebraic Expressions
Halve each exponent.
$\displaystyle \sqrt{a^{2n}} = a^n$
$\displaystyle \sqrt{x^2y^4} = xy^2 \quad (x \ge 0)$
4.11.2
The Square Root of a Perfect Square With Domain Restrictions
With no sign given, $\sqrt{x^2}$ is the distance $\lvert x \rvert$ , because a square root is never negative.
e.g. $\sqrt{(-5)^2} = 5$ , not $-5$
$\displaystyle \sqrt{x^2} = \lvert x \rvert$
$\displaystyle \sqrt{x^6} = \lvert x^3 \rvert$
4.11.3
The Cube Root of a Perfect Cube With Algebraic Expressions
Divide each exponent by 3. No absolute value: a cube root keeps the sign.
$\displaystyle \sqrt[3]{x^3} = x$
$\displaystyle \sqrt[3]{-8a^6} = -2a^2$
4.11.4
Simplifying Square Root Expressions Using the Product Rule
Split off the largest perfect square, including even powers of the variables.
$\displaystyle \sqrt{ab} = \sqrt{a}\,\sqrt{b}$
$\displaystyle \sqrt{50x^3} = \sqrt{25x^2 \cdot 2x} = 5x\sqrt{2x}$
4.11.5
Combining Radical Expressions Using the Product Rule
Multiply the outside numbers together and the insides together, then simplify.
e.g. $2\sqrt3 \cdot 5\sqrt6 = 10\sqrt{18} = 30\sqrt2$
$\displaystyle a\sqrt{b} \cdot c\sqrt{d} = ac\sqrt{bd}$
4.11.6
Simplifying Square Root Expressions Using the Quotient Rule
Split the root over top and bottom, or simplify the fraction inside first.
e.g. $\sqrt{\tfrac{18x^3}{2x}} = \sqrt{9x^2} = 3x$ for $x > 0$
$\displaystyle \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$
4.11.7
Evaluating Algebraic Radical Expressions
Substitute, simplify everything under the root, then take the root.
e.g. $\sqrt{b^2 - 4ac}$ at $a = 1, b = 5, c = 6$ → $\sqrt{1} = 1$
4.11.8
Adding and Subtracting Radical Expressions
Only like radicals combine: same index, same radicand. Simplify each one first.
$\displaystyle 3\sqrt{x} + 5\sqrt{x} = 8\sqrt{x}$
$\displaystyle \sqrt{12} + \sqrt{27} = 2\sqrt3 + 3\sqrt3 = 5\sqrt3$
4.11.9
Rationalizing Denominators of Algebraic Expressions
Multiply top and bottom by the root in the denominator.
$\displaystyle \frac{a}{\sqrt{x}} = \frac{a\sqrt{x}}{x}$
4.11.10
Rationalizing Denominators With Two Terms
Multiply by the conjugate: the same two terms with the middle sign flipped. The difference of squares clears the root.
e.g. $\tfrac{1}{3 + \sqrt2} = \tfrac{3 - \sqrt2}{7}$
$\displaystyle \frac{1}{a + \sqrt{b}} \cdot \frac{a - \sqrt{b}}{a - \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$
4.11.11
Further Rationalizing Denominators of Algebraic Expressions
For a cube root, multiply by enough to make a perfect cube.
$\displaystyle \frac{1}{\sqrt[3]{x}} = \frac{\sqrt[3]{x^2}}{x}$
4.11.12
Solving Radical Equations
Isolate the radical, raise both sides to its power, solve, then check every answer. Squaring can create false solutions.
e.g. Gives $x = 1$ or $6$ ; only $6$ checks
$\displaystyle \sqrt{x + 3} = x - 3$
$\displaystyle \Rightarrow\; x + 3 = x^2 - 6x + 9$

Unit 5 7 topics

Rational Expressions & Equations

Fractions with variables, and the values they forbid.

5.12 Rational Expressions

5.12.1
Equivalent Expressions With Fractions
Multiply or divide top and bottom by the same nonzero expression.
$\displaystyle \frac{a}{b} = \frac{ak}{bk} \quad (k \ne 0)$
5.12.2
Simplifying Rational Expressions
Cancel common factors, never terms. Values that made the original denominator zero stay excluded.
$\displaystyle \frac{6x^2}{9x} = \frac{2x}{3},\ x \ne 0$
$\displaystyle \frac{x + 2}{x} \ne 2$
5.12.3
Simplifying Rational Expressions by Factoring
Factor top and bottom completely, then cancel the shared factors.
$\displaystyle \frac{x^2 - 9}{x^2 + 3x} = \frac{(x - 3)(x + 3)}{x(x + 3)} = \frac{x - 3}{x}$

5.13 Rational Equations

Cross-multiplying is multiplying both sides by both denominators at once. It only works when each side is a single fraction.

5.13.1
Solving Rational Equations Containing One Fractional Term
Isolate the fraction, then multiply both sides by its denominator. Exclude values that make it zero.
e.g. $\tfrac6x + 1 = 4 \Rightarrow \tfrac6x = 3 \Rightarrow x = 2$
5.13.2
Solving Rational Equations Using Cross-Multiplication
When one fraction equals another, multiply diagonally.
$\displaystyle \frac{a}{b} = \frac{c}{d} \;\Rightarrow\; ad = bc$
5.13.3
Solving Rational Equations Containing Binomials Using Cross-Multiplication
Cross-multiply, distribute both sides, solve. Check that no denominator becomes zero.
e.g. $x = 8$
$\displaystyle \frac{x + 1}{3} = \frac{x - 2}{2}$
$\displaystyle \Rightarrow\; 2x + 2 = 3x - 6$
5.13.4
Solving Rational Equations Using the Flip Method
If two fractions are equal, their reciprocals are too. Handy when the variable is in a denominator.
e.g. $\tfrac{3}{x - 1} = \tfrac14 \Rightarrow \tfrac{x - 1}{3} = 4 \Rightarrow x = 13$
$\displaystyle \frac{a}{b} = \frac{c}{d} \;\Rightarrow\; \frac{b}{a} = \frac{d}{c}$

Unit 6 18 topics

Functions

Rules that turn each input into exactly one output.

6.14 Functions

A function sends each input to exactly one output. On a graph, a vertical line that hits twice means some input has two outputs.

6.14.1
Introduction to Functions
A function gives each input exactly one output. $f(x)$ , read “f of x,” is the output at input $x$ .
$\displaystyle f(x) = 2x + 1 \;\Rightarrow\; f(3) = 7$
6.14.2
Visual Representations of Functions
Mapping diagrams, tables, and graphs. It is a function if no input has two arrows, two rows, or two points.
6.14.3
Graphs of Functions
The graph is every point $(x, f(x))$ . The value $f(a)$ is the height of the graph above $x = a$ .
6.14.4
The Domain of a Function
The allowed inputs. Exclude anything that divides by zero or takes an even root of a negative.
$\displaystyle f(x) = \frac{1}{x - 2}:\ x \ne 2$
$\displaystyle g(x) = \sqrt{x + 5}:\ x \ge -5$
6.14.5
The Vertical Line Test
A graph is a function if no vertical line crosses it more than once.
e.g. A circle fails; the parabola $y = x^2$ passes
6.14.6
Global Extrema of Functions
The highest output anywhere is the global maximum; the lowest is the global minimum. Some functions have neither.
6.14.7
End Behavior of Functions
What $f(x)$ does as $x \to \infty$ and as $x \to -\infty$ : rises, falls, or levels off.
e.g. $f(x) = -x^2$ falls at both ends
6.14.8
The Range of a Function
The outputs the function actually produces. Read them off the graph’s lowest and highest points.
e.g. Range of $x^2 + 1$ : $[1, \infty)$
6.14.9
The Range of a Function: Advanced Cases
Watch for values the graph approaches but never reaches, and for gaps.
e.g. $f(x) = \tfrac1x$ : range $(-\infty, 0) \cup (0, \infty)$
6.14.10
The Roots of a Function
Inputs where $f(x) = 0$ : where the graph crosses or touches the $x$ -axis.
e.g. Roots of $x^2 - 4$ : $\pm 2$
$\displaystyle f(x) = 0$
6.14.11
Increasing and Decreasing Functions
Increasing where the graph rises left to right, decreasing where it falls. State the $x$ -interval.
e.g. $x^2$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$
6.14.12
Piecewise Functions
Different rules on different parts of the domain. Find which piece the input belongs to, then use that rule.
e.g. $f(-3) = -1$ , $f(3) = 9$
$\displaystyle f(x) = \begin{cases} x + 2 & x < 0 \\ x^2 & x \ge 0 \end{cases}$
6.14.13
Modeling With Linear Functions
A constant rate of change means a linear model, $f(x) = mx + b$ .
6.14.14
Constructing Linear Functions
Find the slope from two points, then solve for $b$ .
e.g. $f(1) = 5$ , $f(3) = 11$ → $m = 3$ , $f(x) = 3x + 2$
$\displaystyle m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$
6.14.15
The Arithmetic of Functions
Add, subtract, multiply, or divide the outputs at the same input.
$\displaystyle (f + g)(x) = f(x) + g(x)$
$\displaystyle (fg)(x) = f(x)\,g(x)$
$\displaystyle \left(\tfrac{f}{g}\right)(x) = \tfrac{f(x)}{g(x)},\ g(x) \ne 0$
6.14.16
Function Composition
Feed one function’s output into another. Work from the inside out.
e.g. $f(x) = x^2$ , $g(x) = x + 1$ : $f(g(2)) = 9$ , $g(f(2)) = 5$
$\displaystyle (f \circ g)(x) = f(g(x))$
6.14.17
Describing Function Composition
Read $f(g(x))$ as “do $g$ , then $f$ .” Order matters: usually $f(g(x)) \ne g(f(x))$ .
6.14.18
Introduction to Inverse Functions
$f^{-1}$ undoes $f$ . To find it, swap $x$ and $y$ and solve for $y$ . Its graph is the reflection of $f$ in $y = x$ .
$\displaystyle f(f^{-1}(x)) = x$
$\displaystyle f(x) = 2x + 3 \;\Rightarrow\; f^{-1}(x) = \frac{x - 3}{2}$

Unit 7 14 topics

Absolute Value

Distance from zero, in equations, inequalities, and graphs.

7.15 Absolute Value Expressions, Equations & Inequalities

|x − 3| is the distance from x to 3. Less than means close to 3; greater than means far from it.

7.15.1
Absolute Value Expressions
Absolute value is distance from zero, so it is never negative. Evaluate inside first.
e.g. $\lvert 3 - 8 \rvert = 5$
$\displaystyle \lvert x \rvert = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}$
7.15.2
Rules of Absolute Value
It passes through multiplication and division, but not through addition.
$\displaystyle \lvert ab \rvert = \lvert a \rvert\,\lvert b \rvert$
$\displaystyle \left\lvert \tfrac{a}{b} \right\rvert = \tfrac{\lvert a \rvert}{\lvert b \rvert}$
$\displaystyle \lvert a + b \rvert \le \lvert a \rvert + \lvert b \rvert$
7.15.3
Further Rules of Absolute Value
A number and its opposite have the same absolute value.
$\displaystyle \lvert -a \rvert = \lvert a \rvert$
$\displaystyle \lvert a \rvert^2 = a^2$
$\displaystyle \sqrt{a^2} = \lvert a \rvert$
7.15.4
Absolute Value Equations
Isolate the absolute value, then split into two equations. A negative right side has no solution.
e.g. $\lvert 2x - 1 \rvert = 7 \Rightarrow x = 4$ or $-3$
$\displaystyle \lvert A \rvert = k,\ k \ge 0$
$\displaystyle \Rightarrow\; A = k \text{ or } A = -k$
7.15.5
Further Absolute Value Equations
When two absolute values are equal, the insides are equal or opposite.
$\displaystyle \lvert A \rvert = \lvert B \rvert$
$\displaystyle \Rightarrow\; A = B \text{ or } A = -B$
7.15.6
Equations Connecting Absolute Value and Linear Functions
When the other side contains $x$ , split into two cases and keep only answers where that side is $\ge 0$ .
7.15.7
Absolute Value Equations With Extraneous Solutions
Check every answer in the original equation. Some solutions of the split equations fail.
e.g. $\lvert x + 1 \rvert = 2x$ → $x = 1$ works; $x = -\tfrac13$ fails
7.15.8
Absolute Value Inequalities
Less than means between, an “and.” Greater than means outside, an “or.”
$\displaystyle \lvert A \rvert < k \;\Rightarrow\; -k < A < k$
$\displaystyle \lvert A \rvert > k \;\Rightarrow\; A < -k \text{ or } A > k$

7.16 Absolute Value Functions

Every absolute value graph is the V of y = |x| moved, flipped, or stretched. Find the vertex first.

7.16.1
Absolute Value Graphs
$y = \lvert x \rvert$ is a V with its corner at the origin. Shifted, the corner (vertex) is at $(h, k)$ .
$\displaystyle y = a\lvert x - h \rvert + k$
7.16.2
Vertical Reflections of Absolute Value Graphs
A negative sign in front flips the V upside down.
$\displaystyle y = -\lvert x \rvert$
7.16.3
Stretches of Absolute Value Graphs
$\lvert a \rvert > 1$ narrows the V; $0 < \lvert a \rvert < 1$ widens it. The arms have slopes $a$ and $-a$ .
7.16.4
Combining Transformations of Absolute Value Graphs
Plot the vertex $(h, k)$ , then step out with slopes $\pm a$ . Note that $\lvert x + 3 \rvert$ shifts left 3.
e.g. $y = -2\lvert x + 3 \rvert + 4$ : vertex $(-3, 4)$ , opens down
7.16.5
Domain and Range of Absolute Value Functions
The domain is all real numbers. The range starts at $k$ : $[k, \infty)$ if it opens up, $(-\infty, k]$ if down.
7.16.6
Roots of Absolute Value Functions
Set $y = 0$ and solve the absolute value equation. There can be zero, one, or two roots.
e.g. $\lvert x - 2 \rvert - 3 = 0 \Rightarrow x = 5$ or $-1$

Unit 8 22 topics

Exponential Functions

Exponent rules, and quantities that grow or shrink by a fixed factor.

8.17 Rules of Exponents

8.17.1
Writing Radical Expressions Using Fractional Exponents
The denominator is the root; the numerator is the power.
e.g. $x^{3/2} = \sqrt{x^3}$
$\displaystyle a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$
8.17.2
The Product Rule for Exponents With Algebraic Expressions
Multiply the coefficients, add the exponents of matching bases.
e.g. $3x^2 \cdot 4x^5 = 12x^7$
$\displaystyle a^m \cdot a^n = a^{m+n}$
8.17.3
The Quotient Rule for Exponents With Algebraic Expressions
Divide the coefficients, subtract the exponents of matching bases.
e.g. $\tfrac{10x^6y}{2x^2y} = 5x^4$
$\displaystyle \frac{a^m}{a^n} = a^{m-n}$
8.17.4
The Power Rule for Exponents With Algebraic Expressions
A power of a power multiplies the exponents.
e.g. $(x^3)^4 = x^{12}$
$\displaystyle (a^m)^n = a^{mn}$
8.17.5
The Power of Product Rule With Algebraic Expressions
The outside power goes to every factor, including the coefficient and its sign.
e.g. $(-2x^3y)^2 = 4x^6y^2$
$\displaystyle (ab)^n = a^nb^n$
8.17.6
The Power of Quotient Rule With Algebraic Expressions
The outside power goes to the top and the bottom.
e.g. $\left(\tfrac{x^2}{3}\right)^3 = \tfrac{x^6}{27}$
$\displaystyle \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$
8.17.7
Combining the Rules of Exponents With Algebraic Expressions
Apply outside powers, combine matching bases, and finish with positive exponents.
e.g. $\dfrac{(2x^2)^3}{4x^{-1}} = \dfrac{8x^6}{4x^{-1}} = 2x^7$

8.18 Exponential Expressions & Equations

8.18.1
Solving Exponential Equations
Rewrite both sides with the same base, then set the exponents equal.
e.g. $2^{x+1} = 32 = 2^5 \Rightarrow x = 4$
$\displaystyle b^m = b^n \;\Rightarrow\; m = n$
8.18.2
Solving Exponential Equations With Fractional Solutions
Use a base that both sides are powers of.
$\displaystyle 4^x = 8 \;\Rightarrow\; 2^{2x} = 2^3 \;\Rightarrow\; x = \tfrac{3}{2}$
8.18.3
Creating Exponential Growth Expressions
Growth by rate $r$ multiplies by $1 + r$ each period.
e.g. $500 growing 4% a year: $500(1.04)^t$
$\displaystyle a(1 + r)^t$
8.18.4
Creating Exponential Decay Expressions
Decay by rate $r$ multiplies by $1 - r$ each period.
e.g. A $20,000 car losing 15% a year: $20{,}000(0.85)^t$
$\displaystyle a(1 - r)^t$

8.19 Exponential Functions

Equal differences mean linear; equal ratios mean exponential. Any exponential growth eventually passes any linear growth.

8.19.1
Exponential Functions
Starting value $a$ , multiplied by base $b$ at each step. Growth if $b > 1$ , decay if $0 < b < 1$ .
$\displaystyle f(x) = a\,b^x \quad (a \ne 0,\ b > 0,\ b \ne 1)$
8.19.2
Modeling Exponential Growth With Functions
Find the starting amount and the growth factor per time unit. For doubling every $d$ units, use $2^{t/d}$ .
$\displaystyle P(t) = P_0(1 + r)^t$
$\displaystyle P(t) = P_0 \cdot 2^{t/d}$
8.19.3
Interpreting Exponential Growth
In $a(1.06)^t$ , $a$ is the starting value and $1.06$ means 6% growth each period.
e.g. $b = 1.25$ → 25% growth per period
$\displaystyle r = b - 1$
8.19.4
Solving Exponential Growth Problems
Substitute $t$ to find a future value. To find $t$ , match bases or try values.
e.g. 3,000 bacteria doubling hourly → after 4 h, $3000 \cdot 2^4 = 48{,}000$
8.19.5
Modeling Exponential Decay With Functions
Use the decay factor per period, or a half-life $h$ .
$\displaystyle P(t) = P_0(1 - r)^t$
$\displaystyle P(t) = P_0\left(\tfrac{1}{2}\right)^{t/h}$
8.19.6
Interpreting Exponential Decay
$b = 0.9$ means 90% is kept each period, so 10% is lost.
$\displaystyle r = 1 - b$
8.19.7
Solving Exponential Decay Problems
Count the periods, then apply the factor that many times.
e.g. 80 mg with a 6 h half-life → after 18 h, $80\left(\tfrac12\right)^3 = 10$ mg
8.19.8
Linear vs. Exponential Growth and Decay
Linear adds the same amount each step. Exponential multiplies by the same factor.
$\displaystyle \text{linear: } y_{n+1} - y_n = m$
$\displaystyle \text{exponential: } \frac{y_{n+1}}{y_n} = b$
8.19.9
Linear vs. Exponential Growth and Decay Models
In a table, constant differences mean linear and constant ratios mean exponential. Exponential growth eventually passes any linear growth.
e.g. 3, 6, 12, 24: ratio 2 → exponential
8.19.10
Graphing Exponential Growth Functions
Passes through $(0, a)$ , rises ever faster to the right, and flattens toward $y = 0$ on the left without touching it.
$\displaystyle y = a\,b^x,\ b > 1$
8.19.11
Graphing Exponential Decay Functions
Passes through $(0, a)$ and falls toward $y = 0$ on the right without reaching it.
$\displaystyle y = a\,b^x,\ 0 < b < 1$

Unit 9 20 topics

Polynomials

Building, multiplying, and taking apart expressions in powers of x.

9.20 Polynomials

A square with side a + b splits into four pieces. Its area is not a² + b²; the two rectangles make up the difference.

9.20.1
Introduction to Polynomials
A sum of terms $ax^n$ with whole-number exponents. No variables in a denominator or under a root.
e.g. $\tfrac1x$ and $\sqrt{x}$ are not polynomial terms
$\displaystyle 3x^4 - x^2 + 7x - 5$
9.20.2
The Degree of a Polynomial
The highest exponent. For a term with several variables, add their exponents. Standard form lists terms from highest degree down.
$\displaystyle \deg(5x^3 - 2x + 1) = 3$
$\displaystyle \deg(4x^2y^3) = 5$
9.20.3
Simplifying Polynomials
Combine like terms: same variables raised to the same powers.
$\displaystyle 3x^2 + 5x - x^2 + 2x = 2x^2 + 7x$
9.20.4
The Distributive Law for Polynomials
Multiply the outside term by every term inside.
$\displaystyle 2x(x^2 - 3x + 4) = 2x^3 - 6x^2 + 8x$
9.20.5
Adding and Subtracting Polynomials
Combine like terms. When subtracting, flip the sign of every term in the second polynomial.
$\displaystyle (4x^2 + 3x) - (x^2 - 5x) = 3x^2 + 8x$
9.20.6
Monomials, Binomials and Trinomials
One term, two terms, three terms.
$\displaystyle 7x^3 \qquad x + 4 \qquad x^2 - 5x + 6$
9.20.7
Multiplying Binomials
Every term times every term: First, Outer, Inner, Last.
$\displaystyle (a + b)(c + d) = ac + ad + bc + bd$
$\displaystyle (x + 3)(x - 5) = x^2 - 2x - 15$
9.20.8
Multiplying Polynomials
Multiply each term of the first by each term of the second, then combine like terms.
$\displaystyle (x + 2)(x^2 - x + 3)$
$\displaystyle = x^3 + x^2 + x + 6$
9.20.9
Squaring Binomials
Square, twice the product, square. The middle term is the one people forget.
e.g. $(x + 4)^2 = x^2 + 8x + 16$ , not $x^2 + 16$
$\displaystyle (a + b)^2 = a^2 + 2ab + b^2$
$\displaystyle (a - b)^2 = a^2 - 2ab + b^2$
9.20.10
Expanding Binomials Using Pascal's Triangle
Row $n$ gives the coefficients of $(a + b)^n$ ; each entry is the sum of the two above it. Powers of $a$ fall as powers of $b$ rise.
e.g. Rows: 1 · 1 1 · 1 2 1 · 1 3 3 1 · 1 4 6 4 1
$\displaystyle (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
9.20.11
The Difference of Squares Formula
The middle terms cancel.
e.g. $(x + 7)(x - 7) = x^2 - 49$
$\displaystyle (a + b)(a - b) = a^2 - b^2$

9.21 Factoring Polynomials

Factoring runs the box backward: the corners hold x² and the constant, the two middle cells add to the middle term, and the edges are the factors.

9.21.1
The Greatest Common Factor of Two Monomials
GCF of the coefficients, times each shared variable at its lowest power.
$\displaystyle \text{GCF}(12x^3y,\ 18x^2y^4) = 6x^2y$
9.21.2
Factoring Polynomials Using GCFs
Always look for a common factor first.
$\displaystyle 6x^3 - 9x^2 = 3x^2(2x - 3)$
9.21.3
Factoring Perfect Square Trinomials
First and last terms are squares and the middle is twice their product.
$\displaystyle a^2 + 2ab + b^2 = (a + b)^2$
$\displaystyle x^2 - 10x + 25 = (x - 5)^2$
9.21.4
Factoring Perfect Square Trinomials With Leading Coefficients
The same test, with a coefficient on the first term.
$\displaystyle 4x^2 + 12x + 9 = (2x + 3)^2$
9.21.5
Factoring Differences of Squares
Two squares with a minus between them.
e.g. A sum of squares $a^2 + b^2$ doesn’t factor
$\displaystyle a^2 - b^2 = (a + b)(a - b)$
$\displaystyle 9x^2 - 16 = (3x + 4)(3x - 4)$
9.21.6
Factoring Trinomials
For $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ .
e.g. $x^2 + 7x + 12 = (x + 3)(x + 4)$
$\displaystyle x^2 + bx + c = (x + p)(x + q)$
$\displaystyle pq = c,\quad p + q = b$
9.21.7
Factoring Trinomials Using Common Factors
Pull out the GCF first, then factor what is left.
$\displaystyle 2x^2 - 8x - 24 = 2(x^2 - 4x - 12)$
$\displaystyle = 2(x - 6)(x + 2)$
9.21.8
Factoring Trinomials With Leading Coefficients
For $ax^2 + bx + c$ , find two numbers that multiply to $ac$ and add to $b$ . Split the middle term and factor by grouping.
e.g. $2x^2 + 7x + 3$ : $ac = 6$ → 6 and 1 → $2x^2 + 6x + x + 3 = (2x + 1)(x + 3)$
9.21.9
Further Factoring Trinomials With Leading Coefficients
With a negative leading coefficient, factor out $-1$ first. Check by multiplying back.
e.g. $-3x^2 + x + 2 = -(3x^2 - x - 2) = -(3x + 2)(x - 1)$

Unit 10 37 topics

Quadratic Equations & Functions

Equations in x², their solutions, and the parabolas they draw.

10.22 Quadratic Equations

D = b² − 4ac. The solutions are where the parabola meets the x-axis, so D says whether it crosses twice, touches once, or misses.

10.22.1
Introduction to Quadratic Equations
Degree 2. Written in standard form, it has at most two real solutions.
$\displaystyle ax^2 + bx + c = 0 \quad (a \ne 0)$
10.22.2
Solving Perfect Square Quadratic Equations
Write it as a square equal to a number, then take $\pm$ square roots.
e.g. $(x - 2)^2 = 9 \Rightarrow x = 5$ or $-1$
$\displaystyle (x + p)^2 = k \;\Rightarrow\; x = -p \pm \sqrt{k}$
10.22.3
Perfect Square Quadratic Equations With One or No Solutions
$k = 0$ gives one solution. $k < 0$ gives none, since no real number squares to a negative.
$\displaystyle (x + p)^2 = 0 \;\Rightarrow\; x = -p$
$\displaystyle (x + p)^2 = -4 \;\Rightarrow\; \text{no real solution}$
10.22.4
The Zero Product Rule for Solving Quadratic Equations
If a product is zero, one of the factors is zero. One side must be 0 first.
e.g. $(x - 3)(x + 5) = 0 \Rightarrow x = 3$ or $-5$
$\displaystyle ab = 0 \;\Rightarrow\; a = 0 \text{ or } b = 0$
10.22.5
Solving Quadratic Equations Using a Difference of Squares
Factor the difference of squares and use the zero product rule.
e.g. $4x^2 - 25 = 0 \Rightarrow x = \pm\tfrac52$
$\displaystyle x^2 - k^2 = 0$
$\displaystyle \Rightarrow\; (x - k)(x + k) = 0$
$\displaystyle \Rightarrow\; x = \pm k$
10.22.6
Solving Quadratic Equations With No Constant Term
Factor out $x$ . Zero is always a solution; dividing by $x$ would lose it.
$\displaystyle ax^2 + bx = 0$
$\displaystyle \Rightarrow\; x(ax + b) = 0$
$\displaystyle \Rightarrow\; x = 0 \text{ or } x = -\tfrac{b}{a}$
10.22.7
Solving Quadratic Equations by Factoring
Get 0 on one side, factor, then set each factor to zero.
e.g. $x = 3$ or $-2$
$\displaystyle x^2 - x - 6 = 0$
$\displaystyle \Rightarrow\; (x - 3)(x + 2) = 0$
10.22.8
Solving Quadratic Equations With Leading Coefficients by Factoring
Factor $ax^2 + bx + c$ with the $ac$ method, then set each factor to zero.
e.g. $x = -\tfrac13$ or $2$
$\displaystyle 3x^2 - 5x - 2 = 0$
$\displaystyle \Rightarrow\; (3x + 1)(x - 2) = 0$
10.22.9
Completing the Square
Add the square of half the $x$ -coefficient.
e.g. $x^2 + 6x + 9 = (x + 3)^2$
$\displaystyle x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2$
10.22.10
Completing the Square With Odd Linear Terms
The same rule; half of an odd number is a fraction.
$\displaystyle x^2 + 5x + \tfrac{25}{4} = \left(x + \tfrac{5}{2}\right)^2$
10.22.11
Completing the Square With Leading Coefficients
Factor $a$ out of the $x$ -terms, complete the square inside, and adjust the constant.
$\displaystyle 2x^2 + 12x + 5 = 2(x + 3)^2 - 13$
10.22.12
Solving Quadratic Equations by Completing the Square
Move $c$ across, complete the square on both sides, take $\pm$ roots.
e.g. $x^2 + 6x = 7 \Rightarrow (x + 3)^2 = 16 \Rightarrow x = 1$ or $-7$
10.22.13
Solving Quadratic Equations With Leading Coefficients by Completing the Square
Divide every term by $a$ first, then complete the square.
e.g. $2x^2 - 8x - 10 = 0 \Rightarrow x^2 - 4x = 5 \Rightarrow (x - 2)^2 = 9 \Rightarrow x = 5$ or $-1$
10.22.14
The Quadratic Formula
Works for every quadratic in standard form.
e.g. $x^2 - 4x + 1 = 0 \Rightarrow x = 2 \pm \sqrt3$
$\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
10.22.15
The Discriminant of a Quadratic Equation
$b^2 - 4ac$ tells how many real solutions there are without solving.
$\displaystyle D = b^2 - 4ac$
$\displaystyle D > 0:\ \text{two solutions}$
$\displaystyle D = 0:\ \text{one solution}$
$\displaystyle D < 0:\ \text{no real solution}$
10.22.16
Determining Unknown Parameters in Quadratic Equations With One Real Solution
Set the discriminant equal to zero and solve for the parameter.
e.g. $x^2 + kx + 9 = 0$ has one solution when $k^2 = 36$ , $k = \pm 6$
$\displaystyle b^2 - 4ac = 0$
10.22.17
Modeling With Quadratic Equations
Write the area, motion, or number relationship as a quadratic, solve, and reject answers that make no sense, such as negative lengths.
e.g. Length 3 more than width, area 40 → $w(w + 3) = 40 \Rightarrow w = 5$
10.22.18
Equating Polynomial Coefficients
Two polynomials equal for every $x$ have equal coefficients term by term.
e.g. $(x + k)^2 = x^2 + 8x + m \Rightarrow k = 4,\ m = 16$
$\displaystyle ax^2 + bx + c \equiv px^2 + qx + r$
$\displaystyle \Rightarrow\; a = p,\ b = q,\ c = r$
10.22.19
Vieta's Formulas
The sum and product of the roots come straight from the coefficients.
e.g. $x^2 - 7x + 10$ : roots sum to 7 and multiply to 10 (they are 2 and 5)
$\displaystyle r_1 + r_2 = -\frac{b}{a}$
$\displaystyle r_1r_2 = \frac{c}{a}$

10.23 Quadratic Functions

The axis of symmetry runs through the vertex, halfway between the roots. Each form of the equation shows one of these features at a glance.

10.23.1
Graphing Elementary Quadratic Functions
$y = x^2$ is a U-shaped parabola with its vertex at the origin. In $y = ax^2$ , a larger $\lvert a \rvert$ makes it narrower.
$\displaystyle y = ax^2$
10.23.2
Vertical Reflections of Quadratic Functions
A negative $a$ opens the parabola downward, so the vertex becomes a maximum.
$\displaystyle y = -x^2$
10.23.3
Graphs of General Quadratic Functions
$a$ sets direction and width; $c$ is the $y$ -intercept.
$\displaystyle y = ax^2 + bx + c$
10.23.4
Roots of Quadratic Functions
Where the parabola meets the $x$ -axis: solve $ax^2 + bx + c = 0$ .
10.23.5
The Discriminant of a Quadratic Function
Counts the $x$ -intercepts: two, one (vertex on the axis), or none.
$\displaystyle D = b^2 - 4ac$
10.23.6
The Axis of Symmetry of a Parabola
A vertical line through the vertex. The parabola mirrors across it.
e.g. $y = x^2 - 6x + 5$ : axis $x = 3$
$\displaystyle x = -\frac{b}{2a}$
10.23.7
The Average of the Roots Formula
The axis sits exactly halfway between the roots.
e.g. Roots 1 and 5 → vertex at $x = 3$
$\displaystyle x_{\text{vertex}} = \frac{r_1 + r_2}{2}$
10.23.8
The Vertex Form of a Parabola
The vertex is at $(h, k)$ . Watch the sign inside: $(x - h)$ .
e.g. $y = 2(x + 1)^2 - 3$ : vertex $(-1, -3)$
$\displaystyle y = a(x - h)^2 + k$
10.23.9
Writing the Equation of a Parabola in Vertex Form
Complete the square, or find $h = -\tfrac{b}{2a}$ and then $k = f(h)$ .
e.g. $y = x^2 - 6x + 5 = (x - 3)^2 - 4$
10.23.10
Domain and Range of Quadratic Functions
The domain is all real numbers. The range starts at the vertex: $[k, \infty)$ if $a > 0$ , $(-\infty, k]$ if $a < 0$ .
10.23.11
Solving Systems of Nonlinear Equations
Substitute one equation into the other, set it to zero, and solve the quadratic.
e.g. Solutions $(2, 4)$ and $(-1, 1)$
$\displaystyle y = x^2,\ y = x + 2$
$\displaystyle \Rightarrow\; x^2 - x - 2 = 0$
10.23.12
Finding Intersections of Lines and Quadratic Functions
Set the two equal. The discriminant of the result gives 2 crossings, 1 (the line just touches), or 0.
$\displaystyle ax^2 + bx + c = mx + d$

10.24 Modeling With Quadratic Functions

A thrown object’s height is a downward parabola in time. The vertex gives the peak; the positive root gives the landing.

10.24.1
Modeling Downwards Vertical Motion
A dropped object falls $16t^2$ feet in $t$ seconds.
e.g. Dropped from 144 ft → lands when $16t^2 = 144$ , $t = 3$ s
$\displaystyle h(t) = -16t^2 + h_0$
$\displaystyle h(t) = -4.9t^2 + h_0 \;\text{(meters)}$
10.24.2
Modeling Upwards Vertical Motion
Launched upward at speed $v_0$ from height $h_0$ .
$\displaystyle h(t) = -16t^2 + v_0t + h_0$
10.24.3
Vertical Motion
Maximum height is at the vertex. It lands when $h(t) = 0$ ; keep the positive root.
e.g. $h = -16t^2 + 64t$ peaks at $t = 2$ s, 64 ft
$\displaystyle t_{\text{peak}} = -\frac{b}{2a} = \frac{v_0}{32}$
10.24.4
Revenue, Cost, and Profit Functions
Revenue is price times quantity. Profit is revenue minus cost.
$\displaystyle R = p\,q$
$\displaystyle P = R - C$
10.24.5
Constructing Revenue, Cost, and Profit Functions
When price depends on quantity, revenue is quadratic.
$\displaystyle p = 50 - 2q$
$\displaystyle \Rightarrow\; R = q(50 - 2q) = 50q - 2q^2$
10.24.6
Maximizing Profit and Break-Even Points
Maximum profit is at the vertex of $P$ . Break-even is where $R = C$ , which means $P = 0$ .
e.g. $P = -2q^2 + 40q - 150$ : best at $q = 10$ , profit 50
$\displaystyle q_{\max} = -\frac{b}{2a}$

Unit 11 18 topics

Sequences

Ordered lists built by a rule: add each time, or multiply.

11.25 Introduction to Sequences

Arithmetic sequences grow by equal steps, like linear functions. Geometric sequences grow by equal factors, like exponential functions.

11.25.1
Introduction to Sequences
An ordered list of terms. $a_n$ is the $n$ th term. An explicit formula gives any term straight from $n$ .
$\displaystyle a_n = n^2 \;\Rightarrow\; 1, 4, 9, 16, \ldots$
11.25.2
Recursive Sequences
Each term is built from the one before, starting from a given first term.
e.g. 3, 7, 15, 31, …
$\displaystyle a_1 = 3,\quad a_n = 2a_{n-1} + 1$
11.25.3
Fibonacci Sequences
Each term is the sum of the two before it.
e.g. 1, 1, 2, 3, 5, 8, 13, …
$\displaystyle F_n = F_{n-1} + F_{n-2},\quad F_1 = F_2 = 1$

11.26 Arithmetic Sequences

11.26.1
Arithmetic Sequences
Add the same common difference $d$ each time.
e.g. 5, 8, 11, 14: $d = 3$
$\displaystyle d = a_{n+1} - a_n$
11.26.2
Recursive Formulas for Arithmetic Sequences
Give the first term and the rule for the next.
$\displaystyle a_1 = \text{first term},\quad a_n = a_{n-1} + d$
11.26.3
The Nth Term of an Arithmetic Sequence
Start at $a_1$ and take $n - 1$ steps of $d$ .
e.g. 5, 8, 11, … → $a_{20} = 5 + 19 \cdot 3 = 62$
$\displaystyle a_n = a_1 + (n - 1)d$
11.26.4
Translating Between Explicit and Recursive Formulas for Arithmetic Sequences
Both use the same $a_1$ and $d$ .
$\displaystyle a_n = a_{n-1} + d$
$\displaystyle \leftrightarrow\; a_n = a_1 + (n - 1)d$
11.26.5
Finding the Common Difference of an Arithmetic Sequence
Subtract any term from the next. From two terms further apart, divide the gap by the number of steps.
$\displaystyle d = \frac{a_m - a_k}{m - k}$
11.26.6
Finding the Nth Term of an Arithmetic Sequence Given Two Terms
Find $d$ , step back to $a_1$ , then write the formula.
e.g. $a_3 = 10$ , $a_7 = 22$ → $d = 3$ , $a_1 = 4$ , $a_n = 3n + 1$
11.26.7
Determining Indexes of Terms in Arithmetic Sequences
Set the formula equal to the value and solve for $n$ . If $n$ isn’t a positive whole number, the value isn’t in the sequence.
e.g. Is 100 in $a_n = 3n + 1$ ? $n = 33$ , yes
11.26.8
Solving for Variables in Arithmetic Sequences
Three terms in a row are arithmetic when the middle one is the average of the other two.
e.g. $x,\ 2x + 1,\ 5x - 3$ → $2(2x + 1) = 6x - 3$ → $x = \tfrac52$
$\displaystyle b = \frac{a + c}{2}$
11.26.9
Modeling With Arithmetic Sequences
A fixed amount added per step is arithmetic. Check whether counting starts at $n = 0$ or $n = 1$ .
e.g. 20 seats in row 1, 2 more per row → row 15 has $20 + 14 \cdot 2 = 48$

11.27 Geometric Sequences

11.27.1
Introduction to Geometric Sequences
Multiply by the same common ratio $r$ each time.
e.g. 3, 6, 12, 24: $r = 2$
$\displaystyle r = \frac{a_{n+1}}{a_n}$
11.27.2
The Recursive Formula for a Geometric Sequence
Give the first term and multiply by $r$ for the next.
$\displaystyle a_1 = \text{first term},\quad a_n = r\,a_{n-1}$
11.27.3
The Nth Term of a Geometric Sequence
Start at $a_1$ and multiply by $r$ a total of $n - 1$ times.
e.g. 2, 6, 18, … → $a_6 = 2 \cdot 3^5 = 486$
$\displaystyle a_n = a_1r^{n-1}$
11.27.4
Translating Between Explicit and Recursive Formulas for Geometric Sequences
Both use the same $a_1$ and $r$ .
$\displaystyle a_n = r\,a_{n-1}$
$\displaystyle \leftrightarrow\; a_n = a_1r^{n-1}$
11.27.5
Finding the Common Ratio of a Geometric Sequence Given Two Terms
Divide the later term by the earlier one, then take the root for the number of steps between them.
e.g. $a_2 = 6$ , $a_5 = 162$ → $r^3 = 27$ , $r = 3$
$\displaystyle r^{m-k} = \frac{a_m}{a_k}$
11.27.6
Determining Indexes of Terms in Geometric Sequences
Set $a_1r^{n-1}$ equal to the value and match the powers of $r$ .
e.g. 5, 10, 20, …: $640 = 5 \cdot 2^7$ → $n = 8$

Unit 12 6 topics

Bivariate Statistics

Two variables at once: plotting pairs and fitting a line.

12.28 Correlation & Regression

r runs from −1 to 1. The sign gives the direction; the closer to ±1, the tighter the points hug the trend line.

12.28.1
Scatter Plots
Plot paired data $(x, y)$ as points to look for a relationship. The explanatory variable goes on the $x$ -axis.
12.28.2
Trend Lines
A straight line through the middle of the cloud, with about as many points above it as below.
$\displaystyle \hat{y} = mx + b$
12.28.3
Making Predictions Using Trend Lines
Substitute $x$ into the trend line. Predicting inside the data’s range (interpolation) is safer than beyond it (extrapolation).
12.28.4
Interpreting Trend Line Coefficients
The slope is the predicted change in $y$ for each one-unit increase in $x$ . The intercept is the predicted $y$ at $x = 0$ , which may be meaningless.
e.g. $\hat{y} = 2.5x + 60$ , hours studied vs. score: each hour adds about 2.5 points
12.28.5
Linear Correlation
$r$ measures how tightly points hug a line, from $-1$ to $1$ . The sign gives direction; near 0 means no linear pattern. Correlation does not show cause.
$\displaystyle -1 \le r \le 1$
12.28.6
Selecting a Regression Model
Match the shape: linear for steady change, exponential for steady percent change, quadratic for a curve that turns. Leftover patterns in the residuals mean a poor fit.

No topic matches that search. Try a single word, like “vertex” or “slope.”

Topic numbers match the course outline, so a topic here lines up with the same number in the lesson list.

1.1 Linear Equations

Solving Linear Equations With Unknown Coefficients

Solving Linear Equations With Unknown Coefficients by Factoring

Solving Linear Inequalities With Unknown Parameters

Solving Many-Variable Equations

1.2 Modeling With Linear Equations

Interpreting Linear Expressions

Modeling With Linear Equations

Consecutive Integer Problems

Speed, Distance, Time Problems

Further Speed, Distance, Time Problems

Modeling Work Problems

Modeling Mixture Problems

1.3 Linear Inequalities

Solving Compound Inequalities

Interval Notation

Unbounded Intervals

Unions of Intervals

Intersections of Intervals

Compound OR Inequalities

Compound AND Inequalities

Introduction to Modeling With Inequalities

Modeling With One-Step Inequalities

Modeling With Two-Step Inequalities

Modeling With Compound Inequalities

1.4 Nonlinear Equations

Solving Equations Using the Square Root Method

Solving Equations With Odd Exponents Using the Nth Root Method

Solving Equations With Even Exponents Using the Nth Root Method

2.5 Graphs of Linear Equations

Equations of Lines in Slope-Intercept Form

Properties of Lines Given in Slope-Intercept Form

Equations of Lines in Point-Slope Form

Equations of Lines in Standard Form

Properties of Lines Given in Standard Form

Parallel Lines in the Coordinate Plane

2.6 Modeling With Two-Variable Linear Equations

Modeling With Linear Equations in Two Variables

Modeling Totals Using Two-Variable Linear Equations

Analyzing and Interpreting Graphs of Linear Equations

Further Modeling With Linear Equations in Two Variables

Distance-Time Graphs

Speed-Time Graphs

Calculating Distance From a Speed-Time Graph

2.7 Systems of Equations

Introduction to the Elimination Method

Solving Systems of Linear Equations Using Elimination: One Transformation

Solving Systems of Linear Equations Using Elimination: Two Transformations

Systems of Linear Equations With Fractional Coefficients

Systems of Linear Equations With Decimal Coefficients

Systems of Equations With No Solutions and Infinitely Many Solutions

Consistency and Dependency in Linear Systems

Calculating the Intersection of Two Lines

Modeling Using Systems of Linear Equations

Modeling Coin Problems Using Systems of Linear Equations

Solving Systems of Nonlinear Equations Using Graphs

Approximating Solutions to Systems of Linear Equations

Approximating Solutions to Systems of Nonlinear Equations

2.8 Two-Variable Linear Inequalities

Graphing Strict Two-Variable Linear Inequalities

Graphing Non-Strict Two-Variable Linear Inequalities

Further Graphing of Two-Variable Linear Inequalities

Systems of Linear Inequalities

Modeling With Two-Variable Linear Inequalities

Modeling With Systems of Linear Inequalities

Modeling and Solving Systems of Linear Inequalities Using Technology

3.9 Working With Units

Unit Conversions Using Base Units of Mass

Two-Step Unit Conversions

Unit Conversions Using Base Units of Length

Unit Conversions Using Units of Time

Converting Units of Area to Smaller Units

Converting Units of Area to Larger Units

Determining Units in Formulas

Selecting Units for Rates of Change

Converting Between Mixed Units

Degrees of Accuracy

3.10 Direct and Inverse Variation

Inverse Variation

Modeling With Direct Variation