What is on the Algebra 1 final exam?

Most Algebra 1 finals cover linear equations and inequalities, functions and linear models, systems of equations, exponents and exponential functions, polynomials and factoring, quadratics, and basic statistics. A first-semester final usually stops at systems of equations.

How should I study for the Algebra 1 final?

Start about two weeks out. Do the quick checks for every unit without notes to find your weak units, study those lessons first, then switch to mixed practice from all units in the last few days, because the final mixes topics together.

What formulas do I need to know for the Algebra 1 final?

The slope formula, slope-intercept and point-slope form, the exponent rules, the exponential growth and decay models, the vertex formula and the quadratic formula. All of them are on the free StudyQuest formula sheet.

What is the hardest part of the Algebra 1 final?

For most students it is quadratics and factoring, because they build on every earlier skill. If your final is in December, it is usually systems of equations and word problems that ask you to write the equation yourself.

Finals review · Grades 8–10

Algebra 1 Final Exam Review

Everything on a typical Algebra 1 final, one unit at a time: the rules you need, the mistakes that cost the most points, and three practice questions per unit with answers. Start with the quick checks to find your weak units, then use the linked lessons to fix them.

Free from StudyQuest Academy · Updated October 6, 2026 · studyquest.academy/finals/algebra-1

Start the quick checks Printable PDF

For teachers

Share to Google Classroom

What’s on the Algebra 1 final

Most schools teach equations, inequalities, functions and systems in the first semester, then exponents, polynomials, quadratics and statistics in the second. If your final is in December, focus on the Semester 1 units.

Unit	Usually on	Lessons
1. Signed numbers & expressions	Semester 1	Algebra I: Signed Numbers & Absolute Value, Introduction to Algebra
2. Linear equations & formulas	Semester 1	Algebra I: Linear Equations & Formulas
3. Linear inequalities	Semester 1	Algebra I: Systems & Linear Inequalities
4. Functions & linear models	Semester 1	Algebra I: Function Notation & Linear Models, Functions & Patterns
5. Systems of equations	Semester 1	Algebra I: Systems & Linear Inequalities, Systems of Linear Equations
6. Exponents & exponential functions	Semester 2	Exponents & Scientific Notation
7. Polynomials & factoring	Semester 2	Algebra I: Polynomial Operations
8. Quadratic functions & equations	Semester 2	Algebra I: Quadratics
9. Statistics & data	Semester 2	Statistics: Data, Models & Chance, Scatter Plots & Bivariate Data

A two-week study plan

Day 14
Find your weak units
Do the quick checks for all 9 units without notes. Mark each unit solid, shaky or lost. This tells you where your study time goes.
Days 13–6
Fix the lost and shaky units, one a day
Start with the units you marked lost, then shaky. For each one, read the must-know list, work through the linked lesson, and redo that unit's quick checks until you get all three right.
Days 5–3
Mixed practice
Finals mix units together, so practice that way: pick problems from different units, work them without looking at which unit they came from, and check each answer.
Day 2
Formulas and mistakes
Review the formula sheet and every "common mistakes" list. Write out from memory the formulas you still look up.
Day 1
Light review and sleep
Redo only the questions you missed before. Then stop: a full night of sleep helps your memory more than a late cram session.

Unit 1 · Semester 1

Signed numbers & expressions

Must know

Order of operations: grouping symbols, exponents, then multiply and divide left to right, then add and subtract left to right.
Adding signed numbers: same signs add and keep the sign; different signs subtract and take the sign of the number with the larger absolute value.
Multiplying or dividing two negatives gives a positive; one negative gives a negative.
Absolute value is distance from zero, so it is never negative: $|-7| = 7$ .
Distribute to every term, $a(b + c) = ab + ac$ , and combine only like terms: $3x + 5x = 8x$ , but $3x + 5x^2$ cannot be combined.

Common mistakes

Writing $-3^2 = 9$ . Without parentheses the exponent comes first, so $-3^2 = -9$ , while $(-3)^2 = 9$ .
Distributing to only the first term. $-2(x - 4)$ is $-2x + 8$ : the negative multiplies the $-4$ too.

Quick check

1.1 Simplify $-4 + 9 - (-6)$ .
Show answer
$11$ . $-4 + 9 = 5$ , and subtracting $-6$ is the same as adding 6.
1.2 Simplify $3(2x - 5) - 4x$ .
Show answer
$2x - 15$ . Distribute to get $6x - 15 - 4x$ , then combine $6x - 4x$ .
1.3 Evaluate $2x^2 - x$ when $x = -3$ .
Show answer
$21$ . $2(-3)^2 - (-3) = 2(9) + 3 = 21$ .

Review the lesson: Algebra I: Signed Numbers & Absolute Value, Introduction to Algebra

Unit 2 · Semester 1

Linear equations & formulas

Must know

Undo operations in reverse order, doing the same thing to both sides.
With variables on both sides, collect the variable terms on one side first.
Clear fractions by multiplying every term on both sides by the least common denominator.
An equation can have one solution, no solution (you end with something false like $0 = 5$ ) or infinitely many (you end with something true like $0 = 0$ ).
Literal equations are solved the same way for a named variable: $A = lw$ gives $w = \frac{A}{l}$ .

Common mistakes

Multiplying only some terms when clearing fractions. Every term on both sides gets multiplied.
Dropping the negative in front of parentheses: $5 - (x + 2)$ is $3 - x$ , not $7 + x$ .

Quick check

2.1 Solve $4x - 7 = 2x + 9$ .
Show answer
$x = 8$ . Subtract $2x$ to get $2x - 7 = 9$ , add 7 to get $2x = 16$ .
2.2 Solve $\frac{x}{3} + 2 = 6$ .
Show answer
$x = 12$ . Subtract 2 to get $\frac{x}{3} = 4$ , then multiply by 3.
2.3 Solve $P = 2l + 2w$ for $w$ .
Show answer
$w = \frac{P - 2l}{2}$ . Subtract $2l$ from both sides, then divide by 2.

Review the lesson: Algebra I: Linear Equations & Formulas

Unit 3 · Semester 1

Linear inequalities

Must know

Solve an inequality like an equation, except flip the sign when you multiply or divide both sides by a negative number.
On a number line, use an open circle for $<$ or $>$ and a closed circle for $\le$ or $\ge$ .
"And" compound inequalities describe values between two numbers, like $-2 < x \le 5$ ; "or" inequalities describe values outside them.
Absolute value: $|x| < 4$ means $-4 < x < 4$ , and $|x| > 4$ means $x < -4$ or $x > 4$ .
To graph $y > 2x + 1$ , draw a dashed line and shade above it; use a solid line for $\ge$ or $\le$ .

Common mistakes

Forgetting to flip the sign: $-3x > 12$ becomes $x < -4$ , not $x > -4$ .
Drawing a solid boundary line for a strict inequality ( $<$ or $>$ ). Those points are not solutions, so the line is dashed.

Quick check

3.1 Solve $5 - 2x \ge 11$ .
Show answer
$x \le -3$ . Subtract 5 to get $-2x \ge 6$ , then divide by $-2$ and flip the sign.
3.2 Solve $-1 < 2x + 3 \le 9$ .
Show answer
$-2 < x \le 3$ . Subtract 3 from all three parts, then divide all three by 2.
3.3 Solve $|x - 2| \le 5$ .
Show answer
$-3 \le x \le 7$ . Rewrite as $-5 \le x - 2 \le 5$ and add 2 to each part.

Review the lesson: Algebra I: Systems & Linear Inequalities

Unit 4 · Semester 1

Functions & linear models

Must know

A function gives each input exactly one output; its graph passes the vertical line test.
Function notation: $f(3)$ is the output when the input is 3. It does not mean $f$ times 3.
Slope is the rate of change: $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Slope-intercept form is $y = mx + b$ , point-slope form is $y - y_1 = m(x - x_1)$ , and standard form is $Ax + By = C$ .
Parallel lines have equal slopes; perpendicular slopes are negative reciprocals, like $2$ and $-\frac{1}{2}$ .
In a word problem, the slope is the rate (per hour, per ticket) and the $y$ -intercept is the starting amount.

Common mistakes

Putting the change in $x$ on top of the slope fraction. Slope is rise over run: change in $y$ over change in $x$ .
Subtracting the coordinates in a different order on the top and the bottom, which flips the sign of the slope.

Quick check

4.1 If $f(x) = 3x - 4$ , find $f(-2)$ .
Show answer
$-10$ . $3(-2) - 4 = -6 - 4$ .
4.2 Find the slope of the line through $(1, 5)$ and $(4, -1)$ .
Show answer
$-2$ . $\frac{-1 - 5}{4 - 1} = \frac{-6}{3}$ .
4.3 Write the line with slope 3 through $(2, 1)$ in slope-intercept form.
Show answer
$y = 3x - 5$ . Start from $y - 1 = 3(x - 2)$ , distribute, then add 1.

Review the lesson: Algebra I: Function Notation & Linear Models, Functions & Patterns

Unit 5 · Semester 1

Systems of equations

Must know

The solution of a system is the point that makes both equations true: where the lines cross.
Substitution: solve one equation for a variable, then substitute that expression into the other equation.
Elimination: add or subtract the equations, multiplying one or both first if needed, so one variable cancels.
Lines with the same slope and different intercepts are parallel, so there is no solution; the same line has infinitely many solutions.
For a system of inequalities, the solution is the region where the shaded areas overlap.

Common mistakes

Stopping after finding $x$ . The answer to a system is an ordered pair, so substitute back to find $y$ .
Multiplying only one side of an equation, or only some of its terms, before eliminating.

Quick check

5.1 Solve the system $y = 2x + 1$ and $3x + y = 11$ .
Show answer
$(2, 5)$ . Substitute: $3x + 2x + 1 = 11$ , so $x = 2$ and $y = 2(2) + 1 = 5$ .
5.2 Solve the system $x + y = 10$ and $x - y = 4$ .
Show answer
$(7, 3)$ . Adding the equations gives $2x = 14$ , so $x = 7$ , then $y = 3$ .
5.3 How many solutions does the system $y = 4x - 2$ and $y = 4x + 3$ have?
Show answer
None. The lines have the same slope and different $y$ -intercepts, so they are parallel.

Review the lesson: Algebra I: Systems & Linear Inequalities, Systems of Linear Equations

Unit 6 · Semester 2

Exponents & exponential functions

Must know

Exponent rules: $a^m \cdot a^n = a^{m+n}$ , $\frac{a^m}{a^n} = a^{m-n}$ and $(a^m)^n = a^{mn}$ .
Zero and negative exponents: $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ (for $a \ne 0$ ).
An exponential function $y = a \cdot b^x$ shows growth when $b > 1$ and decay when $0 < b < 1$ .
Percent change: growth is $y = a(1 + r)^t$ and decay is $y = a(1 - r)^t$ , with the rate $r$ as a decimal.
Linear patterns add the same amount each step; exponential patterns multiply by the same factor each step.

Common mistakes

Forgetting to apply the outside exponent to the coefficient: $(2x^3)^2 = 4x^6$ , not $2x^6$ .
Using the percent as the base. 5% growth means $b = 1.05$ , not $b = 5$ or $b = 0.05$ .

Quick check

6.1 Simplify $\frac{x^7 \cdot x^{-2}}{x^3}$ .
Show answer
$x^2$ . The top is $x^{7 + (-2)} = x^5$ , and $\frac{x^5}{x^3} = x^2$ .
6.2 A town of 2,000 people grows 3% per year. Write a model for the population $P$ after $t$ years.
Show answer
$P = 2000(1.03)^t$ . Growth of 3% multiplies by $1 + 0.03$ each year.
6.3 Is $y = 50(0.8)^x$ growth or decay, and by what percent per step?
Show answer
Decay of 20% per step, because $0.8 = 1 - 0.2$ .

Review the lesson: Exponents & Scientific Notation

Unit 7 · Semester 2

Polynomials & factoring

Must know

Add and subtract polynomials by combining like terms; to subtract, distribute the negative to every term first.
Multiply binomials by distributing each term (FOIL): $(x + 3)(x - 5) = x^2 - 2x - 15$ .
Always factor out the greatest common factor first: $6x^2 + 9x = 3x(2x + 3)$ .
To factor $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ .
Special patterns: $a^2 - b^2 = (a + b)(a - b)$ and $a^2 + 2ab + b^2 = (a + b)^2$ .

Common mistakes

Squaring a binomial term by term: $(x + 4)^2 = x^2 + 8x + 16$ , not $x^2 + 16$ .
Skipping the GCF, which leaves the answer only partly factored.

Quick check

7.1 Multiply $(2x - 3)(x + 4)$ .
Show answer
$2x^2 + 5x - 12$ . $2x^2 + 8x - 3x - 12$ , then combine $8x - 3x$ .
7.2 Factor $x^2 - 7x + 12$ .
Show answer
$(x - 3)(x - 4)$ . $-3$ and $-4$ multiply to 12 and add to $-7$ .
7.3 Factor $4x^2 - 25$ .
Show answer
$(2x + 5)(2x - 5)$ . It is a difference of squares: $(2x)^2 - 5^2$ .

Review the lesson: Algebra I: Polynomial Operations

Unit 8 · Semester 2

Quadratic functions & equations

Must know

In $y = ax^2 + bx + c$ , the parabola opens up when $a > 0$ and down when $a < 0$ .
The vertex is on the axis of symmetry $x = -\frac{b}{2a}$ ; substitute that $x$ to find the $y$ -coordinate.
Solve by factoring with the zero product property, by taking square roots, or with the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
The discriminant $b^2 - 4ac$ tells you how many real solutions: positive gives two, zero gives one, negative gives none.
Zeros, roots, solutions and $x$ -intercepts all name the same points.

Common mistakes

Factoring before the equation equals zero. Rewrite $x^2 + x = 6$ as $x^2 + x - 6 = 0$ first.
Taking only the positive square root: $x^2 = 49$ gives $x = 7$ or $x = -7$ .

Quick check

8.1 Solve $x^2 - 5x - 14 = 0$ .
Show answer
$x = 7$ or $x = -2$ . It factors as $(x - 7)(x + 2) = 0$ .
8.2 Find the vertex of $y = x^2 - 6x + 5$ .
Show answer
$(3, -4)$ . $x = -\frac{-6}{2(1)} = 3$ , and $3^2 - 6(3) + 5 = -4$ .
8.3 How many real solutions does $2x^2 + 3x + 5 = 0$ have?
Show answer
None. The discriminant is $3^2 - 4(2)(5) = -31$ , which is negative.

Review the lesson: Algebra I: Quadratics

Unit 9 · Semester 2

Statistics & data

Must know

The mean is the average and the median is the middle value; the median is less affected by outliers.
Measures of spread: range, interquartile range ( $IQR = Q_3 - Q_1$ ) and standard deviation.
A box plot shows the five-number summary: minimum, $Q_1$ , median, $Q_3$ and maximum.
Scatter plots show positive, negative or no correlation; the correlation coefficient $r$ runs from $-1$ to $1$ , and values near either end are strong.
A line of best fit is for prediction, and correlation alone does not prove that one thing causes the other.

Common mistakes

Finding the median without putting the data in order first.
Calling $r = -0.9$ weak. The sign gives the direction; the distance from 0 gives the strength, so $-0.9$ is strong.

Quick check

9.1 Find the mean and median of 4, 8, 6, 5, 20. Which better describes a typical value?
Show answer
Mean $8.6$ , median $6$ . The median is better here because the outlier 20 pulls the mean up.
9.2 A box plot has $Q_1 = 12$ and $Q_3 = 30$ . What is the interquartile range?
Show answer
$18$ . $IQR = Q_3 - Q_1 = 30 - 12$ .
9.3 The line of best fit for test score $y$ and hours studied $x$ is $y = 2.5x + 40$ . Predict the score for 6 hours.
Show answer
$55$ . $2.5(6) + 40 = 55$ ; the slope means about 2.5 more points per extra hour.

Review the lesson: Statistics: Data, Models & Chance, Scatter Plots & Bivariate Data

Free tools for studying

Algebra 1 final exam FAQ

What is on the Algebra 1 final exam?: Most Algebra 1 finals cover linear equations and inequalities, functions and linear models, systems of equations, exponents and exponential functions, polynomials and factoring, quadratics, and basic statistics. A first-semester final usually stops at systems of equations.
How should I study for the Algebra 1 final?: Start about two weeks out. Do the quick checks for every unit without notes to find your weak units, study those lessons first, then switch to mixed practice from all units in the last few days, because the final mixes topics together.
What formulas do I need to know for the Algebra 1 final?: The slope formula, slope-intercept and point-slope form, the exponent rules, the exponential growth and decay models, the vertex formula and the quadratic formula. All of them are on the free StudyQuest formula sheet.
What is the hardest part of the Algebra 1 final?: For most students it is quadratics and factoring, because they build on every earlier skill. If your final is in December, it is usually systems of equations and word problems that ask you to write the equation yourself.

For parents & teachers

Get the free 7-Day Study Reset

A research-backed week of short routines that help a student study less and remember more: spacing, self-quizzing, and a calmer week before tests.

Answer key

Unit 1: Signed numbers & expressions

1.1 $11$ . $-4 + 9 = 5$ , and subtracting $-6$ is the same as adding 6.
1.2 $2x - 15$ . Distribute to get $6x - 15 - 4x$ , then combine $6x - 4x$ .
1.3 $21$ . $2(-3)^2 - (-3) = 2(9) + 3 = 21$ .

Unit 2: Linear equations & formulas

2.1 $x = 8$ . Subtract $2x$ to get $2x - 7 = 9$ , add 7 to get $2x = 16$ .
2.2 $x = 12$ . Subtract 2 to get $\frac{x}{3} = 4$ , then multiply by 3.
2.3 $w = \frac{P - 2l}{2}$ . Subtract $2l$ from both sides, then divide by 2.

Unit 3: Linear inequalities

3.1 $x \le -3$ . Subtract 5 to get $-2x \ge 6$ , then divide by $-2$ and flip the sign.
3.2 $-2 < x \le 3$ . Subtract 3 from all three parts, then divide all three by 2.
3.3 $-3 \le x \le 7$ . Rewrite as $-5 \le x - 2 \le 5$ and add 2 to each part.

Unit 4: Functions & linear models

4.1 $-10$ . $3(-2) - 4 = -6 - 4$ .
4.2 $-2$ . $\frac{-1 - 5}{4 - 1} = \frac{-6}{3}$ .
4.3 $y = 3x - 5$ . Start from $y - 1 = 3(x - 2)$ , distribute, then add 1.

Unit 5: Systems of equations

5.1 $(2, 5)$ . Substitute: $3x + 2x + 1 = 11$ , so $x = 2$ and $y = 2(2) + 1 = 5$ .
5.2 $(7, 3)$ . Adding the equations gives $2x = 14$ , so $x = 7$ , then $y = 3$ .
5.3 None. The lines have the same slope and different $y$ -intercepts, so they are parallel.

Unit 6: Exponents & exponential functions

6.1 $x^2$ . The top is $x^{7 + (-2)} = x^5$ , and $\frac{x^5}{x^3} = x^2$ .
6.2 $P = 2000(1.03)^t$ . Growth of 3% multiplies by $1 + 0.03$ each year.
6.3 Decay of 20% per step, because $0.8 = 1 - 0.2$ .

Unit 7: Polynomials & factoring

7.1 $2x^2 + 5x - 12$ . $2x^2 + 8x - 3x - 12$ , then combine $8x - 3x$ .
7.2 $(x - 3)(x - 4)$ . $-3$ and $-4$ multiply to 12 and add to $-7$ .
7.3 $(2x + 5)(2x - 5)$ . It is a difference of squares: $(2x)^2 - 5^2$ .

Unit 8: Quadratic functions & equations

8.1 $x = 7$ or $x = -2$ . It factors as $(x - 7)(x + 2) = 0$ .
8.2 $(3, -4)$ . $x = -\frac{-6}{2(1)} = 3$ , and $3^2 - 6(3) + 5 = -4$ .
8.3 None. The discriminant is $3^2 - 4(2)(5) = -31$ , which is negative.

Unit 9: Statistics & data

9.1 Mean $8.6$ , median $6$ . The median is better here because the outlier 20 pulls the mean up.
9.2 $18$ . $IQR = Q_3 - Q_1 = 30 - 12$ .
9.3 $55$ . $2.5(6) + 40 = 55$ ; the slope means about 2.5 more points per extra hour.

Finals review · Grades 8–10

Algebra 1 Final Exam Review

Free from StudyQuest Academy · Updated October 6, 2026 · studyquest.academy/finals/algebra-1

Start the quick checks Printable PDF

For teachers

Share to Google Classroom

What’s on the Algebra 1 final

Unit	Usually on	Lessons
1. Signed numbers & expressions	Semester 1	Algebra I: Signed Numbers & Absolute Value, Introduction to Algebra
2. Linear equations & formulas	Semester 1	Algebra I: Linear Equations & Formulas
3. Linear inequalities	Semester 1	Algebra I: Systems & Linear Inequalities
4. Functions & linear models	Semester 1	Algebra I: Function Notation & Linear Models, Functions & Patterns
5. Systems of equations	Semester 1	Algebra I: Systems & Linear Inequalities, Systems of Linear Equations
6. Exponents & exponential functions	Semester 2	Exponents & Scientific Notation
7. Polynomials & factoring	Semester 2	Algebra I: Polynomial Operations
8. Quadratic functions & equations	Semester 2	Algebra I: Quadratics
9. Statistics & data	Semester 2	Statistics: Data, Models & Chance, Scatter Plots & Bivariate Data

A two-week study plan

Day 14
Find your weak units
Do the quick checks for all 9 units without notes. Mark each unit solid, shaky or lost. This tells you where your study time goes.
Days 13–6
Fix the lost and shaky units, one a day
Start with the units you marked lost, then shaky. For each one, read the must-know list, work through the linked lesson, and redo that unit's quick checks until you get all three right.
Days 5–3
Mixed practice
Finals mix units together, so practice that way: pick problems from different units, work them without looking at which unit they came from, and check each answer.
Day 2
Formulas and mistakes
Review the formula sheet and every "common mistakes" list. Write out from memory the formulas you still look up.
Day 1
Light review and sleep
Redo only the questions you missed before. Then stop: a full night of sleep helps your memory more than a late cram session.

Unit 1 · Semester 1

Signed numbers & expressions

Must know

Order of operations: grouping symbols, exponents, then multiply and divide left to right, then add and subtract left to right.
Adding signed numbers: same signs add and keep the sign; different signs subtract and take the sign of the number with the larger absolute value.
Multiplying or dividing two negatives gives a positive; one negative gives a negative.
Absolute value is distance from zero, so it is never negative: $|-7| = 7$ .
Distribute to every term, $a(b + c) = ab + ac$ , and combine only like terms: $3x + 5x = 8x$ , but $3x + 5x^2$ cannot be combined.

Common mistakes

Writing $-3^2 = 9$ . Without parentheses the exponent comes first, so $-3^2 = -9$ , while $(-3)^2 = 9$ .
Distributing to only the first term. $-2(x - 4)$ is $-2x + 8$ : the negative multiplies the $-4$ too.

Quick check

1.1 Simplify $-4 + 9 - (-6)$ .
Show answer
$11$ . $-4 + 9 = 5$ , and subtracting $-6$ is the same as adding 6.
1.2 Simplify $3(2x - 5) - 4x$ .
Show answer
$2x - 15$ . Distribute to get $6x - 15 - 4x$ , then combine $6x - 4x$ .
1.3 Evaluate $2x^2 - x$ when $x = -3$ .
Show answer
$21$ . $2(-3)^2 - (-3) = 2(9) + 3 = 21$ .

Review the lesson: Algebra I: Signed Numbers & Absolute Value, Introduction to Algebra

Unit 2 · Semester 1

Linear equations & formulas

Must know

Undo operations in reverse order, doing the same thing to both sides.
With variables on both sides, collect the variable terms on one side first.
Clear fractions by multiplying every term on both sides by the least common denominator.
An equation can have one solution, no solution (you end with something false like $0 = 5$ ) or infinitely many (you end with something true like $0 = 0$ ).
Literal equations are solved the same way for a named variable: $A = lw$ gives $w = \frac{A}{l}$ .

Common mistakes

Multiplying only some terms when clearing fractions. Every term on both sides gets multiplied.
Dropping the negative in front of parentheses: $5 - (x + 2)$ is $3 - x$ , not $7 + x$ .

Quick check

2.1 Solve $4x - 7 = 2x + 9$ .
Show answer
$x = 8$ . Subtract $2x$ to get $2x - 7 = 9$ , add 7 to get $2x = 16$ .
2.2 Solve $\frac{x}{3} + 2 = 6$ .
Show answer
$x = 12$ . Subtract 2 to get $\frac{x}{3} = 4$ , then multiply by 3.
2.3 Solve $P = 2l + 2w$ for $w$ .
Show answer
$w = \frac{P - 2l}{2}$ . Subtract $2l$ from both sides, then divide by 2.

Review the lesson: Algebra I: Linear Equations & Formulas

Unit 3 · Semester 1

Linear inequalities

Must know

Solve an inequality like an equation, except flip the sign when you multiply or divide both sides by a negative number.
On a number line, use an open circle for $<$ or $>$ and a closed circle for $\le$ or $\ge$ .
"And" compound inequalities describe values between two numbers, like $-2 < x \le 5$ ; "or" inequalities describe values outside them.
Absolute value: $|x| < 4$ means $-4 < x < 4$ , and $|x| > 4$ means $x < -4$ or $x > 4$ .
To graph $y > 2x + 1$ , draw a dashed line and shade above it; use a solid line for $\ge$ or $\le$ .

Common mistakes

Forgetting to flip the sign: $-3x > 12$ becomes $x < -4$ , not $x > -4$ .
Drawing a solid boundary line for a strict inequality ( $<$ or $>$ ). Those points are not solutions, so the line is dashed.

Quick check

3.1 Solve $5 - 2x \ge 11$ .
Show answer
$x \le -3$ . Subtract 5 to get $-2x \ge 6$ , then divide by $-2$ and flip the sign.
3.2 Solve $-1 < 2x + 3 \le 9$ .
Show answer
$-2 < x \le 3$ . Subtract 3 from all three parts, then divide all three by 2.
3.3 Solve $|x - 2| \le 5$ .
Show answer
$-3 \le x \le 7$ . Rewrite as $-5 \le x - 2 \le 5$ and add 2 to each part.

Review the lesson: Algebra I: Systems & Linear Inequalities

Unit 4 · Semester 1

Functions & linear models

Must know

A function gives each input exactly one output; its graph passes the vertical line test.
Function notation: $f(3)$ is the output when the input is 3. It does not mean $f$ times 3.
Slope is the rate of change: $m = \frac{y_2 - y_1}{x_2 - x_1}$ .
Slope-intercept form is $y = mx + b$ , point-slope form is $y - y_1 = m(x - x_1)$ , and standard form is $Ax + By = C$ .
Parallel lines have equal slopes; perpendicular slopes are negative reciprocals, like $2$ and $-\frac{1}{2}$ .
In a word problem, the slope is the rate (per hour, per ticket) and the $y$ -intercept is the starting amount.

Common mistakes

Putting the change in $x$ on top of the slope fraction. Slope is rise over run: change in $y$ over change in $x$ .
Subtracting the coordinates in a different order on the top and the bottom, which flips the sign of the slope.

Quick check

4.1 If $f(x) = 3x - 4$ , find $f(-2)$ .
Show answer
$-10$ . $3(-2) - 4 = -6 - 4$ .
4.2 Find the slope of the line through $(1, 5)$ and $(4, -1)$ .
Show answer
$-2$ . $\frac{-1 - 5}{4 - 1} = \frac{-6}{3}$ .
4.3 Write the line with slope 3 through $(2, 1)$ in slope-intercept form.
Show answer
$y = 3x - 5$ . Start from $y - 1 = 3(x - 2)$ , distribute, then add 1.

Review the lesson: Algebra I: Function Notation & Linear Models, Functions & Patterns

Unit 5 · Semester 1

Systems of equations

Must know

The solution of a system is the point that makes both equations true: where the lines cross.
Substitution: solve one equation for a variable, then substitute that expression into the other equation.
Elimination: add or subtract the equations, multiplying one or both first if needed, so one variable cancels.
Lines with the same slope and different intercepts are parallel, so there is no solution; the same line has infinitely many solutions.
For a system of inequalities, the solution is the region where the shaded areas overlap.

Common mistakes

Stopping after finding $x$ . The answer to a system is an ordered pair, so substitute back to find $y$ .
Multiplying only one side of an equation, or only some of its terms, before eliminating.

Quick check

5.1 Solve the system $y = 2x + 1$ and $3x + y = 11$ .
Show answer
$(2, 5)$ . Substitute: $3x + 2x + 1 = 11$ , so $x = 2$ and $y = 2(2) + 1 = 5$ .
5.2 Solve the system $x + y = 10$ and $x - y = 4$ .
Show answer
$(7, 3)$ . Adding the equations gives $2x = 14$ , so $x = 7$ , then $y = 3$ .
5.3 How many solutions does the system $y = 4x - 2$ and $y = 4x + 3$ have?
Show answer
None. The lines have the same slope and different $y$ -intercepts, so they are parallel.

Review the lesson: Algebra I: Systems & Linear Inequalities, Systems of Linear Equations

Unit 6 · Semester 2

Exponents & exponential functions

Must know

Exponent rules: $a^m \cdot a^n = a^{m+n}$ , $\frac{a^m}{a^n} = a^{m-n}$ and $(a^m)^n = a^{mn}$ .
Zero and negative exponents: $a^0 = 1$ and $a^{-n} = \frac{1}{a^n}$ (for $a \ne 0$ ).
An exponential function $y = a \cdot b^x$ shows growth when $b > 1$ and decay when $0 < b < 1$ .
Percent change: growth is $y = a(1 + r)^t$ and decay is $y = a(1 - r)^t$ , with the rate $r$ as a decimal.
Linear patterns add the same amount each step; exponential patterns multiply by the same factor each step.

Common mistakes

Forgetting to apply the outside exponent to the coefficient: $(2x^3)^2 = 4x^6$ , not $2x^6$ .
Using the percent as the base. 5% growth means $b = 1.05$ , not $b = 5$ or $b = 0.05$ .

Quick check

6.1 Simplify $\frac{x^7 \cdot x^{-2}}{x^3}$ .
Show answer
$x^2$ . The top is $x^{7 + (-2)} = x^5$ , and $\frac{x^5}{x^3} = x^2$ .
6.2 A town of 2,000 people grows 3% per year. Write a model for the population $P$ after $t$ years.
Show answer
$P = 2000(1.03)^t$ . Growth of 3% multiplies by $1 + 0.03$ each year.
6.3 Is $y = 50(0.8)^x$ growth or decay, and by what percent per step?
Show answer
Decay of 20% per step, because $0.8 = 1 - 0.2$ .

Review the lesson: Exponents & Scientific Notation

Unit 7 · Semester 2

Polynomials & factoring

Must know

Add and subtract polynomials by combining like terms; to subtract, distribute the negative to every term first.
Multiply binomials by distributing each term (FOIL): $(x + 3)(x - 5) = x^2 - 2x - 15$ .
Always factor out the greatest common factor first: $6x^2 + 9x = 3x(2x + 3)$ .
To factor $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ .
Special patterns: $a^2 - b^2 = (a + b)(a - b)$ and $a^2 + 2ab + b^2 = (a + b)^2$ .

Common mistakes

Squaring a binomial term by term: $(x + 4)^2 = x^2 + 8x + 16$ , not $x^2 + 16$ .
Skipping the GCF, which leaves the answer only partly factored.

Quick check

7.1 Multiply $(2x - 3)(x + 4)$ .
Show answer
$2x^2 + 5x - 12$ . $2x^2 + 8x - 3x - 12$ , then combine $8x - 3x$ .
7.2 Factor $x^2 - 7x + 12$ .
Show answer
$(x - 3)(x - 4)$ . $-3$ and $-4$ multiply to 12 and add to $-7$ .
7.3 Factor $4x^2 - 25$ .
Show answer
$(2x + 5)(2x - 5)$ . It is a difference of squares: $(2x)^2 - 5^2$ .

Review the lesson: Algebra I: Polynomial Operations

Unit 8 · Semester 2

Quadratic functions & equations

Must know

In $y = ax^2 + bx + c$ , the parabola opens up when $a > 0$ and down when $a < 0$ .
The vertex is on the axis of symmetry $x = -\frac{b}{2a}$ ; substitute that $x$ to find the $y$ -coordinate.
Solve by factoring with the zero product property, by taking square roots, or with the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
The discriminant $b^2 - 4ac$ tells you how many real solutions: positive gives two, zero gives one, negative gives none.
Zeros, roots, solutions and $x$ -intercepts all name the same points.

Common mistakes

Factoring before the equation equals zero. Rewrite $x^2 + x = 6$ as $x^2 + x - 6 = 0$ first.
Taking only the positive square root: $x^2 = 49$ gives $x = 7$ or $x = -7$ .

Quick check

8.1 Solve $x^2 - 5x - 14 = 0$ .
Show answer
$x = 7$ or $x = -2$ . It factors as $(x - 7)(x + 2) = 0$ .
8.2 Find the vertex of $y = x^2 - 6x + 5$ .
Show answer
$(3, -4)$ . $x = -\frac{-6}{2(1)} = 3$ , and $3^2 - 6(3) + 5 = -4$ .
8.3 How many real solutions does $2x^2 + 3x + 5 = 0$ have?
Show answer
None. The discriminant is $3^2 - 4(2)(5) = -31$ , which is negative.

Review the lesson: Algebra I: Quadratics

Unit 9 · Semester 2

Statistics & data

Must know

The mean is the average and the median is the middle value; the median is less affected by outliers.
Measures of spread: range, interquartile range ( $IQR = Q_3 - Q_1$ ) and standard deviation.
A box plot shows the five-number summary: minimum, $Q_1$ , median, $Q_3$ and maximum.
Scatter plots show positive, negative or no correlation; the correlation coefficient $r$ runs from $-1$ to $1$ , and values near either end are strong.
A line of best fit is for prediction, and correlation alone does not prove that one thing causes the other.

Common mistakes

Finding the median without putting the data in order first.
Calling $r = -0.9$ weak. The sign gives the direction; the distance from 0 gives the strength, so $-0.9$ is strong.

Quick check

9.1 Find the mean and median of 4, 8, 6, 5, 20. Which better describes a typical value?
Show answer
Mean $8.6$ , median $6$ . The median is better here because the outlier 20 pulls the mean up.
9.2 A box plot has $Q_1 = 12$ and $Q_3 = 30$ . What is the interquartile range?
Show answer
$18$ . $IQR = Q_3 - Q_1 = 30 - 12$ .
9.3 The line of best fit for test score $y$ and hours studied $x$ is $y = 2.5x + 40$ . Predict the score for 6 hours.
Show answer
$55$ . $2.5(6) + 40 = 55$ ; the slope means about 2.5 more points per extra hour.

Review the lesson: Statistics: Data, Models & Chance, Scatter Plots & Bivariate Data

Free tools for studying

Algebra 1 final exam FAQ

What is on the Algebra 1 final exam?: Most Algebra 1 finals cover linear equations and inequalities, functions and linear models, systems of equations, exponents and exponential functions, polynomials and factoring, quadratics, and basic statistics. A first-semester final usually stops at systems of equations.
How should I study for the Algebra 1 final?: Start about two weeks out. Do the quick checks for every unit without notes to find your weak units, study those lessons first, then switch to mixed practice from all units in the last few days, because the final mixes topics together.
What formulas do I need to know for the Algebra 1 final?: The slope formula, slope-intercept and point-slope form, the exponent rules, the exponential growth and decay models, the vertex formula and the quadratic formula. All of them are on the free StudyQuest formula sheet.
What is the hardest part of the Algebra 1 final?: For most students it is quadratics and factoring, because they build on every earlier skill. If your final is in December, it is usually systems of equations and word problems that ask you to write the equation yourself.

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Answer key

Unit 1: Signed numbers & expressions

1.1 $11$ . $-4 + 9 = 5$ , and subtracting $-6$ is the same as adding 6.
1.2 $2x - 15$ . Distribute to get $6x - 15 - 4x$ , then combine $6x - 4x$ .
1.3 $21$ . $2(-3)^2 - (-3) = 2(9) + 3 = 21$ .

Unit 2: Linear equations & formulas

2.1 $x = 8$ . Subtract $2x$ to get $2x - 7 = 9$ , add 7 to get $2x = 16$ .
2.2 $x = 12$ . Subtract 2 to get $\frac{x}{3} = 4$ , then multiply by 3.
2.3 $w = \frac{P - 2l}{2}$ . Subtract $2l$ from both sides, then divide by 2.

Unit 3: Linear inequalities

3.1 $x \le -3$ . Subtract 5 to get $-2x \ge 6$ , then divide by $-2$ and flip the sign.
3.2 $-2 < x \le 3$ . Subtract 3 from all three parts, then divide all three by 2.
3.3 $-3 \le x \le 7$ . Rewrite as $-5 \le x - 2 \le 5$ and add 2 to each part.

Unit 4: Functions & linear models

4.1 $-10$ . $3(-2) - 4 = -6 - 4$ .
4.2 $-2$ . $\frac{-1 - 5}{4 - 1} = \frac{-6}{3}$ .
4.3 $y = 3x - 5$ . Start from $y - 1 = 3(x - 2)$ , distribute, then add 1.

Unit 5: Systems of equations

5.1 $(2, 5)$ . Substitute: $3x + 2x + 1 = 11$ , so $x = 2$ and $y = 2(2) + 1 = 5$ .
5.2 $(7, 3)$ . Adding the equations gives $2x = 14$ , so $x = 7$ , then $y = 3$ .
5.3 None. The lines have the same slope and different $y$ -intercepts, so they are parallel.

Unit 6: Exponents & exponential functions

6.1 $x^2$ . The top is $x^{7 + (-2)} = x^5$ , and $\frac{x^5}{x^3} = x^2$ .
6.2 $P = 2000(1.03)^t$ . Growth of 3% multiplies by $1 + 0.03$ each year.
6.3 Decay of 20% per step, because $0.8 = 1 - 0.2$ .

Unit 7: Polynomials & factoring

7.1 $2x^2 + 5x - 12$ . $2x^2 + 8x - 3x - 12$ , then combine $8x - 3x$ .
7.2 $(x - 3)(x - 4)$ . $-3$ and $-4$ multiply to 12 and add to $-7$ .
7.3 $(2x + 5)(2x - 5)$ . It is a difference of squares: $(2x)^2 - 5^2$ .

Unit 8: Quadratic functions & equations

8.1 $x = 7$ or $x = -2$ . It factors as $(x - 7)(x + 2) = 0$ .
8.2 $(3, -4)$ . $x = -\frac{-6}{2(1)} = 3$ , and $3^2 - 6(3) + 5 = -4$ .
8.3 None. The discriminant is $3^2 - 4(2)(5) = -31$ , which is negative.

Unit 9: Statistics & data

9.1 Mean $8.6$ , median $6$ . The median is better here because the outlier 20 pulls the mean up.
9.2 $18$ . $IQR = Q_3 - Q_1 = 30 - 12$ .
9.3 $55$ . $2.5(6) + 40 = 55$ ; the slope means about 2.5 more points per extra hour.

What’s on the Algebra 1 final

A two-week study plan

Find your weak units

Fix the lost and shaky units, one a day

Mixed practice

Formulas and mistakes

Light review and sleep

Signed numbers & expressions

Must know

Common mistakes

Quick check

Linear equations & formulas

Must know

Common mistakes

Quick check

Linear inequalities

Must know

Common mistakes

Quick check

Functions & linear models

Must know

Common mistakes

Quick check

Systems of equations

Must know

Common mistakes

Quick check

Exponents & exponential functions

Must know

Common mistakes

Quick check

Polynomials & factoring

Must know

Common mistakes

Quick check

Quadratic functions & equations

Must know

Common mistakes

Quick check

Statistics & data

Must know

Common mistakes

Quick check

Free tools for studying

Algebra 1 final exam FAQ

Get the free 7-Day Study Reset

Answer key

Unit 1: Signed numbers & expressions

Unit 2: Linear equations & formulas

Unit 3: Linear inequalities

Unit 4: Functions & linear models

Unit 5: Systems of equations

Unit 6: Exponents & exponential functions

Unit 7: Polynomials & factoring

Unit 8: Quadratic functions & equations

Unit 9: Statistics & data

What’s on the Algebra 1 final

A two-week study plan

Find your weak units

Fix the lost and shaky units, one a day

Mixed practice

Formulas and mistakes

Light review and sleep

Signed numbers & expressions

Must know

Common mistakes

Quick check

Linear equations & formulas

Must know

Common mistakes

Quick check

Linear inequalities

Must know

Common mistakes

Quick check

Functions & linear models

Must know

Common mistakes

Quick check

Systems of equations