Algebra 1 Formula Sheet Algebra 1
Reference & Formula Sheet
Linear Equations
Slope Formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Slope-Intercept Form: \[ y = mx + b \]
Point-Slope Form: \[ y - y_1 = m(x - x_1) \]
Standard Form: \[ Ax + By = C \]
Exponential Functions
General Form: \[ y = a \cdot b^x \]
Growth/Decay: \[ A = P(1 \pm r)^t \]
Polynomial Operations
Difference of Squares: \[ a^2 - b^2 = (a - b)(a + b) \]
Perfect Square Trinomials: \[ a^2 + 2ab + b^2 = (a + b)^2 \]
\[ a^2 - 2ab + b^2 = (a - b)^2 \]
Quadratic Functions
Standard Form: \[ y = ax^2 + bx + c \]
Vertex Form: \[ y = a(x - h)^2 + k \]
Quadratic Formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Axis of Symmetry: \[ x = -\frac{b}{2a} \]
Discriminant: \[ D = b^2 - 4ac \]
Properties of Exponents
Product: \( a^m \cdot a^n = a^{m+n} \)
Power: \( (a^m)^n = a^{m \cdot n} \)
Quotient: \( \frac{a^m}{a^n} = a^{m-n} \)
Negative: \( a^{-n} = \frac{1}{a^n} \)
Zero: \( a^0 = 1 \)
Notes Area Algebra 1 Final Exam ALGEBRA 1 ASSESSMENT
Final Examination
NAME: __________________________
DATE: __________________________
Section 1: Solving Equations
Solve each equation. Show all algebraic steps clearly for full credit.
1
\( 4(x - 3) + 7 = 23 \)
2
\( \frac{2}{3}x - 5 = \frac{1}{2}x + 1 \)
3
\( 5x - 2(x + 4) = 3x - 8 \)
4
Solve for \( w \): \( P = 2l + 2w \)
Section 2: Linear Functions & Slope
5
Find the slope of the line passing through \( (-4, 7) \) and \( (2, -5) \).
6
Write the equation of a line in slope-intercept form that passes through \( (3, 1) \) and is perpendicular to \( y = -\frac{1}{2}x + 4 \).
Section 3: Graphing & Systems
7
Graph the linear equation:
Label the x and y intercepts.
8
Solve the system by graphing:
Solution:
( , )
9
A theater charges $8 for adult tickets and $5 for student tickets. A total of 120 tickets were sold, and the total revenue was $810. How many adult tickets and how many student tickets were sold?
Define Variables & Set up Equations:
Show Work:
Section 4: Exponents & Polynomials
Simplify Exponents
10
\( (3x^2 y^3)(4x y^5) \)
11
\( \frac{15a^6 b^{-2}}{5a^2 b^3} \)
Exponential Functions
12
A population of 500 bacteria triples every hour. Write an equation to represent the population \( y \) after \( x \) hours.
How many bacteria will there be after 4 hours?
Polynomial Operations & Factoring
13
Multiply: \( (2x - 5)(x + 4) \)
14
Simplify: \( (3x^2 - 5x + 7) - (x^2 - 2x - 4) \)
15
Factor completely: \( x^2 - 9x + 20 \)
16
Factor completely: \( 3x^2 - 27 \)
Section 5: Quadratic Functions
17
Solve the quadratic equation using the Quadratic Formula. Give exact answers.
\( x^2 - 6x + 7 = 0 \)
18
Analyze and graph the function:
Vertex: (____ , ____)
Axis of Symmetry: x = ____
y-intercept: (____ , ____)
x-intercepts: (____ , ____), (____ , ____)
Mathematics Excellence
Algebra 1 Answer Key TEACHER ANSWER KEY
Algebra 1 Assessment
Solutions & Grading Guide
Section 1: Solving Equations
1. \( 4(x - 3) + 7 = 23 \)
\( 4x - 12 + 7 = 23 \)
\( 4x - 5 = 23 \)
\( 4x = 28 \)
\( x = 7 \)
2. \( \frac{2}{3}x - 5 = \frac{1}{2}x + 1 \)
Multiply by 6: \( 4x - 30 = 3x + 6 \)
\( 4x - 3x = 6 + 30 \)
\( x = 36 \)
3. \( 5x - 2(x + 4) = 3x - 8 \)
\( 5x - 2x - 8 = 3x - 8 \)
\( 3x - 8 = 3x - 8 \)
Identity: Infinite Solutions
4. Solve for \( w \): \( P = 2l + 2w \)
\( P - 2l = 2w \)
\( w = \frac{P - 2l}{2} \)
Section 2: Linear Functions & Slope
5. Find slope: \( (-4, 7) \) and \( (2, -5) \)
\( m = \frac{-5 - 7}{2 - (-4)} = \frac{-12}{6} \)
\( m = -2 \)
6. Perp to \( y = -\frac{1}{2}x + 4 \) through \( (3, 1) \)
New slope \( m = 2 \)
\( y - 1 = 2(x - 3) \)
\( y - 1 = 2x - 6 \)
\( y = 2x - 5 \)
Section 3: Graphing & Systems
7. \( 2x - 3y = 6 \)
x-int: (3, 0), y-int: (0, -2)
8. System: \( y = 2x - 3 \) and \( y = -x + 3 \)
Solution: (2, 1)
9. Theater Word Problem
Variables: \( a \) = adult tickets, \( s \) = student tickets
Equations: \( a + s = 120 \) and \( 8a + 5s = 810 \)
Steps: \( s = 120 - a \)
\( 8a + 5(120 - a) = 810 \)
\( 8a + 600 - 5a = 810 \)
\( 3a = 210 \)
Adult: 70 tickets, Student: 50 tickets
Section 4: Exponents & Polynomials
10. \( (3x^2 y^3)(4x y^5) \)
\( 12x^3 y^8 \)
11. \( \frac{15a^6 b^{-2}}{5a^2 b^3} \)
\( \frac{3a^4}{b^5} \)
12. Bacteria Population
Equation: \( y = 500(3)^x \)
After 4 hours: \( 500(3)^4 = 500(81) = 40,500 \)
Total: 40,500 bacteria
13. \( (2x - 5)(x + 4) \)
\( 2x^2 + 3x - 20 \)
14. \( (3x^2 - 5x + 7) - (x^2 - 2x - 4) \)
\( 2x^2 - 3x + 11 \)
15. Factor \( x^2 - 9x + 20 \)
\( (x - 4)(x - 5) \)
16. Factor \( 3x^2 - 27 \)
\( 3(x - 3)(x + 3) \)
Section 5: Quadratic Functions
17. Solve \( x^2 - 6x + 7 = 0 \) using Quadratic Formula
\( a = 1, b = -6, c = 7 \)
