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Algebra 2 | Free Practice Problems & Explanations

I. Expressions, Equations, and Inequalities

Patterns and Expressions

Example: What is the next number in the pattern: 3, 6, 9, 12, _ ?

Properties of Real Numbers

Example: Which property justifies the equation 3(2x-5y)+1=6x-15y+1?

Lesson: Algebraic Expressions

Example: Which algebraic expression models the phrase “10 less than a number m”?

Solving Equations

Example: What is the value of x in 3x-7=11?

Solving Inequalities

Example: Write the inequality that represents “3 less than a number is greater than 5”

Absolute Value Equations and Inequalities

Example: Evaluate |-x|=12

II. Functions, Equations, and Graphs

Relations and Functions

Example: What is the output of the function f(x)=x^2 -3x -2 if x=5?

Direct Variation

Example: If y varies directly with x, and y=5 when x=2, then what is the value of y when x=6?

Linear Functions and Slope-Intercept Form

Example: Find the slope of the line passing through the points (2,4) and (4,-2)

More About Linear Equations

Example: Write the equation of the line with a slope of 3 that goes through the point (5,0).

Using Linear Models

Example: Can the values 2, 4, 6, 8, 10 be modeled by a linear function?

Families of Functions

Example: What is the equation of the line y=x after it undergoes a transformation 2 units up and 3 units left?

Absolute Value Functions and Graphs

Example: Solve for x: |4x-1|=15

Two Variable Inequalities

Example: Draw the graph of y \textless 3x-1

III. Linear Systems

Lesson: Solving Systems of Equations Using Tables and Graphs

Example: Solve the following system of equations graphically: y=3x-1 and y=-x+5

Solving Systems Algebraically

Example: Solve the following system of equations algebraically: 3x-4y=-3 and y=2x+1

Lesson: System of Inequalities

Example: Solve the following system of inequalities: y \textless 2x-1 and y \geq -x

Linear Programming

Example: What values for x and y maximize the objective function P=2x+4y?

Systems with Three Variables

Example: Solve the following system of equations: 3x-2y+z=0, -2x+5y+4z=12, and 6x-y-z=-8

Solving Systems Using Matrices

Example: Solve the following system of equations by using matrices: 3x-2y=9 and -x+6y=-1

IV. Quadratic Functions and Equations

Quadratic Functions and Transformations

Example: What is the vertex of the function y=3(x-2)^2 +3?

Standard Form of a Quadratic Equation

Example: What is the vertex of the function y=x^2-5x-24?

Modeling with Quadratic Functions

Example: Do the points (3,-1), (4,0), and (5,-1) lie on the function y=-(x-4)^2?

Lesson: Factoring Quadratic Expressions

Example: Factor the expression x^2 - 5x -36

Lesson: Quadratic Equations

Example: Solve for x: x^2 +15 +50

Lesson: Completing the Square

Example: What are the solutions to y=x^2 -12x -14 = 9

The Quadratic Formula

Example: Solve for x using the quadratic formula: x^2+2x+14=0

Complex Numbers

Example: What is equivalent to \sqrt{-25}?

Lesson: Quadratic Systems

Example: Solve the following system of equations: y=2x-1 and y=x^2+3x-4

V. Polynomials and Polynomial Functions

Polynomial Functions

Example: Classify the following polynomial by its degree and number of terms: 3x^5 -2x+1

Polynomials, Linear Factors, and Zeros

Example: What is the factored form of x^3 -x^2 -12x

Solving Polynomial Equations

Example: Solve for x: x^3 -8 =0

Lesson: Dividing Polynomials

Example: What is the solution to \dfrac{x^2+20+44}{x-2}?

Rational Root Theorem

Example: What are the possible rational zeroes for f(x)=x^3 -2x^2 +12x -1?

The Fundamental Theorem of Algebra

Example: Solve for x: x^4 +x^3 +4x^2 -8x +25=0

The Binomial Theorem

Example: How can you expand (3x-2y)^3?

Polynomial Models in the Real World

Example: What are three points that lie on f(x)=x^3 -7x^2 +4?

Transforming Polynomial Functions

Example: What is the equation of y=x^3 after it is reflected in the y-axis and then translated 4 units down and 3 units left?

VI. Radical Functions and Rational Exponents

Roots and Radical Expressions

Example: What values are square roots of 25?

Lesson: Multiplying and Dividing Radical Expressions

Example: Evaluate \sqrt{288} \times \sqrt{4}

Binomial Radical Expressions

Example: Evaluate \sqrt{50} + \sqrt{8}

Rational Expressions

Example: Evaluate (27x^6)^{\frac{1}{3}}

Lesson: Solving Square Root and Other Rational Expressions

Example: Solve for x: \sqrt{x-2} +12 = 14

Function Operations

Example: Find (f-g)(x) if f(x)=12x-12 and g(x)=-x^2 +2x +8

Inverse Relations and Functions

Example: What is the inverse function of f(x)=\dfrac{3x-5}{2} -3?

Lesson: Graphing Radical Functions

Example: Draw the graph of y=\sqrt{3x}

VII. Exponential and Logarithmic Functions

Exploring Exponential Models

Example: How can you describe the function y=3(0.6)^x?

Properties of Exponential Models

Example: What transformation changed y=4^x to y=4^{x-1}?

Logarithmic Functions as Inverses

Example: What is the inverse function of y=2+4^x?

Properties of Logarithms

Example: Expand log_3 (x\sqrt{y} - z)

Exponenital and Logarithmic Equations

Example: Solve for x: 3^x -1.8 = 5.2

Natural Logarithms

Example: Evaluate ln 4

VIII. Rational Funtions

Lesson: Inverse Variation

Example: Suppose x and y vary inversely. Write a function for when x=4 and y=20

The Reciprocal Family Functions

Example: What is the y-intercept of y=\dfrac{2}{x}

Lesson: Rational Functions and their Graphs

Example: Draw the graph of y=\sqrt{x-1}

Rational Expressions

Example:

Lesson: Adding and Subtracting Rational Expressions

Example: Simplify \dfrac{3x}{2x^2} + \dfrac{3+x}{x}

Lesson: Solving Rational Equations

Example: Solve for x: \dfrac{1}{2x^2} = \dfrac{3}{x} + \dfrac{5}{2}

IX. Sequences and Series

Mathematical Patterns

Example: Continue the pattern with the next three values: 4, 12, 36, 108, …

Arithmetic Sequences

Example: What is the explicit formula for the sequence 12, 15, 18, 21, …?

Geometric Sequences

Example: What is the explicit formula for the sequence 2, -6, 18, -54, …?

Arithmetic Series

Example: What is the sum of the series 32, 43, 54, 65, 76?

Geometric Series

Example: What is the sum of the series -3 -6 -12 -24 …, n=6?

X. Quadratic Relations and Conic Sections

Exploring Conic Sections

Example: Which conic section is associated with the equation 4x^2 +36y^2 = 16?

Parabolas

Example: What is the vertex of the function y=3x^2 -6x +9?

Circles

Example: What is the radius of the circle (x-4)^2 + (y+1)^2 = 12?

Ellipses

Example: What are the vertices of the ellipse \dfrac{(x-9)^2}{9} + (y-9)^2 = 1?

Hyperbolas

Example: What are the vertices of the hyperbola \dfrac{x^2}{9} - \dfrac{(y-1)^2}{9} = 1?

Translating Conic Sections

Example: Write the standard form equation of an ellipse with vertices at (-2,3) and (5,3), and a focus at (2,3)

XI. Probability and Statistics

Lesson: Permutation and Combinations

Example: Evaluate _7 P _2

Lesson: Probability

Example: Bob flipped heads 6 out of 8 times. What is the experimental probability that he flips tails on his next coin toss?

Lesson: Probability of Multiple Events

Example: What is the probability of A and B if they are independent events and the probability of A is \dfrac{3}{10} and the probability of B is \dfrac{2}{3}?

Conditional Probability

Example: If the freshman students are 45% male and 55% female, and 86% of the males passed this semester while 89% of the females did, what is the probability that a student is male who failed this semester?

Probability Models

Example: The weather channel tells you that there is a 30% chance of rain on each of the next three days. What is the chance that it does not rain on all three days?

Analyzing Data

Example: What is the median of {28 26, 40, 37, 39, 29, 31, 33, 33, 33, 31, 36}?

Standard Deviations

Example: What is the standard deviation of {28 26, 40, 37, 39, 29, 31, 33, 33, 33, 31, 36}?

Lesson: Samples and Surveys

Example: If a principal wants to conduct a survey to determine which subject is the favorite in his school, so he samples the physics class to get results, is this a biased sample?

Binomial Distributions

Example: What is the fourth term in (2x-y)^5?

Normal Distributions

Example: What percent of data is in the interval from 45 to 50 in a data set with a mean of 55 and a standard deviation of 5?

XII. Matrices

Lesson: Adding and Subtracting Matrices

Example: Simplify \begin{bmatrix} 4 & 1 & -8 & -2 \\ \end{bmatrix} + \begin{bmatrix} 3 & 6 & -5 & -1 \\ \end{bmatrix}

Lesson: Matrix Multiplication

Example: Simplify 6 \begin{bmatrix} 1 & 3 & 0 & -9 \\ \end{bmatrix}

Determinants and Inverses

Example: \begin{vmatrix} 4 & -2\\ 8 & -6\\ \end{vmatrix}

Inverse Matrices and Systems

Example: Solve the following system of equations using matrices: 4x-2y=12 and 3x+y=1

Geometric Transformations

Example: Dilate the coordinates forming a traingle given in the following matrix by a scale factor of 3. \begin{bmatrix} -3 & -2 & -1 \\ 5 & -3 & -3 \\ \end{bmatrix}

Vectors

Example: If a vector starts at the point (2, 3) and extends to (4, 6), what is its component form?

XIII. Periodic Functions and Trigonometry

Exploring Periodic Data

Example: What is the amplitude and the period of y=2 \sin{x}?

Angles and the Unit Circle

Example: If an angle extends outwards from the x-axis and touches the x-axis on the other side, what is the degree measure of the angle formed?

Radian Measures

Example: What is 50 degrees in terms of radians?

The Sine Function

Example: What is the amplitude and the period of y=3 \sin{4x}?

The Cosine Function

Example: What is the amplitude and the period of y= 2 \cos{2x}?

The Tangent Function

Example: What is the amplitude and the period of y= 4 \tan{x}?

Translating Sine and Cosine Functions

Example: What is the amplitude and the period of y= 5 \sin{3x} -3?

Reciprocal Trigonometric Functions

Example: Evaluate \csc{\dfrac{2\pi}{3}}

XIV. Trigonometric Identities and Equations

Trigonometric Identities

Example: What is equivalent to \sin ^2 \theta + \cos ^2 \theta

Solving Trigonometric Equations Using Inverses

Example: Solve for \theta: \dfrac{\sqrt{2}}{2} - 2\sin \theta = -\sin \theta

Right Triangles and Trigonometric Ratios

Example: In a 30-60-90 triangle, what is \tan 30 \textdegree?

Area and Law of Sines

Example: In \triangle ABC, \angle A = 34 \textdegree, b=12, and c=30. Find m\angle C

The Law of Cosines

Example: In \triangle ABC, a=5, b=13, and m\angle C = 57 \textdegree. Find c

Angle Identities

Example: Evaluate \sin 75\textdegree

Double-Angle and Half-Angle Identities

Example: Find \sin 2\theta is 0 \textless \theta \textless 90 and \cos \theta = \dfrac{\sqrt{5}}{4}