# 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 ?

### Lesson: Algebraic Expressions

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

### Solving Equations

*Example: * What is the value of in ?

### Solving Inequalities

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

### Absolute Value Equations and Inequalities

*Example: * Evaluate

#### II. Functions, Equations, and Graphs

### Relations and Functions

*Example: * What is the output of the function if ?

### Direct Variation

*Example: * If varies directly with , and when , then what is the value of when ?

### 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 after it undergoes a transformation 2 units up and 3 units left?

### Absolute Value Functions and Graphs

*Example: * Solve for :

### Two Variable Inequalities

*Example: * Draw the graph of

#### III. Linear Systems

### Lesson: Solving Systems of Equations Using Tables and Graphs

*Example: * Solve the following system of equations graphically: and

### Solving Systems Algebraically

*Example: * Solve the following system of equations algebraically: and

### Lesson: System of Inequalities

*Example: * Solve the following system of inequalities: and

### Linear Programming

*Example: * What values for and maximize the objective function ?

### Systems with Three Variables

*Example: * Solve the following system of equations: , , and

### Solving Systems Using Matrices

*Example: * Solve the following system of equations by using matrices: and

#### IV. Quadratic Functions and Equations

### Quadratic Functions and Transformations

*Example: * What is the vertex of the function ?

### Standard Form of a Quadratic Equation

*Example: * What is the vertex of the function ?

### Modeling with Quadratic Functions

*Example: * Do the points (3,-1), (4,0), and (5,-1) lie on the function ?

### Lesson: Factoring Quadratic Expressions

*Example: * Factor the expression

### Lesson: Quadratic Equations

*Example: * Solve for :

### Lesson: Completing the Square

*Example: * What are the solutions to

### The Quadratic Formula

*Example: * Solve for using the quadratic formula:

### Complex Numbers

*Example: * What is equivalent to ?

### Lesson: Quadratic Systems

*Example: * Solve the following system of equations: and

#### V. Polynomials and Polynomial Functions

### Polynomial Functions

*Example: * Classify the following polynomial by its degree and number of terms:

### Polynomials, Linear Factors, and Zeros

*Example: * What is the factored form of

### Solving Polynomial Equations

*Example: * Solve for :

### Lesson: Dividing Polynomials

*Example: * What is the solution to ?

### Rational Root Theorem

*Example: * What are the possible rational zeroes for ?

### The Fundamental Theorem of Algebra

*Example: * Solve for :

### The Binomial Theorem

*Example: * How can you expand ?

### Polynomial Models in the Real World

*Example: * What are three points that lie on ?

### Transforming Polynomial Functions

*Example: * What is the equation of 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

### Binomial Radical Expressions

*Example: * Evaluate

### Rational Expressions

*Example: * Evaluate

### Lesson: Solving Square Root and Other Rational Expressions

*Example: * Solve for :

### Function Operations

*Example: * Find if and

### Inverse Relations and Functions

*Example: * What is the inverse function of ?

### Lesson: Graphing Radical Functions

*Example: * Draw the graph of

#### VII. Exponential and Logarithmic Functions

### Exploring Exponential Models

*Example: * How can you describe the function ?

### Properties of Exponential Models

*Example: * What transformation changed to ?

### Logarithmic Functions as Inverses

*Example: * What is the inverse function of ?

### Properties of Logarithms

*Example: * Expand

### Exponenital and Logarithmic Equations

*Example: * Solve for :

### Natural Logarithms

*Example: * Evaluate

#### VIII. Rational Funtions

### Lesson: Inverse Variation

*Example: * Suppose and vary inversely. Write a function for when and

### The Reciprocal Family Functions

*Example: * What is the y-intercept of

### Lesson: Rational Functions and their Graphs

*Example: * Draw the graph of

### Rational Expressions

*Example: *

### Lesson: Adding and Subtracting Rational Expressions

*Example: * Simplify

### Lesson: Solving Rational Equations

*Example: * Solve for :

#### 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 ?

### Parabolas

*Example: * What is the vertex of the function ?

### Circles

*Example: * What is the radius of the circle ?

### Ellipses

*Example: * What are the vertices of the ellipse ?

### Hyperbolas

*Example: * What are the vertices of the hyperbola ?

### 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

### 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 and the probability of B is ?

### 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 ?

### 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

### Lesson: Matrix Multiplication

*Example: * Simplify

### Determinants and Inverses

*Example: *

### Inverse Matrices and Systems

*Example: * Solve the following system of equations using matrices: and

### Geometric Transformations

*Example: * Dilate the coordinates forming a traingle given in the following matrix by a scale factor of 3.

### 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 ?

### 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 ?

### The Cosine Function

*Example: * What is the amplitude and the period of ?

### The Tangent Function

*Example: * What is the amplitude and the period of ?

### Translating Sine and Cosine Functions

*Example: * What is the amplitude and the period of ?

### Reciprocal Trigonometric Functions

*Example: * Evaluate

#### XIV. Trigonometric Identities and Equations

### Trigonometric Identities

*Example: * What is equivalent to

### Solving Trigonometric Equations Using Inverses

*Example: * Solve for :

### Right Triangles and Trigonometric Ratios

*Example: * In a 30-60-90 triangle, what is ?

### Area and Law of Sines

*Example: * In , , , and . Find

### The Law of Cosines

*Example: * In , , , and . Find

### Angle Identities

*Example: * Evaluate

### Double-Angle and Half-Angle Identities

*Example: * Find is and

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