Algebra 2 Problems

A set of algebra 2 problems with their detailed solutions to self test and diagnose your background and review and gain deep understanding on the following topics:
  1. Complex Numbers
    1. Adding and Subtracting Complex Numbers
    2. Conjugate of Complex Numbers
    3. Multiplying and Dividing Complex Numbers
  2. Quadratic Equations
    1. Solving Quadratic Equations with Complex Solutions
    2. Number and Nature of Solutions of Quadratic Equations
  3. Functions
    1. Evaluate a function
    2. Domain and Range of a function
    3. Operations on functions
    4. One-to-one functions
    5. Inverse of a function
    6. Absolute Value function
    7. Even and Odd functions
    8. Transformations of functions
    9. Piecewise functions
    10. Rate of Change of a Function
  4. Polynomials Functions
    1. Dividing Polynomials
    2. Factoring Polynomials
    3. Factoring Polynomials Using Special Polynomials
    4. Solving Polynomial Equations
    5. Solving Polynomial Inequalities
    6. Graphing Polynomials Functions
  5. Rational Expressions, Equations, Inequalities and Functions
    1. Simplifying Rational Expressions
    2. Solving Rational Equations
    3. Solving Rational Inequalities
    4. Graphing Rational Functions
  6. Trigonometry and Trigonometric Functions
    1. Converting Degrees into Radians and Vice Versa
    2. Evaluating Exact Values of Trigonometric Ratios
    3. Using Trigonometric Identities to Simplify Trigometric Expressions
    4. Solving Trigonometric Equations
    5. Graphing Trigonometric Functions
  7. Logarithmic and Exponential Functions
    1. Simplifying Exponnential Expressions
    2. Using Relationship Between Logarithmic and Exponential Expressions
    3. Simplifying Logarithmic Expressions
    4. Solving Logarithmic and Exponential Equations
    5. Graphing Logarithmic and Exponential Functions

Algebra 2 problems with detailed solutions

  1. Complex Numbers

  2. Problem 1-1
    Let z = 2 - 3 i where i is the imaginary unit. Evaluate z z* , where z* is the conjugate of z , and write the answer in standard form.

    Detailed Solution.



    Problem 1-2
    Evaluate and write in standard form \( \dfrac{1-i}{2-i} \) , where i is the imaginary unit.

    Detailed Solution.



  3. Quadratic Equations

  4. Problem 2-1
    Find all solutions of the equation \( x(x + 3) = - 5 \).

    Detailed Solution.



    Problem 2-2
    Find all values of the parameter m for which the equation \( -2 x^2 + m x = 2 m \) has complex solutions.

    Detailed Solution.



  5. Functions

  6. Problem 3-1
    Let \( f(x) = - x^2 + 3(x - 1) \). Evaluate and simplify \( f(a-1)\).

    Detailed Solution.



    Problem 3-2
    Write, in interval notation, the domain of function \(f\) given by \(f(x) = \sqrt{x^2-16} \).

    Detailed Solution.



    Problem 3-3
    Find and write, in interval notation, the range of function \(f\) given by \(f(x) = - x^2 - 2x + 6 \).

    Detailed Solution.



    Problem 3-4
    Let \(f(x) = \sqrt{x - 2} \) and \(g(x) = x^2 + 2 \); evaluate \( (f_o g)(a - 1) \) for \( a \lt 1 \).

    Detailed Solution.



    Problem 3-5
    Which of the following is a one-to-one function?(There may be more than one answer).
    a) \(f(x) = - 2 \)     b) \(g(x) = \ln(x^2 - 1) \)     c) \(h(x) = |x| + 2 \)     d) \(j(x) = 1/x + 2 \)     e) \(k(x) = \sin(x) + 2 \)     f) \(l(x) = ln(x - 1) + 1 \)

    Detailed Solution.



    Problem 3-6
    What is the inverse of function f given by \(f(x) = \dfrac{-x+2}{x-1}\)?

    Detailed Solution.



    Problem 3-7
    Classify the following functions as even, odd or neither.
    a) \(f(x) = - x^3 \)     b) \(g(x) = |x|+ 2 \)     c) h(x) = \( \ln(x - 1) \)

    Detailed Solution.



    Problem 3-8
    Function \(f \) has one zero only at \(x = -2\). What is the zero of the function \(2f(2x - 5) \)?

    Detailed Solution.



    Problem 3-9
    Which of the following piecewise functions has the graph shown below?
    a) \( f(x) = \begin{cases} x^2 & \text{if} \; x \ge 0 \\ 2 & \text{if} \; -2 \lt x \lt 0\\ - x + 1& \text{if} \; x \le -2 \end{cases} \)     b) \( g(x) = \begin{cases} x^2 & \text{if} \; x \gt 0 \\ 2 & \text{if} \; -2 \lt x \le 0\\ - x + 1& \text{if} \; x \le -2 \end{cases} \)     c) \( h(x) = \begin{cases} x^2 & \text{if} \; x \gt 0 \\ 2 & \text{if} \; -2 \lt x \lt 0\\ - x + 1 & \text{if} \; x \lt -2 \end{cases} \)

    Graph of a piecewise function in problem 3-9

    Detailed Solution.



    Problem 3-10
    Calculate the average rate of change of function \( f(x) = \dfrac{1}{x} \) as x changes from \( x = a\) to \( x = a + h \).

    Detailed Solution.



  7. Polynomials

  8. Problem 4-1
    Find the quotient and the remainder of the division \( \dfrac{-x^4+2x^3-x^2+5}{x^2-2} \).

    Detailed Solution.



    Problem 4-2
    Find \( k \) so that the remainder of the division \( \dfrac{4 x^2+2x-3}{2 x + k} \) is equal to \( -1 \)?

    Detailed Solution.



    Problem 4-3
    \( (x - 2) \) is one of the factors of \( p(x) = -2x^4-8x^3+2x^2+32x+24 \). Factor \(p\) completely.

    Detailed Solution.



    Problem 4-4
    Factor \( 16 x^4 - 81 \) completely.

    Detailed Solution.



    Problem 4-5
    Find all solutions to the equation \( (x - 3)(x^2 - 4) = (- x + 3)(x^2 + 2x) \)

    Detailed Solution.



    Problem 4-6
    Solve the inequality \( (x + 2)(x^2-4x-5) \ge (-x - 2)(x+1)(x-3)\)

    Detailed Solution.



    Problem 4-7
    The graph of a polynomial function is shown below. Which of the following functions can possibly have this graph?
    a) \( y = -(x+2)^5(x-1)^2 \)     b) \( y = 0.5(x+2)^3(x-1)^2 \)     c) \( y = -0.5(x+2)^3 (x-1)^2 \)     d) \( y = -(x+2)^3(x-1)^2 \)

    Graph of polynomial in problem 4-7

    Detailed Solution.



    Problem 4-8
    Which of the following graphs could possibly be that of the function f given by \( f(x) = k (x - 1)(x^2 + 4) \) where k is a negative constant? Find k if possible.

    Graph of polynomial in problem 4-8

    Detailed Solution.



  9. Rational Expressions, Equations, Inequalities and Functions

    Problem 5-1
    Write as a single rational expression: \( \dfrac{x^2+3x-5}{(x-1)(x+2)} - \dfrac{2}{x+2} - 1 \).

    Detailed Solution.



    Problem 5-2
    Solve the equation: \( \dfrac{- x^2+5}{x-1} = \dfrac{x-2}{x+2} - 4 \).

    Detailed Solution.



    Problem 5-3
    Solve the inequality: \( \dfrac{1}{x-1}+\dfrac{1}{x+1} \ge \dfrac{3}{x^2-1} \).

    Detailed Solution.



    Problem 5-4
    Find the horizontal and vertical asymptotes of the function: \( y = \dfrac{3x^2}{5 x^2 - 2 x - 7} + 2 \).

    Detailed Solution.



    Problem 5-5
    Which of the following rational functions has an oblique asymptote? Find the point of intersection of the oblique asymptote with the function.
    a) \( y = -\dfrac{x-1}{x^2+2} \)     b) \( y = -\dfrac{x^4-1}{x^2+2} \)     c) \( y = -\dfrac{x^3 + 2x ^ 2 -1}{x^2- 2} \)     d) \( y = -\dfrac{x^2-1}{x^2+2} \)

    Detailed Solution.



    Problem 5-6
    Which of the following graphs could be that of function \( f(x) = \dfrac{2x-2}{x-1} \)?


    Graph of rational function in problem 5-6

    Detailed Solution.



  10. Trigonometry and Trigonometric Functions

  11. Problem 6-1
    A rotating wheel completes 1000 rotations per minute. Determine the angular speed of the wheel in radians per second.

    Detailed Solution.



    Problem 6-2
    Determine the exact value of \( sec(-11\pi/3) \).

    Detailed Solution.



    Problem 6-3
    Convert 1200 in radians giving the exact value.

    Detailed Solution.



    Problem 6-4
    Convert \( \dfrac{-7\pi}{9} \) in degrees giving the exact value.

    Detailed Solution.



    Problem 6-5
    What is the range and the period of the the function \( f(x) = -2\sin(-0.5(x - \pi/5)) - 6 \)?

    Detailed Solution.



    Problem 6-6
    Which of the following graphs could be that of function given by: \( y = - \cos(2x - \pi/4) + 2 \)?

    Graph of trigonometric  functions in problem 6-6

    Detailed Solution.



    Problem 6-7
    Find a possible equation of the form \( y = a \sin(b x + c) + d \) for the graph shown below.(there are many possible solutions)

    Graph of trigonometric  function in problem 6-7

    Detailed Solution.



    Problem 6-8
    Find the smallest positive value of x, in radians, such that \( - 4 \cos (2x - \pi/4) + 1 = 3 \)

    Detailed Solution.



    Problem 6-9
    Simplify the expression: \( \dfrac{\cot(x)\sin(x) + \cos(x) \sin^2(x)+\cos^3(x)}{\cos(x)} \)

    Detailed Solution.



  12. Logarithmic and Exponential Functions

  13. Problem 7-1
    Simplify the expression \( \dfrac{4x^2 y^8}{8 x^3 y^5} \) using positive exponents in the final answer.

    Detailed Solution.



    Problem 7-2
    Evaluate the expression \( \dfrac{3^{1/3} 9^{1/3}}{4^{1/2}} \).

    Detailed Solution.



    Problem 7-3
    Rewrite the expression \( \log_b(2x - 4) = c \) in exponential form.

    Detailed Solution.



    Problem 7-4
    Simplify the expressiomn: \( \log_a(9) \cdot \log_3(a^2) \)

    Detailed Solution.



    Problem 7-5
    Solve the equation \( \log(x + 1) - log(x - 1) = 2 \log(x + 1) \).

    Detailed Solution.



    Problem 7-6
    Solve the equation \( e^{2x} + e^x = 6 \).

    Detailed Solution.



    Problem 7-7
    What is the horizontal asymptote of the graph of \( f(x) = 2 ( - 2 - e^{x-1}) \)?

    Detailed Solution.



    Problem 7-8
    What is the vertical asymptote of the graph of \( f(x) = log(2x - 6) + 3 \)?

    Detailed Solution.



    Problem 7-9
    Match the given functions with the graph shown below?
    A) \( y = 2 - 0.5^{2x-1} \)     B) \( y = 0.5^{2x-1} \)     C) \( y = 2 - 0.5^{-2x+1} \)     D) \( y = 0.5^{-2x+1} \)
    Graph of Exponential  functions in problem 7-9

    Detailed Solution.



    Problem 7-10
    Match the given functions with the graph shown below?
    A) \( y = 2+ln(x-2) \)     B) \( y=-log_2(x+1)-1 \)     C) \( y = -ln(-x) \)     D) \( y = y=-log_3(x+1)-1 \)
    Graph of Logarithmic functions in problem 7-10

    Detailed Solution.



More References and Links

Algebra Problems
Step by Step Math Worksheets Solvers
Math Problems and Online Self Tests.
Basic Rules and Properties of Algebra.
More Intermediate and College Algebra Questions and Problems with Answers .