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Quizzes > High School Quizzes > Mathematics

Absolute Value Functions Practice Quiz

Practice absolute value questions with clear explanations

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Paper art depicting trivia for The Absolute Value Challenge, a high school math practice quiz.

What is the value of |7|?
7
-7
 
0
The absolute value of a number is its distance from zero, which is always nonnegative. Therefore, |7| equals 7.
What is the value of |-3|?
 
0
3
-3
Absolute value converts any negative number to its positive counterpart. Thus, |-3| is 3.
If x = 4, what is the value of |x - 10|?
6
4
-6
14
Substituting x = 4 into |x - 10| gives |4 - 10| = |-6|. The absolute value turns -6 into 6.
What is the unique solution to the equation |x| = 0?
Any negative number
0
No solution
0 and -0
An absolute value equals zero only when the input is zero because nonzero values always yield a positive result. Hence, x must be 0.
What is the value of |-12|?
0
-12
12
Cannot be determined
Absolute value removes the sign of a number. Therefore, |-12| equals 12.
Solve the equation |x - 5| = 3 for x.
x = 2
x = -2 or x = 8
x = 8
x = 2 or x = 8
The equation |x - 5| = 3 produces two cases: x - 5 = 3 leading to x = 8, and x - 5 = -3 leading to x = 2. Both values satisfy the equation.
Solve the equation |2x + 1| = 7 for x.
x = 4 or x = -3
x = 3
x = 3 or x = -4
x = -4
Setting up two cases, 2x + 1 = 7 gives x = 3 and 2x + 1 = -7 gives x = -4. Both solutions satisfy the original absolute value equation.
Which equation represents a horizontal translation 3 units to the right of y = |x|?
y = |x + 3|
y = |x - 3|
y = |x| + 3
y = |x| - 3
A horizontal translation of a graph by h units to the right is achieved by replacing x with (x - h) in the function. Hence, y = |x - 3| is the correct transformation.
Solve the inequality |x - 4| < 3.
x > 7 or x < 1
1 ≤ x ≤ 7
1 < x < 7
x < 1
Rewriting |x - 4| < 3 as -3 < x - 4 < 3 and then adding 4 to all parts yields 1 < x < 7. This interval represents the solution set.
Solve the inequality |3x - 9| ≥ 6.
1 < x < 5
x ≤ 1 or x ≥ 5
x ≥ 1 and x ≤ 5
x > 1 or x < 5
Dividing |3x - 9| ≥ 6 by 3 gives |x - 3| ≥ 2. This inequality splits into x - 3 ≥ 2 (x ≥ 5) or x - 3 ≤ -2 (x ≤ 1).
Find the vertex of the function y = |x + 2| - 4.
(-2, 4)
(2, -4)
(2, 4)
(-2, -4)
The vertex form of an absolute value function is y = |x - h| + k. Rewriting y = |x + 2| - 4 as y = |x - (-2)| - 4 shows that the vertex is at (-2, -4).
What transformation does the function f(x) = |x| + 5 represent?
Vertical shift upward by 5
Vertical shift downward by 5
Horizontal shift left by 5
Horizontal shift right by 5
Adding a constant outside the absolute value function results in a vertical shift. Here, f(x) = |x| + 5 moves the graph of y = |x| upward by 5 units.
Solve the equation |x + 4| = |2x - 1| for x.
x = 5
x = 5 or x = 1
x = 5 or x = -1
x = -1
To solve |x + 4| = |2x - 1|, consider the two cases: x + 4 = 2x - 1, which gives x = 5, and x + 4 = -(2x - 1), which gives x = -1.
Which of the following properties is always true for absolute value expressions?
|a + b| = |a| + |b|
|a · b| = |a| · |b|
|a - b| = |a| - |b|
|a / b| = |a| + |b|
The multiplicative property |a · b| = |a| · |b| holds for all real numbers. The other properties listed are not true in general for absolute value expressions.
Solve for x: |x/2 - 3| = 4.
x = 2 or x = -14
x = 14
x = -2
x = 14 or x = -2
The equation |x/2 - 3| = 4 gives two cases: x/2 - 3 = 4 which leads to x = 14, and x/2 - 3 = -4 which leads to x = -2. Both solutions satisfy the equation.
Solve the inequality |2x - 3| > 5.
-1 < x < 4
x < -1 or x > 4
x > -1 and x < 4
x > -1 or x < 4
To solve |2x - 3| > 5, set up two inequalities: 2x - 3 > 5 and 2x - 3 < -5, which give x > 4 and x < -1 respectively. Thus, the solution is x < -1 or x > 4.
Solve the equation |x² - 4| = 0 for real x.
x = -2
x = 0
x = 2
x = 2 or x = -2
An absolute value equals zero only when its argument is zero. Setting x² - 4 = 0 and factoring gives (x - 2)(x + 2) = 0, leading to x = 2 or x = -2.
Determine the equation of an absolute value function with vertex (3, -2) and with right arm slope 2.
y = 2|x - 3| - 2
y = |x - 3| - 2
y = 2|x + 3| - 2
y = 2|x - 3| + 2
The vertex form y = a|x - h| + k has vertex (h, k). With vertex (3, -2) and a = 2 to ensure the arms have a slope of 2 (and -2), the function is y = 2|x - 3| - 2.
If f(x) = |3x + 6|, what is the value of f(-4)?
-6
6
0
12
Substituting x = -4 into f(x) gives |3(-4) + 6| = |-12 + 6|, which simplifies to |-6|. The absolute value of -6 is 6.
Solve the equation |x + 1| + |x - 1| = 4.
x = 2 or x = -2
x = 2
x = -2
x = 2, -2, or 0
Analyzing the equation in different regions shows that when x ≥ 1, the equation simplifies to 2x = 4 yielding x = 2, and when x ≤ -1, it simplifies to -2x = 4 yielding x = -2. No solution exists between -1 and 1.
0
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Study Outcomes

  1. Understand the definition and properties of absolute value functions.
  2. Apply the concept of absolute value to solve linear equations and inequalities.
  3. Analyze piecewise representations of absolute value functions.
  4. Graph absolute value functions and identify key features such as vertex and axis of symmetry.
  5. Evaluate and interpret transformations of absolute value functions, including shifts and reflections.

Absolute Value Functions Worksheet Cheat Sheet

  1. Distance from Zero - The absolute value of a number is like its personal GPS, showing exactly how far it is from zero on the number line - always a non-negative result. Imagine flexing your math muscles with |−5| = 5 and |5| = 5, proving that negatives can't hide their distance! OpenStax: Absolute Value Functions
  2. Splitting Equations - To solve equations with absolute values, you split them into two cases: one where the inside is positive and one where it's negative. For example, |x − 3| = 7 becomes x − 3 = 7 or x − 3 = −7, leading to x = 10 or x = −4. ChiLiMath: Practice Problems
  3. V‑Shaped Graph - Absolute value functions always graph as a snazzy "V" shape, with the point (vertex) sitting where the inside expression equals zero. Plot y = |x − 2| and watch the vertex pop up at (2, 0), splitting the graph into two linear rays. MathBits Notebook: Graph Practice
  4. Graph Transformations - Shifts, reflections, and stretches let you customize that "V" to your heart's content. For instance, y = 2|x + 1| − 3 scoots left 1 unit, doubles the steepness, then drops down 3 units - like giving your graph a superhero cape! Symbolab: Transformation Guide
  5. Domain & Range - The domain of any absolute value function is all real numbers - your graph can wander from −∞ to +∞ on the x-axis. But the range depends on vertical moves: for y = |x| alone, outputs never dip below 0, so y ≥ 0. Intellectual Math: Domain & Range
  6. Tackling Inequalities - Absolute value inequalities mean two worlds: less than (<) gives you a "sandwich" between two values, while greater than (>) splits into two outside regions. Solving |x − 4| < 2 means 4 − 2 < x < 4 + 2, so 2 < x < 6 - piece of cake! OpenStax: Inequality Strategies
  7. Piecewise Power - Breaking absolute value functions into piecewise definitions reveals their true dual nature: y = x when x ≥ 0 and y = −x when x < 0. This two‑part rulebook is your secret weapon for sketching and understanding every twist and turn. OpenStax: Piecewise Definitions
  8. Vertex & Symmetry - Every "V" has a vertex (h, k) and an axis of symmetry at x = h - think of it as the balance beam of your graph. Spotting these features first makes graphing faster than a calculator! MathBits Notebook: Vertex & Symmetry
  9. Real‑World Relevance - Absolute value isn't just theory - it models real situations like measuring distances or deviations from expected values. Next time you calculate the absolute difference between actual and predicted scores, you're flexing your absolute value skills! OpenStax: Applications
  10. Practice Makes Perfect - The more you tackle varied problems - equations, inequalities, graphing - the more intuitive absolute values become. Regular quizzes and exercises are like gym workouts for your brain, building confidence and speed! ChiLiMath: Quiz Yourself
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