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Unit 9: Absolute Value and Piecewise Functions

Absolute value measures distance from zero, which is why its equations split into two cases and its graph folds into a V.

5 topics · 17 questions

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What Absolute Value Actually Measures

  1. How it worksHow does absolute value strip away a number's sign while keeping its size?
  2. Worked exampleFind the value of the absolute value of -8 plus the absolute value of 3.
  3. Common mistakeWhy doesn't the absolute value of a variable expression always end up positive?
  4. Ties togetherHow does absolute value connect to measuring distance between two points on a number line?
Your turn1 of 2
Find the value of the absolute value of -14 plus the absolute value of 13.

Solving Absolute Value Equations

  1. How it worksHow does splitting an absolute value equation into two separate cases capture both solutions?
  2. Worked exampleSolve the absolute value of 2x - 3 equals 7 for x.
  3. Common mistakeWhy doesn't the absolute value of x equals -4 have any solution at all?
  4. Worked exampleSolve the absolute value of x + 4 equals 10 for x.
Your turn1 of 2
Solve the absolute value of 5x - 6 equals 29 for x. Find the positive solution.

Graphing Absolute Value Functions

  1. How it worksHow does an absolute value graph reflect the negative half of a line back up above the x-axis?
  2. Worked exampleGraph the function y equals the absolute value of x minus 3, plus 2.
  3. Going furtherWhat happens to the V-shaped graph when a negative sign sits in front of the absolute value?
Your turn1 of 2
The absolute value function y = |x| is shifted right by 7 and up by 6. Write the resulting function y(x).

Piecewise Functions

  1. How it worksHow does a piecewise function switch between different formulas depending on which interval x falls in?
  2. Worked exampleFind f(-3) and f(3) for the piecewise function that uses x + 2 when x is less than 0, and x2 when x is at least 0.
  3. Common mistakeWhy can't you use both pieces of a piecewise function's formula for the same input?
Your turn1 of 2
Find f(-4) for the piecewise function that uses x + 7 when x is less than 0, and x2 when x is at least 0, where -4 is negative.

Absolute Value Inequalities

  1. How it worksHow does an absolute value inequality split into a compound inequality to solve it?
  2. Worked exampleSolve the absolute value of x - 2 is greater than 4 for x.
  3. Ties togetherHow does solving an absolute value inequality connect back to solving compound inequalities?
Your turn1 of 2
Solve the absolute value of x - 7 is greater than 13 for x. Find the upper boundary value.