WRITING ABSOLUTE VALUE FUNCTIONS AS PIECEWISE FUNCTIONS WORKSHEET

For each of the functions, create a piecewise definition.

Problem 1 :

f(x) = |x - 1|

Solution

Problem 2 :

f(x) = |x + 2|

Solution

Problem 3 :

f(x) = |2x - 1|

Solution

Problem 4 :

f(x) = |5 - 2x|

Solution

Problem 5 :

f(x) = |1 - 3x|

Solution

Problem 6 :

f(x) = |2x + 1|

Solution

Problem 7 :

f(x) = x - |x|

Solution

Problem 8 :

f(x) = x + |x - 1|

Solution

Problem 9 :

In holography, light from a laser beam is split into two beams, a reference beam and an object beam. Light from the object beam reflects off an object and is recombined with the reference beam to form images on film that can be used to create three-dimensional images.

a. Write an absolute value function that represents the path of the reference beam.

b. Write the function in part (a) as a piecewise function.

Solution

Problem 10 :

You are sitting on a boat on a lake. You can get a sunburn from the sunlight that hits you directly and also from the sunlight that reflects off the water.

a. Write an absolute value function that represents the path of the sunlight that reflects off the water.

b. Write the function in part (a) as a piecewise function.

Solution

Answer Key

1) 

2) 

3) 

4) 

5) 

6) 

7) 

8) 

9) a) the function g(x) = −1.6 ∣ x − 5 ∣ + 8 represents the path of the reference beam.

b)

a piecewise function for 

g(x) = 1.6x    if x < 5

g(x) = -1.6x + 16    if  x ≥ 5

10) a) y = 2|x - 3|

b)

f(x) = 2[-(x - 3)]  if x < 3

f(x) = 2[(x - 3)]  if x ≥ 3

Create a piecewise definition for the given absolute value function.

Problem 1 :

f(x) = |x + 1|

Solution

Problem 2 :

f(x) = |x - 4|

Solution

Problem 3 :

f(x) = |4 - 5x|

Solution

Problem 4 :

f(x) = |3 - 2x|

Solution

Problem 5 :

You are trying to make a hole in one on the miniature golf green.

a. Write an absolute value function that represents the path of the golf ball.

b. Write the function in part (a) as a piecewise function.

Solution

Problem 6 :

Michelle likes riding her bike to and from her favorite lake on Wednesdays. She created the following graph to represent the distance she is away from the lake while biking.

1. Interpret the graph by writing three observations about Michelle’s bike ride.

2. Write a piecewise function for this situation, with each linear function being in point-slope form using the point (3,0). What do you notice?

3. This particular piecewise function is called a linear absolute value function. What are the traits you are noticing about linear absolute value functions?

Solution

Problem 7 :

You are trying to make a hole in one on the miniature golf green.

a. Write an absolute value function that represents the path of the golf ball.

b. Write the function in part (a) as a piecewise function.

Solution

Problem 8 :

You are sitting on a boat on a lake. You can get a sunburn from the sunlight that hits you directly and also from the sunlight that reflects off the water.

a. Write an absolute value function that represents the path of the sunlight that reflects off the water.

b. Write the function in part (a) as a piecewise function.

Solution

Answer Key

1) 

2) 

3) 

4) 

5) a) y = (-2/3) |x - 6| + 4

b) 

• f(x) = (-2/3) (6 - x) + 4  when x < 6
• f(x) = (-2/3) (x - 6) + 4 when x ≥ 6

6) 

• By observing the point (3, 0),  Michelle can reach the lake in 3 minutes.
• At 4 minutes, he is away from the lake for 2 blocks.
• By observing the point (0, 6), Michelle starts 6 blocks away from the lake

f(x) = -2x + 6 when x < 3

f(x) = 2x - 6 when x ≥ 3

7) a) y = -1|x - 5| + 4

b) y = x - 1 when x < 6

y = -x + 9 when x ≥ 6

8) a) y = 2|x - 3|

b) y = -2x + 6 when x < 3

y = 2x - 6 when x ≥ 3

Solve the following absolute value equations.

Problem 1 :

|x + 4| = |2x - 7|

Solution

Problem 2 :

|3x + 5| = |x - 6|

Solution

Problem 3 :

|x - 9| = |x + 6|

Solution

Problem 4:

|x + 4| = |x - 3|

Solution

Problem 5 :

|5t + 7| = |4t + 3|

Solution

Problem 6 :

|3a - 1| = |2a + 4|

Solution

Problem 7 :

|n - 3| = |3 - n|

Solution

Problem 8 :

|y - 2| = |2 - y|

Solution

Problem 9 :

|7 - a| = |a + 5|

Solution

Problem 10 :

|6 - t| = |t + 7|

Solution

Problem 11 :

|1/2x - 5| = |1/4x + 3|

Solution