CREATE PIEWISE DEFINITION FOR ABSOLUTE VALUE FUNCTION

Create a piecewise definition for the given absolute value function.

Problem 1 :

f(x) = |x + 1|

Solution :

f(x) = |x - 1|

f(x) = x - 1  and  f(x) = -(x - 1)

Case 1 :

f(x) = x - 1

when x ≥ 1, f(x) is positive

Case 2 :

f(x) = -(x - 1)

when x < 1, f(x) is negative

Distributing negative, we get

f(x) = - x + 1

So, the required definition is

Problem 2 :

f(x) = |x - 4|

Solution :

f(x) = |x - 4|

f(x) = x - 4 and  f(x) = -(x - 4)

Case 1 :

f(x) = x - 4

when x ≥ 4, f(x) is positive

Case 2 :

f(x) = -(x - 4)

when x < 4, f(x) is negative

Distributing negative, we get

f(x) = - x + 4

So, the required definition is

Problem 3 :

f(x) = |4 - 5x|

Solution :

f(x) = |4 - 5x|

f(x) = 4 - 5x and  f(x) = -(4 - 5x)

Case 1 :

f(x) = 4 - 5x

when x < 4/5, f(x) is positive

Case 2 :

f(x) = -(4 - 5x)

when x ≥ 4/5, f(x) is negative

Distributing negative, we get

f(x) = - 4 + 5x

So, the required definition is

Problem 4 :

f(x) = |3 - 2x|

Solution :

f(x) = |3 - 2x|

f(x) = 3 - 2x and  f(x) = -(3 - 2x)

Case 1 :

f(x) = 3 - 2x

when x < 3/2, f(x) is positive

Case 2 :

f(x) = -(3 - 2x)

when x ≥ 3/2, f(x) is negative

Distributing negative, we get

f(x) = - 3 + 2x

So, the required definition is

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 :

a) The vertex of the given absolute value function is at (6, 4).

y = a|x - h| + k

Here (h, k) is at (6, 4)

y = a|x - 6| + 4

Applying the point (3, 2), we get

2 = a|3 - 6| + 4

2 = a|-3| + 4

2 = 3a + 4

2 - 4 = 3a 

3a = -2

a = -2/3

y = (-2/3) |x - 6| + 4

b) Defining as piecewise function.

f(x) = (-2/3) (-(x - 6)) + 4

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

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 :

1.

• 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

2. By observing the point (3, 0),  Michelle can reach the lake in 3 minutes.

(0, 6) and (2, 2)

Slope = (2 - 6) / (2 - 0)

= -4/2

= -2

Equation of the line using the slope -2 and the point (3, 0)

(y - y₁) = m(x - x₁)

(y - 0) = -2(x - 3)

y = -2x + 6

(4, 2) and (6, 6)

Slope = (6 - 2) / (6 - 4)

= 4/2

= 2

Equation of the line using the slope 2 and the point (3, 0)

(y - y₁) = m(x - x₁)

(y - 0) = 2(x - 3)

y = 2x - 6

3.

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

f(x) = 2x - 6 when x ≥ 3

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 :

a) y = a|x - h| + k

(h, k) is at (5, 4)

y = a|x - 5| + 4

Applying the point (3, 2) which lies on the absolute value function, we get

2 = a|3 - 5| + 4

2 = a|-2| + 4

2 - 4 = a|-2|

-2 = 2a

a = -2/2

a = -1

y = -1|x - 5| + 4

b) Writing piecewise function, 

y = -1[-(x - 5)] + 4

= -1[-x + 5] + 4

= x - 5 + 4

y = x - 1 when x < 6

y = -1[(x - 5)] + 4

= -1x + 5 + 4

= -x + 9

y = -x + 9 when x ≥ 6

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 :

a) y = a|x - h| + k

(h, k) is at (3, 0)

y = a|x - 3| + 0

Applying the point (2, 2) which lies on the absolute value function, we get

2 = a|2 - 3|

2 = a|-1|

a = 2

y = 2|x - 3|

b) Writing piecewise function, 

y = 2[-(x - 3)]

= 2(-x + 3)

y = -2x + 6 when x < 3

y = 2(x - 3)

= 2x - 6

y = 2x - 6 when x ≥ 3