Exercise 47 Page 223

Practice makes perfect

a We have been given a graph that shows the path of sunlight as it reflects in water.

Diagram showing the path of the sunlight

To write this as an absolute value function, the point of reflection is going to be the vertex. Recall the vertex form of an absolute value function.

f (x) = a ∣ x - h ∣ + k

Here the coordinate

(h, k)

is the vertex. Let's substitute our vertex

(3, 0)

into this equation.

f (x) = a ∣ x - 3 ∣ + 0 \Leftrightarrow f (x) = a ∣ x - 3 ∣

Now we can solve for

a

by substituting one other known point into the equation. Let's use

(0, 6) .

f (x) = a ∣ x - 3 ∣

SubstituteII

$x = 0$ , $f (x) = 6$

6 = a ∣ 0 - 3 ∣

▼

Solve for

a

SubTerm

Subtract term

6 = a ∣ - 3 ∣

AbsNeg

$∣ - 3 ∣ = 3$

6 = a (3)

DivEqn

$LHS / 3 = RHS / 3$

2 = a

RearrangeEqn

Rearrange equation

a = 2

We can now write our final equation.

f (x) = 2 ∣ x - 3 ∣

b To write the absolute value function as a piecewise function, we first need to determine where the function changes direction, as this is where our domains will split. In this case, this occurs when

x = 3 .

Diagram showing the path of the sunlight. x=3 is marked.

We will let the domain of the left piece be all real numbers less than

3 .

Domain Left Piece : 3 < x

The right piece of the function will then have a domain that includes all real numbers greater than or equal to

3 .

Domain Right Piece : x \geq 3

Next, we can write the function as two separate equations by splitting our absolute value into its two cases: one positive and one negative.

f (x) = {2 [- (x - 3)], 2 (x - 3), x < 3 x \geq 3

We can simplify these equations a little bit. Let's start with the negative case.

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

▼

Simplify right-hand side

Multiply

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

Distr

Distribute $- 2$

f (x) = - 2 x + 6

Now we can simplify the positive case.

f (x) = 2 (x - 3) \Leftrightarrow f (x) = 2 x - 6

Finally, we can write the equation of the piecewise function.

f (x) = {- 2 x + 6, - 2 x - 6, if x < 3 if x \geq 3

Subchapter links

Communicate Your Answer p.217

Monitoring Progress p.218-221

Exercises p.222-224

Exercise 47 Page 223

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