EXAMPLE 4 Write an absolute value function

HologramsIn 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. Write an equation for the path of the reference beam.

EXAMPLE 4 Write an absolute value function

SOLUTION

The vertex of the path of the reference beam is (5, 8). So, the equation has the form y = a x – 5 + 8. Substitute the coordinates of the point (0, 0) into the equation and solve for a.

0 = a 0 – 5 + 8 Substitute 0 for y and 0 for x.

– 1.6 = a Solve for a.

An equation for the path of the reference beam is y = – 1.6 x – 5 + 8.

ANSWER

Apply transformations to a graph

EXAMPLE 5

The graph of a function y = f (x) is shown.Sketch the graph of the given function.

a. y = 2 f (x)

b. y = – f (x + 2) + 1

Apply transformations to a graph

EXAMPLE 5

The graph of y = 2 f (x) is the graph of y = f (x) stretched vertically by a factor of 2. (There is no reflection or translation.) To draw the graph, multiply the y-coordinate of each labeled point on the graph of y = f (x) by 2 and connect their images.

a .

SOLUTION

Apply transformations to a graphEXAMPLE 5

The graph of y = – f (x + 2) +1 is the graph of y = f (x) reflected in the x-axis, then translated left 2 units and up 1 unit. To draw the graph, first reflect the labeled points and connect their images. Then translate and connect these points to form the final image.

b .

SOLUTION

GUIDED PRACTICE for Examples 4 and 5

4. WHAT IF? In Example 4, suppose the reference beam originates at (3, 0) and reflects off a mirror at (5, 4). Write an equation for the path of the beam.

y = –2| x – 5 | + 4ANSWER

GUIDED PRACTICE for Examples 4 and 5

Use the graph of y = f (x) from Example 5 to graph the given function.

5. y = 0.5 f (x)

SOLUTION

GUIDED PRACTICE for Examples 4 and 5

6. y = – f (x – 2) – 5

Use the graph of y = f (x) from Example 5 to graph the given function.

SOLUTION

GUIDED PRACTICE for Examples 4 and 5

7. y = 2 f (x + 3) – 1

Use the graph of y = f (x) from Example 5 to graph the given function.

SOLUTION