14.2 Translate and Reflect Trigonometric Graphs

Goal · Translate and reflect trigonometric graphs.

Your Notes

TRANSLATION OF SINE AND COSINE GRAPHS

To graph y = a sin b(x - h) + k or y = a cos b(x - h) + k where a > 0 and b > 0, follow these steps:

Step 1 Identify the amplitude a, the period the horizontal shift h, and the vertical shift k of the graph.

Step 2 Draw the horizontal line y = k, called the _midline_ of the graph.

Step 3 Find the five key points by translating the key points of y = a sin bx or y = a cos bx _horizontally_ h units and _vertically_ k units.

Step 4 Draw the graph through the five translated key points.

Your Notes

Example 1

Graph a vertical translation

Graph y = 3 sin 2x + 1

Solution

Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.

Amplitude: a = _3_	Horizontal shift: h = _0_
Period:	Vertical shift: k = _1_
= _p_

Step 2 Draw the midline of the graph, y = _1_.

Step 3 Find the five key points.

On y = k: (0, 0 + 1) = _(0, 1)_ ;

Because the graph is shifted up 1 unit, the y-coordinates of the five key points will be increased by 1.

Maximum:

Minimum:

Step 4 Draw the graph through the key points.

Checkpoint Graph the function.

1. y = 4 sin 2x + 3

Your Notes

Example 2

Combine a translation and a reflection

Graph y = -3 sin

Step 1 Identify the amplitude, period, horizontal shift, and vertical shift.

Amplitude: ½ a ½ = _3_	Horizontal shift: h =
Period:	Vertical shift: k = _0_
= _8p_

Step 2 Draw the midline of the graph. Because _k = 0_ , the midline is the _x-axis_.

Step 3 Find the five key points of y = ½ -3 ½ sin

On y = k:

Because the graph is shifted to the right units, the x-coordinates of the five key points will be increased by

Maximum:

Minimum:

Step 4 Reflect the graph. Because a < 0, the graph is reflected in the midline y = 0. So, becomes and becomes

The minimum and maximum of the original graph become the maximum and minimum, respectively, of the reflected graph.

Step 5 Draw the graph through the key points found.

Your Notes

Example 5

Model with a tangent function

Flagpole You watch a classmate lower a flag on a 20-foot flagpole. You are standing 15 feet from the base of the flagpole. Write and graph a model that gives the flag's distance d (in feet) from the top of the flagpole as a function of the angle of elevation q.

Solution

Use a tangent function to write an equation relating d and q.

tan q =	Definition of tangent
__15 tan q __= __20 - d__	Multiply each side by 15.
__15 tan q - 20__ = _-d_	Subtract 20 from each side.
__15 tan q + 20__ = _d_	Solve for d.

The graph is shown at the right.

Checkpoint Complete the following exercises.

2. Graph

y = -2 cos (x + p).

3. Write and graph a model for Example 3 if you stand 10 feet from the flagpole.