UNIT 7 Selected Answers MODULE 18

7. Period: 90; Midline; h(t) = 24 Amplitude: |a| = 16;

Lesson 18.3 Translating Trigonometric Graphs

Maximum: 40 Minimum: 8

Your Turn

(0, 8) is a minimum

1 1 1 3. Period: = 2, so b = ; period = 2π i = π b 2 2

Second minimum: (90, 8) Maximum: (45, 40)

Midline: y = k, or y = 1

Midline crossings: (22.5, 24) and (67.5, 24)

Amplitude: a = 2 The point (0, 1) on the graph of y = cos x is a local maximum, and lies 1 unit above the midline. The corresponding point for π⎞ ⎛ f(x) = 2cos2 ⎜ x − ⎟ + 1 will lie a(1) = 2 2⎠ ⎝ (1) = 2 units above the midline, or at y = 1 π + 2 = 3. There is also a translation h = 2 ⎛π ⎞ units to the right. So, ⎜ , 3 ⎟ is on the 2 ⎝ ⎠ graph. The next local maximum is one cycle to the right, or at ⎛π ⎞ ⎛ 3π ⎞ ⎜ 2 + π , 3 ⎟ = ⎜ 2 , 3 ⎟. ⎝ ⎠ ⎝ ⎠

(0, 8) means the car is 8 feet above the ground at time t = 0; (22.5, 24) means it is at the wheel’s center height of 24 feet after 22.5 seconds; at (45, 40) it is at its maximum height of 40 feet; at (67.5, 24) it is back at the wheel’s center height; (90, 8) means it is again at the minimum height of 8 feet after a 90-second cycle.

Evaluate 1. Period: 2π i 1 = 2π; Midline: y = k, or y = 1 Amplitude: |a| = 3 h = −π, k = 1 ⇒ image of (0, 0) from y = sin x

Minimum: a = 2 units below the midline, or k − a = 1 − 2 = −1; occurs one half cycle after the first maximum, or at x = π π + = π . So, (π, −1) is on the graph. 2 2

is (−π, 1) ; other midline crossings are at cycle halfway point, (−π + π, 1) = (0, 1) and cycle endpoint, (−π + 2π, 1) = (π, 1) . Plot (−π, 1) , (0, 1) , and (π, 1).

The graph crosses its midline halfway between the first maximum and the minimum, and halfway between the minimum and the second maximum, or at π 3π π+ +π 3π 2 2 = 5π . So, = x= and x = 2 4 2 4 ⎛ 3π ⎞ ⎛ 5π ⎞ ⎜ 4 , 1⎟ and ⎜ 4 , 1⎟ are on the graph. ⎝ ⎠ ⎝ ⎠

a < 0, so the graph is reflected in its midline, and the minimum occurs before the maximum.

Minimum: k − |a| = 1 − 3 = −2; occurs −π + 0 π at x = =− . 2 2 Maximum: k + |a| = 1 + 3 = 4; occurs at 0+π π ⎛ π ⎞ = . Plot ⎜ − , − 2 ⎟ and x= 2 2 ⎝ 2 ⎠ ⎛π ⎞ ⎜ 2 , 4 ⎟. ⎝ ⎠

π⎞ ⎛ 6. f(x) = 2sin2 ⎜ x − ⎟ − 1 2⎠ ⎝ π⎞ ⎛ (Note that f(x) = 2sin2 ⎜ x + ⎟ − 1 also 2⎠ ⎝ represents the graph.)

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⎛ π⎞ h(0) = 10 sin ⎜ − ⎟ + 36 = 26 ⇒ (0, 26) is ⎝ 2⎠ on the graph.

This is a minimum. The next minimum is one cycle later at (2, 26). A maximum occurs halfway through the cycle at (1, 46). Midline crossings: halfway between first minimum and maximum at (0.5, 36) and halfway between maximum and second minimum at (1.5, 36). (0, 26) means the toy is at its lowest point 26 inches above the floor t = 0; (0.5, 36) means it is at its midline (resting) height of 36 inches after 0.5 second; at (1, 46) it is at its maximum height of 46 inches, then back to the midline height at (1.5, 36) , and finally back to the minimum height after 2 seconds.

3.

Period: b = 2 ⇒ π i b = 2π; Midline: y = k, or y = 3; h = π ⇒ midline crossing is translated π units to the right to (π, 3); asymptotes are half a cycle to the left and right at x = π − π = 0 and x = π + π = 2π. 11. The key points are translated h units right for positive h, h units left for negative h, k units up for positive k, and k units down for negative k.

Halfway points are halfway between the asymptotes and midline crossing, or at x=

π 2

3π . The points are 2 ⎛ 3π ⎞ and ⎜ , 3 ⎟. 2 ⎝ ⎠

and x =

⎛π ⎞ ⎜ 2 , 3⎟ ⎝ ⎠

13. Yes, a sine curve with a negative coefficient a is the same as a curve with a positive coefficient a translated horizontally by half a cycle. For example, the graphs of y = −sin x and y = sin (x + π) are the same. Also, any cosine curve is the same as a sine curve translated by one fourth of a cycle, so a similar result holds for cosine graphs.

5. f(x) = 2sin(x − π) + 1. (Note that f(x) = 2sin(x + π) + 1 also represents the graph.) 7. f(x) = 3tan 9. Period:

1 x+2 2

1 1 = π, so b = , and 2π i b = 2; π b

Midline: k = 36, so the midline is h(t) = 36. Amplitude: a = 10; Maximum: 36 + 10 = 46; Minimum: 36 − 10 = 26

Original content Copyright © by Houghton Mifflin Harcourt. Additions and changes to the original content are the responsibility of the instructor.

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