Lesson 15

Features of Trigonometric Graphs (Part 1)

  • Let’s compare graphs and equations of trigonometric functions.

Problem 1

Here is a graph of a trigonometric function. Which equation could define this function?

graph of cosine function. amplitude = 1 point 5. midline = y = -4. period = 2 pi. 
A:

\(y = 1.5\sin(x) - 4\)

B:

\(y = 1.5\cos(x) - 4\)

C:

\(y = \text-4\sin(1.5x)\)

D:

\(y = \text-4\cos(1.5x)\)

Problem 2

Select all the functions that have period \(\pi\).

A:

\(y = \cos\left(\frac{x}{2}\right)\)

B:

\(y = \sin\left(\frac{x}{2}\right)\)

C:

\(y = \cos(x)\)

D:

\(y = \cos(2x)\)

E:

\(y = \sin(2x)\)

Problem 3

  1. Sketch a graph of \(a(\theta) = \cos(3\theta)\).

  2. Compare the graph of \(a\) to the graph of \(b(\theta)=\cos(\theta)\). How are the two graphs alike? How are they different?

horizontal axis, theta, scale 0 to 2 pi, by the fraction pi over 3's. vertical axis, y, scale -1 to 1, by 1's. 

Problem 4

The functions \(f\) and \(g\) are given by \(f(x) = \cos(x)\) and \(g(x) = \cos(5x)\). How are the graphs of \(f\) and \(g\) related? 

Problem 5

Here is a point at the tip of a windmill blade. The center of the windmill is 6 feet off the ground and the blades are 1.5 feet long.

Write an equation giving the height \(h\) of the point \(P\) after the windmill blade rotates by an angle of \(a\). Point \(P\) is currently rotated \(\frac{\pi}{4}\) radians from the point directly to the right of the center of the windmill.

windmill with 4 equivalent blades. upper right blade has P on the end of blade. 
(From Unit 6, Lesson 14.)

Problem 6

The coordinates of \(P\) are \((1,0)\).

A circle on a coordinate plane. The circle is divided by 12 congruent central angles. Point P is where the circle meets the x axis.
  1. If the wheel makes a \(\frac{1}{3}\) rotation counterclockwise around its center, what radian angle does \(P\) rotate through?
  2. If the wheel makes a \(1 \frac{1}{4}\) rotation counterclockwise around its center, what radian angle does \(P\) rotate through?
(From Unit 6, Lesson 3.)

Problem 7

A Ferris wheel has a radius of 95 feet and its center is 105 feet above the ground. Which statement is true about a point on the Ferris wheel as it goes around in a circle?

A:

It is 85 feet off the ground once in quadrant 1 and once in quadrant 2. 

B:

It is is 85 feet off the ground once in quadrant 2 and once in quadrant 3. 

C:

It is 85 feet off the ground once in quadrant 3 and once in quadrant 4. 

D:

It is 85 feet off the ground once in quadrant 4 and once in quadrant 1. 

(From Unit 6, Lesson 7.)