Lesson 17

Comparing Transformations

  • Let's ask questions to figure out transformations of trigonometric functions.

Problem 1

Here is the graph of a trigonometric function.

Which equation has this graph? Select all that apply.

graph of y = sine of x reflected over x axis. amplitude = the fraction 3 over 2. period = 1. 
A:

\(y=\frac{3}{2} \cos\left (2\pi x - \frac{\pi}{2}\right)\)

B:

\(y=\text-\frac{3}{2}\sin(2\pi x)\)

C:

\(y=\frac{3}{2}\cos(2\pi x)\)

D:

\(y=\frac{3}{2} \cos\left (2\pi x + \frac{\pi}{2}\right)\)

E:

\(y=\frac{3}{2} \sin(2\pi x + \pi)\)

Problem 2

Here is the graph of a trigonometric function.

Which equation has this graph?

graph of cosine function with midline = 1, amplitude = the fraction 1 over 2, period 2. 
A:

\(y=\cos(x) + 1\)

B:

\(y=\frac{1}{2}\cos(x) + 1\)

C:

\(y=\frac{1}{2}\cos(\pi \boldcdot x) + 1\)

D:

\(y=\frac{1}{2} \cos(2\pi \boldcdot x) + 1\)

Problem 3

Here is the graph of a trigonometric function.

graph of sine function with midline = 3. period = pi. amplitude = 1. 
  1. Find a trigonometric function \(f\) that has this graph. Explain your reasoning.
  2. The graph is translated right by \(\frac{\pi}{2}\) so it has a minimum value at \(x=0\), then stretched horizontally so its period is 3 times greater than the period of \(f\). Find a trigonometric function \(g\) that has this new graph. Explain your reasoning.

Problem 4

The function \(f\) is given by \(f(x) = 4 + 2\sin\left(\pi x\right)\). The graph of \(g\) is the graph of \(f\) translated left by \(\frac{\pi}{2}\) and translated vertically by -1. Which expression defines \(g\)? 

A:

\(5 + 2\sin\left(\pi x + \frac{\pi}{2}\right)\)

B:

\(3 + 2\sin\left(\pi x + \frac{\pi}{2}\right)\)

C:

\(3 + 2\sin\left(\pi \left(x-\frac{\pi}{2}\right)\right)\)

D:

\(3 + 2\sin\left(\pi \left( x + \frac{\pi}{2}\right)\right)\)

Problem 5

Here are graphs of trigonometric functions \(f\) and \(g\). What transformations can be applied to the graph of \(f\) to get the graph of \(g\)? Make sure to list them in the order they are applied.

graph of functions f and g. g is f translated left by the fraction 1 over 4 and down 1 point 5, vertical stretch by factor 1 point 5, horizontal stretch by factor the fraction one half.

 

Problem 6

The table shows the vertical position of a point at the tip of a windmill blade after the blade has rotated through different angles. The point starts at the location furthest to the right.

  1. How long is the windmill blade? Explain how you know.
  2. What is the height of the windmill? Explain how you know.
rotation angle
of windmill		 vertical position
of \(P\) in feet
\(\frac{\pi}{6}\) 11.25
\(\frac{\pi}{3}\) 10 + \(\frac{2.5\sqrt{3}}{2}\)
\(\frac{\pi}{2}\) 12.5
\(\pi\) 10
\(\frac{3\pi}{2}\) 7.5
(From Unit 6, Lesson 13.)

Problem 7

The function \(f\) is given by \(f(\theta) = 6 +5\cos\left(\theta + \frac{\pi}{2}\right)\). Which of the following are true of \(f\)? Select all that apply.

A:

The amplitude of \(f\) is 6.

B:

The function \(f\) takes its maximum value when \(x = 0\).

C:

The midline of \(f\) is 6.

D:

The graph of \(f\) is the same as the graph of \(g(\theta) = 6+ 5\cos(\theta)\) translated to the right by \(\frac{\pi}{2}\).

E:

The graph of \(f\) is the same as the graph of \(g(\theta) = 6+ 5\cos(\theta)\) translated to the left by \(\frac{\pi}{2}\).

(From Unit 6, Lesson 14.)