Lesson 17

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

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

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

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

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

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

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

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

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

Here is the graph of a trigonometric function.

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.

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 \(g\)? 

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

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

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

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

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.

 

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.

The amplitude of \(f\) is 6.

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

The midline of \(f\) is 6.

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

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