Your best friend asks you to describe how the graph of
Guidance
Just like other functions, sine and cosine curves can be moved to the left, right, up and down. The general equation for a sine and cosine curve is
Example A
Graph
Solution: This function will be shifted
Example B
Graph
Solution: Because 2 is not written in terms of
Example C
Find the equation of the sine curve below.
Solution: First, we know the amplitude is 1 because the average between 2 and 0 (the maximum and minimum) is 1. Next, we can find the vertical shift. Recall that the maximum is usually 1, in this equation it is 2. That means that the function is shifted up 1 unit
Because
To determine the value of the horizontal shift, you might have to estimate. For example, we estimated that the negative shift was 3 because the maximum value of the parent graph is at
Concept Problem Revisit
If you compare
Therefore, the graph of
Guided Practice
Graph the following functions from
1.
2.
3. Find the equation of the cosine curve below.
Answers
1. Shift the parent graph down one unit.
2. Shift the parent graph to the left
3. The parent graph is in green. It moves up 3 units and to the right
If you moved the cosine curve backward, then the equation would be
Explore More
For questions 14, match the equation with its graph.

y=sin(x−π2)  \begin{align*}y=\cos \left(x\frac{\pi}{4}\right)+3\end{align*}
 \begin{align*}y=\cos \left(x+\frac{\pi}{4}\right)2\end{align*}
 \begin{align*}y=\sin \left(x\frac{\pi}{4}\right)+2\end{align*}
Which graph above also represents these equations?
 \begin{align*}y=\cos(x\pi)\end{align*}
 \begin{align*}y=\sin \left(x+\frac{3 \pi}{4}\right)2\end{align*}
 Write another sine equation for graph A.
 Writing How many sine (or cosine) equations can be generated for one curve? Why?
 Fill in the blanks below.
 \begin{align*}\sin x=\cos(x\underline{\;\;\;\;\;})\end{align*}
 \begin{align*}\cos x=\sin(x\underline{\;\;\;\;\;})\end{align*}
For questions 1015, graph the following equations from \begin{align*}[2\pi, 2\pi]\end{align*}.
 \begin{align*}y=\sin \left(x+\frac{\pi}{4}\right)\end{align*}
 \begin{align*}y=1+\cos x\end{align*}
 \begin{align*}y=\cos(x+\pi)2\end{align*}
 \begin{align*}y=\sin(x+3)4\end{align*}
 \begin{align*}y=\sin \left(x\frac{\pi}{6}\right)\end{align*}
 \begin{align*}y=\cos(x1)3\end{align*}
 Critical Thinking Is there a difference between \begin{align*}y=\sin x +1\end{align*} and \begin{align*}y=\sin(x+1)\end{align*}? Explain your answer.