Your math teacher has decided to give you a quiz to see if you recognize how to combine changes to graphs of sine and cosine functions. You recall that you've learned about shifting graphs, as well as stretching/dilating them. But now your teacher wants to see if you know how to combine both of these effects into one graph. She gives you the equation:
and asks you to plot the equation, and then identify what each part of the above equation does to change the graph.
Can you accomplish this task?
Read on, and at the conclusion of this Concept, you'll know how to plot this equation and identify which parts of it make changes to the graph.
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James Sousa Example: Graphing a Transformation of Sine and Cosine
Guidance
In other Concepts, you learned how to translate and dilate sine and cosine waves both horizontally and vertically. Combining all the information learned, the general equations are:
Recall the relationship between period,
With this knowledge, we should be able to sketch any sine or cosine function as well as write an equation given its graph.
Example A
Given the function:
a. Identify the period, amplitude, and frequency.
b. Explain any vertical or horizontal translations present in the equation.
c. Sketch the graph from
Solution: a. From the equation, the amplitude is 2 and the frequency is also 2. To find the period we use:
So, there are two complete waves from
b.
c. To sketch the graph, start with the graph of
Translate the graph
Next, move the graph 1 unit up (
Now we can add the dilations. Remember that the “starting point” of the wave is
Using symmetry, each interval needs to cross the line
One sine wave contains a “mountain” and a “valley”. The mountain “peak” and the valley low point must occur halfway between the points above.
Extend the curve through the domain.
Finally, extend the minimum and maximum points to match the amplitude of 2.
Example B
Given the function:
a. Identify the period, amplitude, and frequency.
b. Explain any vertical or horizontal translations present in the equation.
c. Sketch the graph from
Solution: a. From the equation, the amplitude is 3 and the frequency is
So, there is only one half of a cosine curve from 0 to
b.
c. To sketch the graph, start with the graph of
Adjust the amplitude so the cosine wave reaches up to 3 and down to negative three. This affects the maximum points, but the points on the
According to the period, we should see one of these shapes every
Now, shift the graph
Finally, we need to adjust for the vertical shift by moving it up 3 units.
Example C
Find the equation of the sinusoid graphed here.
Solution: First of all, remember that either sine or cosine could be used to model these graphs. However, it is usually easier to use cosine because the horizontal shift is easier to locate in most cases. Therefore, the model that we will be using is
First, if we think of the graph as a cosine function, it has a horizontal translation of zero. The maximum point is also the
\begin{align*}D=center=\frac{60+20}{2}=\frac{40}{2}=20\end{align*}
The amplitude is the height from the center line, or vertical shift, to either the minimum or the maximum. So, \begin{align*}A=6020=40\end{align*}
The last value to find is the frequency. In order to do so, we must first find the period. The period is the distance required for one complete wave. To find this value, look at the horizontal distance between two consecutive maximum points.
On our graph, from maximum to maximum is 3.
Therefore, the period is 3, so the frequency is \begin{align*}B=\frac{2\pi}{3}\end{align*}
We have now calculated each of the four parameters necessary to write the equation. Replacing them in the equation gives:
\begin{align*}y=20+40 \cos \frac{2\pi}{3}x\end{align*}
If we had chosen to model this curve with a sine function instead, the amplitude, period and frequency, as well as the vertical shift would all be the same. The only difference would be the horizontal shift. The sine wave starts in the middle of an upward sloped section of the curve as shown by the red circle.
This point intersects with the vertical translation line and is a third of the distance back to 3. So, in this case, the sine wave has been translated 1 unit to the left. The equation using a sine function instead would have been: \begin{align*}y=20+40 \sin \left(\frac{2\pi}{3}(x+1) \right)\end{align*}
Guided Practice
1. Identify the amplitude, period, frequency, maximum and minimum points, vertical shift, and horizontal shift of \begin{align*}y=2+3 \sin(2(x1)).\end{align*}
2. Identify the amplitude, period, frequency, maximum and minimum points, vertical shift, and horizontal shift of \begin{align*}y=1+ \sin \left(\pi (x+\frac{\pi}{3}\right)).\end{align*}
3. Identify the amplitude, period, frequency, maximum and minimum points, vertical shift, and horizontal shift of \begin{align*}y=\cos (40(x120))+5.\end{align*}
Solutions:
1. This is a sine wave that has been translated 1 unit to the right and 2 units up. The amplitude is 3 and the frequency is 2. The period of the graph is \begin{align*}\pi\end{align*}
2. This is a sine wave that has been translated 1 unit down and \begin{align*}\frac{\pi}{3}\end{align*}
3. This is a cosine wave that has been translated 5 units up and 120 radians to the right. The amplitude is 1 and the frequency is 40. The period of the graph is \begin{align*}\frac{\pi}{20}\end{align*}
Concept Problem Solution
With your advanced knowledge of sinusoidal equations, you can identify in the equation:
\begin{align*}f(x) = 3 + 7 \sin(4(x + \frac{\pi}{2}))\end{align*}
The vertical shift of the graph is 3 units up. The amplitude of the graph is 7. The horizontal shift of the graph is \begin{align*}\frac{\pi}{2}\end{align*}
Since the frequency is 4, the period can be calculated:
\begin{align*} p = \frac{2\pi}{f}\\ p = \frac{2\pi}{4}\\ p = \frac{\pi}{2}\\ \end{align*}
This means that the graph takes \begin{align*}\frac{\pi}{2}\end{align*}
The graph of this equation looks like this:
Explore More
For each equation below, identify the period, amplitude, frequency, and any vertical/horizontal translations.

\begin{align*}y=24\cos(\frac{2}{3}(x3))\end{align*}
y=2−4cos(23(x−3)) 
\begin{align*}y=3+\frac{1}{2}\sin(\frac{1}{2}(x\pi))\end{align*}
y=3+12sin(12(x−π)) 
\begin{align*}y=1+5\cos(4(x+\frac{\pi}{2}))\end{align*}
y=1+5cos(4(x+π2)) 
\begin{align*}y=4\cos(2(x+1))\end{align*}
y=4−cos(2(x+1)) 
\begin{align*}y=3+2\sin(x4)\end{align*}
y=3+2sin(x−4)
Graph each of the following equations from \begin{align*}2\pi\end{align*}

\begin{align*}y=13\sin(\frac{1}{3}(x\pi))\end{align*}
y=1−3sin(13(x−π)) 
\begin{align*}y=5+\frac{1}{2}\sin(\frac{1}{2}(x2))\end{align*}
y=5+12sin(12(x−2)) 
\begin{align*}y=2+\cos(4(x+\frac{\pi}{2}))\end{align*}
y=2+cos(4(x+π2))  \begin{align*}y=4+2\cos(2(x+3))\end{align*}
 \begin{align*}y=23\sin(x\frac{3\pi}{2})\end{align*}
Find the equation of each sinusoid.