Objectives
- Determine the basic properties of the graphs of sine and cosine, including amplitude, period, and phase shift.
Section 7.1 Properties of Sine and Cosine Graphs (PF1)
Subsection 7.1.1 Activities
Remark 7.1.1.
In the last module, we learned about finding values of trigonometric functions. Now, we will learn about the graphs of these functions.
Activity 7.1.2.
We’ll begin with the graph of the sine function, \(f(x)=\sin x\text{.}\)
(a)
Fill in the following table for \(f(x)=\sin x\text{.}\) Find the exact values, then approximate each to two decimal places. (Notice that the values in the table are all the standard angles found on the unit circle!)
NOTE FROM ABBY: this table looks terrible. How can we make it look better? Also, should we give the exact values to lighten the load some, and have them approximate so they can graph???
| \(x\) | \(f(x)\) (exact) | \(f(x)\) (approximate) |
|---|---|---|
| \(0\) | ||
| \(\dfrac{\pi}{6}\) | ||
| \(\dfrac{\pi}{4}\) | ||
| \(\dfrac{\pi}{3}\) | ||
| \(\dfrac{\pi}{2}\) | ||
| \(\dfrac{2\pi}{3}\) | ||
| \(\dfrac{3\pi}{4}\) | ||
| \(\dfrac{5\pi}{6}\) | ||
| \(\pi\) |
| \(x\) | \(f(x)\) (exact) | \(f(x)\) (approximate) |
|---|---|---|
| \(\dfrac{7\pi}{6}\) | ||
| \(\dfrac{5\pi}{4}\) | ||
| \(\dfrac{4\pi}{3}\) | ||
| \(\dfrac{3\pi}{2}\) | ||
| \(\dfrac{5\pi}{3}\) | ||
| \(\dfrac{7\pi}{4}\) | ||
| \(\dfrac{11\pi}{6}\) | ||
| \(2\pi\) |
Answer.
Answers coming once the table is formatted and looks better!
(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\sin x\text{.}\)
Answer.
(c)
What is the range of the function?
Answer.
\([-1,1]\)
Activity 7.1.3.
Let’s change our function a bit and look at \(g(x)=3\sin x\text{.}\)
(a)
Fill in the table below.
| \(x\) | \(f(x)=\sin x\) | \(g(x)=3\sin x\) |
|---|---|---|
| \(0\) | ||
| \(\dfrac{\pi}{6}\) | ||
| \(\dfrac{\pi}{4}\) | ||
| \(\dfrac{\pi}{3}\) | ||
| \(\dfrac{\pi}{2}\) |
Answer.
Answers coming!
(b)
Which of the following best describes how \(g(x)\) is related to \(f(x)=\sin x\text{?}\)
- The \(x\)-values in \(g(x)\) are three times the \(x\)-values of \(f(x)\text{.}\)
- The \(x\)-values in \(g(x)\) are one third of the \(x\)-values of \(f(x)\text{.}\)
- The \(y\)-values in \(g(x)\) are three times the \(y\)-values of \(f(x)\text{.}\)
- The \(y\)-values in \(g(x)\) are one third of the \(y\)-values of \(f(x)\text{.}\)
Answer.
C
(c)
What is the range of \(g(x)\) ?
Answer.
\([-3,3]\)
Definition 7.1.4.
The amplitude of a sine curve is vertical distance from the center of the curve to the maximum (or minimum) value.
We can also think of the amplitude as the value of the vertical stretch or compression.
When written as a function \(f(x)=A\sin x\text{,}\) the amplitude is \(|A|\text{.}\)
Activity 7.1.5.
(a)
We only found \(f(x)=\sin x\) for some values of \(x\) in the table in Activity 7.1.2, but those did not represent the entire domain. For which values of \(x\) can you find \(\sin x\text{?}\) (That is, what is the domain of \(f(x)=\sin x\text{?}\))
Answer.
\((-\infty,\infty)\)
(b)
Coterminal angles will have the same sine values. How do we know if two angles are coterminal?
- The difference between them is a multiple of \(\dfrac{\pi}{2}\text{.}\)
- The difference between them is a multiple of \(\pi\text{.}\)
- The difference between them is a multiple of \(\dfrac{3\pi}{2}\text{.}\)
- The difference between them is a multiple of \(2\pi\text{.}\)
Answer.
D
(c)
How often will the sine values repeat?
- Every \(\dfrac{\pi}{2}\) radians.
- Every \(\pi\) radians.
- Every \(\dfrac{2\pi}{2}\) radians.
- Every \(2\pi\) radians.
Answer.
D
(d)
Extend the graph you made in Activity 7.1.2 in both the positive and negative direction to show the repeated sine values.
Answer.
coming soon!
Definition 7.1.6.
The period of a sine function is the minimum value for which the \(y\)-values begin repeating.
The period for \(f(x)=\sin x\text{,}\) the standard sine curve, is \(2\pi\text{.}\)
Activity 7.1.7.
Now let’s look at \(h(x)=\sin 2x\text{.}\)
(a)
Think back to the types of transformations a function can have. (See Section 2.4 if you need a refresher!) What kind of transformation is happening in \(h(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
- A vertical stretch/compression.
- A horizontal stretch/compression.
- A vertical shift.
- A horizontal shift.
Answer.
B
(b)
Which of the following graphs is likely to be \(h(x)=\sin 2x\text{.}\) (To help compare the functions, \(f(x)=\sin x\) is shown as a dashed line on each graph.)
- placeholder for graph of \(\sin \frac{1}{2}x\)
- placeholder for graph of \(\sin (x-2)\)
- placeholder for graph of \(2 \sin x\)
Answer.
update once the order is fixed!
Activity 7.1.8.
Consider \(j(x)=\sin \frac{1}{2}x\text{.}\)
(a)
What type of transformation is happening in \(j(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
- A vertical stretch/compression.
- A horizontal stretch/compression.
- A vertical shift.
- A horizontal shift.
Answer.
B
(b)
Which of the following graphs is likely to be \(j(x)=\sin \frac{1}{2}x\text{.}\) (To help compare the functions, \(f(x)=\sin x\) is shown as a dashed line on each graph.)
- placeholder for graph of \(\sin \frac{1}{2}x\)
- placeholder for graph of \(\sin (x-2)\)
- placeholder for graph of \(2 \sin x\)
Answer.
update once the order is fixed!
Remark 7.1.9.
When written as a function \(f(x)=\sin Bx\text{,}\) the period is \(\dfrac{2\pi}{|B|}\text{.}\)
Activity 7.1.10.
Consider \(k(x)=\sin \left(x+\dfrac{\pi}{2}\right)\text{.}\)
(a)
What type of transformation is happening in \(k(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
- A vertical stretch/compression.
- A horizontal stretch/compression.
- A vertical shift.
- A horizontal shift.
Answer.
D
(b)
Which of the following graphs is likely to be \(k(x)=\sin \left(x+\dfrac{\pi}{2}\right)\text{.}\) (To help compare the functions, \(f(x)=\sin x\) is shown as a dashed line on each graph.)
- placeholder for graph of \(\sin \frac{1}{2}x\)
- placeholder for graph of \(\sin (x-2)\)
- placeholder for graph of \(2 \sin x\)
Answer.
update once the order is fixed!
Definition 7.1.11.
The phase shift is the amount which a sine function is shifted horizontally from the standard sine curve.
The phase shift for \(f(x)=\sin (x+C)\) is \(C\) units to the left. The phase shift for \(f(x)=\sin (x-C)\) is \(C\) units to the right.
Activity 7.1.12.
Let’s now turn our focus to the cosine function, \(f(x)=\cos x\text{.}\)
(a)
Fill in the following table for \(f(x)=\cos x\text{.}\) Find the exact values, then approximate each to two decimal places. (Notice that the values in the table are all the standard angles found on the unit circle!)
NOTE FROM ABBY: This table is also terrible looking, same as with the first activity. Update both the same way!
| \(x\) | \(f(x)\) (exact) | \(f(x)\) (approximate) |
|---|---|---|
| \(0\) | ||
| \(\dfrac{\pi}{6}\) | ||
| \(\dfrac{\pi}{4}\) | ||
| \(\dfrac{\pi}{3}\) | ||
| \(\dfrac{\pi}{2}\) | ||
| \(\dfrac{2\pi}{3}\) | ||
| \(\dfrac{3\pi}{4}\) | ||
| \(\dfrac{5\pi}{6}\) | ||
| \(\pi\) |
| \(x\) | \(f(x)\) (exact) | \(f(x)\) (approximate) |
|---|---|---|
| \(\dfrac{7\pi}{6}\) | ||
| \(\dfrac{5\pi}{4}\) | ||
| \(\dfrac{4\pi}{3}\) | ||
| \(\dfrac{3\pi}{2}\) | ||
| \(\dfrac{5\pi}{3}\) | ||
| \(\dfrac{7\pi}{4}\) | ||
| \(\dfrac{11\pi}{6}\) | ||
| \(2\pi\) |
Answer.
Answers coming once the table is formatted and looks better!
(b)
Plot these values on a graph.
Answer.
insert pic
(c)
What is the range of the function?
Answer.
\([-1,1]\)
Observation 7.1.13.
The cosine function, \(f(x)=\cos x\text{,}\) is equivalent to the sine function shifted to the left \(\dfrac{\pi}{2}\) units, \(g(x)=\sin\left(x+ \dfrac{\pi}{2}\right)\text{.}\)
Because of this, all of the methods we used to find amplitude, period, and phase shift for the sine function apply to the cosine function as well.
Exercises 7.1.2 Exercises
Exercises available at
https://tbil.org/precalculus/preview/exercises/#/bank/PF1/.
