Skip to main content

Precalculus for Team-Based Inquiry Learning: PREVIEW Edition — Instructor Version

TBIL Community

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.
\(x\) \(f(x)\) (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.
\([-1,1]\)

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 \(f(x)=\sin x\text{?}\)
  1. The \(x\)-values in \(g(x)\) are three times the \(x\)-values of \(f(x)\text{.}\)
  2. The \(x\)-values in \(g(x)\) are one third of the \(x\)-values of \(f(x)\text{.}\)
  3. The \(y\)-values in \(g(x)\) are three times the \(y\)-values of \(f(x)\text{.}\)
  4. 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?
  1. The difference between them is a multiple of \(\dfrac{\pi}{2}\text{.}\)
  2. The difference between them is a multiple of \(\pi\text{.}\)
  3. The difference between them is a multiple of \(\dfrac{3\pi}{2}\text{.}\)
  4. The difference between them is a multiple of \(2\pi\text{.}\)
Answer.
D
(c)
How often will the sine values repeat?
  1. Every \(\dfrac{\pi}{2}\) radians.
  2. Every \(\pi\) radians.
  3. Every \(\dfrac{2\pi}{2}\) radians.
  4. 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 \(h(x)=\sin \left( \dfrac{1}{2}x \right)\text{.}\)
(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 \(f(x)=\sin x\text{?}\)
  1. A vertical stretch/compression.
  2. A horizontal stretch/compression.
  3. A vertical shift.
  4. A horizontal shift.
Answer.
B

Exercises 7.1.2 Exercises

PrevTopNext