1.
(a)
What is the period of the sine function? Of the cosine function?
(b)
What is the range of the sine function? Of the cosine function?
Section 6.1 Graphs of Sine and Cosine Functions
In this section, we’ll graph transformations of \(y = sin(x)\) and \(y = cos(x)\) and examine phase shifts of sine and cosine curves.
Subsection Textbook Reference
This relates to content in §8.1 of Algebra and Trigonometry 2e.
Exercises Preparation Exercises
Answer the following without using a calculator. These questions are intended to check the prerequisite skills needed to complete the rest of the lab.
2.
Imagine a function \(y = h(x)\text{,}\) with some points given in the table below. Fill in the table to show how the points are transformed for the given function values.
| Points on \(y=h(x)\) | \((0,0)\) | \((1,1) \) | \((-1,-1)\) | \((-3,2) \) | \((2,-3) \) |
|---|---|---|---|---|---|
| Points on \(y=3h(x)\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) |
| Points on \(y=3h(2x)\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) |
| Points on \(y=3h(2(x-1))\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) |
| Points on \(y=3h(2(x-1))+4\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) | \(\qquad\) |
3.
(a)
(b)
Exercises Practice Exercises
1.
Make a graph of \(y = sin(x)\) by plotting the points that you know from the table. You need to know that \(\frac{\sqrt{2}}{2} \approx 0.71\) and \(\frac{\sqrt{3}}{2} \approx 0.87\text{.}\)
| \(\theta\) | \(0\) | \(\frac{\pi}{6}\) | \(\frac{\pi}{4}\) | \(\frac{\pi}{3}\) | \(\frac{\pi}{2}\) | \(\frac{2\pi}{3}\) | \(\frac{3\pi}{4}\) | \(\frac{5\pi}{6}\) | \(\pi\) | \(\frac{7\pi}{6}\) | \(\frac{5\pi}{4}\) | \(\frac{4\pi}{3}\) | \(\frac{3\pi}{2}\) | \(\frac{5\pi}{3}\) | \(\frac{7\pi}{4}\) | \(\frac{11\pi}{6}\) | \(2\pi\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\sin(\theta)\) | \(0\) | \(\frac{1}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(1\) | \(\frac{\sqrt{3}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(\frac{1}{2}\) | \(0\) | \(-\frac{1}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(-1\) | \(-\frac{\sqrt{3}}{2}\) | \(-\frac{\sqrt{2}}{2}\) | \(-\frac{1}{2}\) | \(0\) |
2.
Compared to \(y=\sin(x)\text{...}\)
(a)
What is the phase shift for the function \(2\sin(3x-\pi)+4\text{?}\) Interpret this as a fraction of a period shift.
(b)
What is the horizontal shift for the function \(2\sin(3x-\pi)+4\text{?}\)
3.
(a)
What is the range of \(G\text{?}\)
(b)
(c)
(d)
(e)
(f)
If we think of \(G\) as a transformation of \(\sin(x)\text{,}\) how much of a horizontal shift would have happened to create \(G\text{?}\)
(g)
(h)
If we think of \(G\) as a transformation of \(\cos(x)\text{,}\) how much of a horizontal shift would have happened to create \(G\text{?}\)
(i)
4.
Answer the questions about the function \(T(x) = 3\sin(\frac{\pi}{3}(x+1))-5\text{.}\)
(a)
What is the midline of the function \(T\text{?}\)
(b)
(c)
What is the range of the function \(T\text{?}\)
(d)
(e)
(f)
(g)
Make a graph of \(T\) with what you know about the shape of \(\sin(x)\) and the information that you gained in the above parts. Check your result on a calculator after you’ve tried everything by hand.
Subsection Definitions
Definition 6.1.3. Even/Odd Function Properties of Sine and Cosine.
Sine is an odd function, which means that it is symmetrical over the origin. Cosine is an even function, which means that it is symmetrical over the \(y\)-axis.
Definition 6.1.4. Period.
The period of a function is the smallest number, \(P\text{,}\) such that a horizontal shift of \(P\) units results in a graph that perfectly overlaps the graph of the original function. Try this Desmos link to see a visual on the topic.
Definition 6.1.5. Midline.
The midline of a periodic function is the horizontal line, \(y= D\text{,}\) that is exactly halfway between the highest and lowest \(y\)-values on the graph. You can find this equation by averaging the highest and lowest \(y\)-values on the graph with the formula \(y= \frac{\text{highest y-value } + \text{lowest y-value}}{2}\text{.}\)
Definition 6.1.6. Amplitude.
The amplitude of a periodic function is the vertical distance from the midline to the highest \(y\)-value on the graph or the vertical distance from the midline to the lowest \(y\)-value on the graph. You can find this distance with the formula \(|A|= \frac{\text{highest y-value } - \text{lowest y-value}}{2}\text{.}\)
Definition 6.1.7. Sinusoidal Function.
A sinusoidal function is a transformation of a sine or cosine function. Sinusoidal functions have the form \(f(x) = Asin(B(x−C)) + D\) or \(f(x) = Acos(B(x−C)) + D\text{.}\)
Definition 6.1.8. Phase Shift.
The phase shift of a periodic function represents the fraction of a period that the graph has been shifted from the base function. You can find the phase shift of a sinusoidal function (\(f(x) = Asin(B(x−C)) + D\) or \(f(x) = Acos(B(x−C)) + D\)) by evaluating \(\frac{BC}{2\pi}\text{.}\) Please note that this definition is different from the one that the book gives, but is consistent with most other physics, engineering, and technical mathematics texts. The book defines the phase shift to be the same as the horizontal shift, for which we already have a name.
Definition 6.1.9. Horizontal Shift.
A sinusoidal function, \(f(x) = Asin(B(x−C)) + D\) or \(f(x) = Acos(B(x−C)) + D\text{,}\) is horizontally shifted C units from the base function position.
Definition 6.1.10. Relationship between the B-value and the Period.
The B value of a sinusoidal function, \(f(x) = Asin(B(x−C)) + D\) or \(f(x) = Acos(B(x−C)) + D\text{,}\) has a relationship with the period, \(P\text{,}\) of the function. Since the period is a positive number, we will use absolute values on B, since B might be negative. That relationship is that \(|B|=\frac{2\pi}{P}\text{,}\) which is the same as saying \(P = \frac{2\pi}{|B|}\text{.}\)
Exercises Exit Exercises
1.
Consider the function \(W(x) = 5\sin\left(\frac{4\pi}{5}(x-1)\right)+3\text{:}\)
(a)
What is the period of the function \(W(x)\text{?}\)
(b)
What is the midline of the function \(W(x)\text{?}\)
(c)
What is the amplitude of the function \(W(x)\text{?}\)
(d)
By hand, make a graph of the function \(W(x)\) using the information above.
A blank graph.
2.
Find two possible formulas, one using sine and the other using cosine, for the graph shown in Figure 6.1.11.
Reflection Reflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
