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Precalculus for Team-Based Inquiry Learning 2025 Edition — Instructor Version

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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.
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Activity 7.1.2.

We’ll begin with the graph of the sine function, \(f(x)=\sin x\text{.}\)
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(a)
Fill in the missing values in the table below for \(f(x)=\sin x\text{.}\) Find the exact values, then express as a decimal, approximated to two decimal places if needed. (Notice that the values in the table are all the standard angles found on the unit circle!)
\(x\) \(\sin x\) (exact) \(\sin x\) (as a decimal)
\(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{1}{2}\)
\(\dfrac{\pi}{4}\) \(\approx 0.71\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{2}\) \(1\)
\(\dfrac{2\pi}{3}\) \(\dfrac{\sqrt{3}}{2}\)
\(\dfrac{3\pi}{4}\)
\(\dfrac{5\pi}{6}\) \(0.5\)
\(\pi\)
\(\dfrac{7\pi}{6}\) \(-\dfrac{1}{2}\)
\(\dfrac{5\pi}{4}\) \(\dfrac{\sqrt{2}}{2}\)
\(\dfrac{4\pi}{3}\) \(\approx -0.87\)
\(\dfrac{3\pi}{2}\)
\(\dfrac{5\pi}{3}\) \(-\dfrac{\sqrt{3}}{2}\)
\(\dfrac{7\pi}{4}\)
\(\dfrac{11\pi}{6}\) \(-0.5\)
\(2\pi\)
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Answer.
\(x\) \(\sin x\) (exact) \(\sin x\) (as a decimal)
\(0\) \(0\) \(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{1}{2}\) \(0.5\)
\(\dfrac{\pi}{4}\) \(\dfrac{\sqrt{2}}{2}\) \(\approx 0.71\)
\(\dfrac{\pi}{3}\) \(\dfrac{\sqrt{3}}{2}\) \(\approx 0.87\)
\(\dfrac{\pi}{2}\) \(1\) \(1\)
\(\dfrac{2\pi}{3}\) \(\dfrac{\sqrt{3}}{2}\) \(\approx 0.87\)
\(\dfrac{3\pi}{4}\) \(\dfrac{\sqrt{2}}{2}\) \(\approx 0.71\)
\(\dfrac{5\pi}{6}\) \(\dfrac{1}{2}\) \(0.5\)
\(\pi\) \(0\) \(0\)
\(\dfrac{7\pi}{6}\) \(-\dfrac{1}{2}\) \(-0.5\)
\(\dfrac{5\pi}{4}\) \(-\dfrac{\sqrt{2}}{2}\) \(\approx -0.71\)
\(\dfrac{4\pi}{3}\) \(-\dfrac{\sqrt{3}}{2}\) \(\approx -0.87\)
\(\dfrac{3\pi}{2}\) \(-1\) \(-1\)
\(\dfrac{5\pi}{3}\) \(-\dfrac{\sqrt{3}}{2}\) \(\approx -0.87\)
\(\dfrac{7\pi}{4}\) \(-\dfrac{\sqrt{2}}{2}\) \(\approx -0.71\)
\(\dfrac{11\pi}{6}\) \(-\dfrac{1}{2}\) \(-0.5\)
\(2\pi\) \(0\) \(0\)
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(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\sin x\text{.}\) Then sketch in the graph of the sine curve using the points as a guide.
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Answer.
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Activity 7.1.3.

Let’s change our function a bit and look at \(g(x)=3\sin x\text{.}\)
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(a)
Fill in the table below.
\(x\) \(f(x)=\sin x\) \(g(x)=3\sin x\)
\(0\)
\(\dfrac{\pi}{6}\)
\(\dfrac{\pi}{4}\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{2}\)
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Answer.
\(x\) \(f(x)=\sin x\) \(g(x)=3\sin x\)
\(0\) \(0\) \(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{1}{2}\) \(\dfrac{3}{2}\)
\(\dfrac{\pi}{4}\) \(\dfrac{\sqrt{2}}{2}\) \(\dfrac{3\sqrt{2}}{2}\)
\(\dfrac{\pi}{3}\) \(\dfrac{\sqrt{3}}{2}\) \(\dfrac{3\sqrt{3}}{2}\)
\(\dfrac{\pi}{2}\) \(1\) \(3\)
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(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 \(f(x)\text{.}\)
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  2. The \(x\)-values in \(g(x)\) are one third of the \(x\)-values of \(f(x)\text{.}\)
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  3. The \(y\)-values in \(g(x)\) are three times the \(y\)-values of \(f(x)\text{.}\)
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  4. The \(y\)-values in \(g(x)\) are one third of the \(y\)-values of \(f(x)\text{.}\)
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Answer.
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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.
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We can also think of the amplitude as the value of the vertical stretch or compression.
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When written as a function \(f(x)=A\sin x\text{,}\) the amplitude is \(|A|\text{.}\)
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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 \(f(x)=\sin x\text{?}\))
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Answer.
\((-\infty,\infty)\)
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(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 \(\dfrac{\pi}{2}\text{.}\)
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  2. The difference between them is a multiple of \(\pi\text{.}\)
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  3. The difference between them is a multiple of \(\dfrac{3\pi}{2}\text{.}\)
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  4. The difference between them is a multiple of \(2\pi\text{.}\)
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Answer.
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Definition 7.1.6.

The period of a sine function is the minimum value for which the \(y\)-values begin repeating.
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The period for \(f(x)=\sin x\text{,}\) the standard sine curve, is \(2\pi\text{.}\)
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Activity 7.1.7.

Now let’s look at \(h(x)=\sin 2x\text{.}\)
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(a)
Think back to the types of transformations a function can have. (See Section 2.4 if you need a reminder!) What kind of transformation is happening in \(h(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
  1. A vertical stretch/compression.
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  2. A horizontal stretch/compression.
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  3. A vertical shift.
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  4. A horizontal shift.
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Answer.
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(b)
Which of the following graphs represents one cycle of \(h(x)=\sin 2x\text{.}\) (To help compare the functions, one cycle of \(f(x)=\sin x\) is shown as a dashed line on each graph.)
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Answer.
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Activity 7.1.8.

Consider \(j(x)=\sin \frac{1}{2}x\text{.}\)
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(a)
What type of transformation is happening in \(j(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
  1. A vertical stretch/compression.
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  2. A horizontal stretch/compression.
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  3. A vertical shift.
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  4. A horizontal shift.
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Answer.
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(b)
Which of the following graphs represents one cycle of \(j(x)=\sin \frac{1}{2}x\text{.}\) (To help compare the functions, one cycle of \(f(x)=\sin x\) is shown as a dashed line on each graph.)
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Answer.
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Remark 7.1.9.

When written as a function \(f(x)=\sin Bx\text{,}\) the period is \(\dfrac{2\pi}{|B|}\text{.}\)
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Activity 7.1.10.

Consider \(k(x)=\sin \left(x+\dfrac{\pi}{2}\right)\text{.}\)
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(a)
What type of transformation is happening in \(k(x)\) compared the parent function \(f(x)=\sin x\text{?}\)
  1. A vertical stretch/compression.
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  2. A horizontal stretch/compression.
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  3. A vertical shift.
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  4. A horizontal shift.
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Answer.
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(b)
Which of the following graphs represents one cycle of \(k(x)=\sin \left(x+\dfrac{\pi}{2}\right)\text{.}\) (To help compare the functions, one cycle of \(f(x)=\sin x\) is shown as a dashed line on each graph.)
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Answer.
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Definition 7.1.11.

The phase shift is the amount which a sine function is shifted horizontally from the standard sine curve.
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The phase shift for \(f(x)=\sin (x-C)\) is \(C\text{,}\) or \(C\) units to the right. The phase shift for \(f(x)=\sin (x+C)\) is \(-C\text{,}\) or \(C\) units to the left.
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A function can have both a horizontal shift and a change in period. In that case, it could be written as \(f(x)=\sin(Bx-C)\text{.}\) Here the phase shift would be \(\dfrac{C}{B}\text{.}\) You can think of solving the equation \(Bx-C=0\) for \(x\text{.}\) A positive value would represent a shift to the right and a negative value would represent a shift to the left.
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Activity 7.1.12.

Let’s now turn our focus to the cosine function, \(f(x)=\cos x\text{.}\)
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(a)
Fill in the missing values in the table below for \(f(x)=\cos x\text{.}\) Find the exact values, then express as a decimal, approximated to two decimal places if needed. (Notice that the values in the table are all the standard angles found on the unit circle!)
\(x\) \(\cos x\) (exact) \(\cos x\) (as a decimal)
\(0\)
\(\dfrac{\pi}{6}\) \(\dfrac{\sqrt{3}}{2}\)
\(\dfrac{\pi}{4}\) \(\approx 0.71\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{2}\) \(0\)
\(\dfrac{2\pi}{3}\) \(\dfrac{-1}{2}\)
\(\dfrac{3\pi}{4}\)
\(\dfrac{5\pi}{6}\) \(\approx -.87\)
\(\pi\)
\(\dfrac{7\pi}{6}\) \(-\dfrac{\sqrt{3}}{2}\)
\(\dfrac{5\pi}{4}\) \(\dfrac{-\sqrt{2}}{2}\)
\(\dfrac{4\pi}{3}\) \(-0.5\)
\(\dfrac{3\pi}{2}\)
\(\dfrac{5\pi}{3}\) \(\dfrac{1}{2}\)
\(\dfrac{7\pi}{4}\)
\(\dfrac{11\pi}{6}\) \(\approx 0.87\)
\(2\pi\)
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Answer.
\(x\) \(\cos x\) (exact) \(\cos x\) (as a decimal)
\(0\) \(1\) \(1\)
\(\dfrac{\pi}{6}\) \(\dfrac{\sqrt{3}}{2}\) \(\approx 0.87\)
\(\dfrac{\pi}{4}\) \(\dfrac{\sqrt{2}}{2}\) \(\approx 0.71\)
\(\dfrac{\pi}{3}\) \(\dfrac{1}{2}\) \(0.5\)
\(\dfrac{\pi}{2}\) \(0\) \(0\)
\(\dfrac{2\pi}{3}\) \(\dfrac{-1}{2}\) \(-0.5\)
\(\dfrac{3\pi}{4}\) \(-\dfrac{\sqrt{2}}{2}\) \(\approx 0.71\)
\(\dfrac{5\pi}{6}\) \(-\dfrac{\sqrt{3}}{2}\) \(\approx -.87\)
\(\pi\) \(-1\) \(-1\)
\(\dfrac{7\pi}{6}\) \(-\dfrac{\sqrt{3}}{2}\) \(\approx -0.87\)
\(\dfrac{5\pi}{4}\) \(\dfrac{-\sqrt{2}}{2}\) \(\approx -0.71\)
\(\dfrac{4\pi}{3}\) \(-\dfrac{1}{2}\) \(-0.5\)
\(\dfrac{3\pi}{2}\) \(0\) \(0\)
\(\dfrac{5\pi}{3}\) \(\dfrac{1}{2}\) \(0.5\)
\(\dfrac{7\pi}{4}\) \(\dfrac{2}{2}\) \(\approx 0.71\)
\(\dfrac{11\pi}{6}\) \(\dfrac{\sqrt{3}}{2}\) \(\approx 0.87\)
\(2\pi\) \(1\) \(1\)
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(b)
Plot these values on a coordinate plane to approximate the graph of \(f(x)=\cos x\text{.}\) Then sketch in the graph of the cosine curve using the points as a guide.
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Answer.
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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, \(g(x)=\sin\left(x+ \dfrac{\pi}{2}\right)\text{.}\)
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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.
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Observation 7.1.14.

Now that we can graph both the standard sine and cosine curves, we can add them to our list of parent functions in Section A.1. We also show them graphed below on the interval \([0,2\pi]\text{.}\)
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\(f(x)=\sin x\)
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\(f(x)=\cos x\)
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Activity 7.1.15.

Find the amplitude, period, and phase shift of each of the following sine functions shown as a solid line. To help, \(f(x)=\sin x\) is shown as a dotted line.
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(a)
Answer.
Amplitude: \(2\)
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Period: \(2\pi\)
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Phase shift: \(\dfrac{\pi}{2}\) or \(-\dfrac{3\pi}{2}\)
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Activity 7.1.16.

Find the amplitude, period, and phase shift of each of the following functions.
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(a)
\(f(x)=\dfrac{3}{2}\cos(x+\pi)\)
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Answer.
Amplitude: \(\dfrac{3}{2}\)
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Period: \(2\pi\)
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Phase shift: \(-\pi\) (or \(\pi\) to the left)
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(b)
\(f(x)=3\sin(2x-\pi)\)
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Answer.
Amplitude: \(3\)
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Period: \(\pi\)
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Phase shift: \(\dfrac{\pi}{2}\) (or \(\dfrac{\pi}{2}\) to the right)
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(c)
\(f(x)=4\sin\left(3x+\dfrac{\pi}{2}\right)\)
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Answer.
Amplitude: \(4\)
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Period: \(\dfrac{2\pi}{3}\)
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Phase shift: \(-\dfrac{\pi}{6}\) (or \(\dfrac{\pi}{6}\) to the left)
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(d)
\(f(x)=\dfrac{1}{2}\cos\left(2x+\dfrac{\pi}{4}\right)\)
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Answer.
Amplitude: \(\dfrac{1}{2}\)
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Period: \(\pi\)
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Phase shift: \(-\dfrac{\pi}{8}\) (or \(\dfrac{\pi}{8}\) to the left)
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Activity 7.1.17.

Find an expression for each of the following transformations of the sine and cosine functions.
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(a)
\(\sin(x)\) with a phase shift left by \(\pi\) and an amplitude of \(3\text{.}\)
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Answer.
\(3\sin(x+\pi)\)
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(b)
\(\cos(x)\) with a phase shift right by \(\dfrac{\pi}{2}\) and a period of \(\pi\text{.}\)
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Answer.
\(\cos\left(2\left(x-\dfrac{\pi}{2}\right)\right)\)
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(c)
\(\sin(x)\) with a phase shift right by \(\dfrac{\pi}{6}\text{,}\) a period of \(\dfrac{\pi}{3}\text{,}\) and an amplitude of \(5\text{.}\)
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Answer.
\(5\sin\left(6\left(x+\dfrac{\pi}{6}\right)\right)\)
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Activity 7.1.18.

To graph \(f(x)=3\sin(2x-\pi)-3\text{,}\) lets apply one transformation at a time.
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(a)
Which of the following is equivalent to \(f(x)=3\sin(2x-\pi)-3\text{?}\)
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  1. \(\displaystyle 3\sin\left(2\left(x-\pi\right)\right)-3\)
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  2. \(\displaystyle 3\sin\left(2\left(x-\dfrac{\pi}{2}\right)\right)-3\)
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  3. \(\displaystyle 3\sin\left(2\left(x-\dfrac{\pi}{3}\right)\right)-2\)
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  4. \(\displaystyle 3\sin\left(2\left(x-2\pi\right)\right)-3\)
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Answer.
B.
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(b)
Graph the function \(f_1(x) = \sin\left(2x\right)\text{.}\)
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Answer.
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(c)
Graph the function \(f_2(x) = \sin\left(2\left(x-\dfrac{\pi}{2}\right)\right)\text{.}\)
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Answer.
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(d)
Graph the function \(f_3(x) = 3\sin\left(2\left(x-\dfrac{\pi}{2}\right)\right)\text{.}\)
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Answer.
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(e)
Graph the function \(f_4(x) = 3\sin\left(2\left(x-\dfrac{\pi}{2}\right)\right)-3\text{.}\)
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Answer.
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Activity 7.1.19.

Graph each of the following functions.
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(c)
\(-3\sin\left(4x-\dfrac{\pi}{2}\right)+1\)
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Answer.
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Subsection 7.1.2 Exercises

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