14.1: Graphing Trigonometric Functions
Objective
To graph, translate, and reflect the sine, cosine, and tangent functions.
Review Queue
Find the exact values of the following expressions.
1.
2.
3.
4.
Graphing Sine and Cosine
Objective
To graph and stretch the sine and cosine functions.
Guidance
In this concept, we will take the unit circle, introduced in the previous chapter, and graph it on the Cartesian plane.
To do this, we are going to “unravel” the unit circle. Recall that for the unit circle the coordinates are
Notice that the curve ranges from 1 to 1. The maximum value is 1, which is at
Now, look at the domain. It seems that, if we had continued the curve, it would repeat. This means that the sine curve is periodic. Look back at the unit circle, the sine value changes until it reaches
Similarly, when we expand the cosine curve,
Notice that the range is also between 1 and 1 and the domain will be all real numbers. The cosine curve is also periodic, with a period of
Comparing
If we shift either curve
Example A
Identify the highlighted points on
,
Solution: For each point, think about what the sine or cosine value is at those values. For point
For the cosine curve, point
More Guidance
In addition to graphing
Example B
Graph
Solution: Start with the basic sine curve. Recall that one period of the parent graph,
Notice that the
Example C
Graph
Solution: Now, the amplitude will be
Example D
Graph
Solution: The last two examples dealt with changing
Guided Practice
1. Is the point
Graph the following functions for two full periods.
2.
3.
4.
Answers
1. Substitute in the point for
This is true, so
2. Stretch the cosine curve so that the maximum is 6 and the minimum is 6.
3. The graph is reflected over the
4. The fraction is equivalent to 1.5, making 1.5 the amplitude.
Vocabulary
 Trigonometric Function

When the sine, cosine, or tangent of an angle is plotted in the
x−y plane such that(x,f(x)) , wherex is the central angle from the unit circle andf(x) is the sine, cosine, or tangent of that angle.
 Amplitude

The height of a sine or cosine curve. In the equation,
y=asinx ory=acosx , the amplitude isa .
 Periodic

When a function repeats its
y values over a regular interval.
 Period
 The regular interval over which a periodic function repeats itself.
 Phase Shift
 The horizontal shift of a trigonometric function.
Problem Set
 Determine the exact value of each point on
y=sinx ory=cosx .
 List all the points in the interval
[0,4π] wheresinx=cosx . Use the graph from #1 to help you.  Draw from
y=sinx from[0,2π] . Findf(π3) andf(5π3) . Plot these values on the curve.
For questions 412, graph the sine or cosine curve over two periods.

y=2sinx 
y=−5cosx 
y=14cosx 
y=−23sinx 
y=4sinx 
y=−1.5cosx 
y=53cosx 
y=10sinx 
y=−7.2sinx  Graph
y=sinx andy=cosx on the same set of axes. How many units would you have to shift the sine curve (to the left or right) so that it perfectly overlaps the cosine curve?  Graph
y=sinx andy=−cosx on the same set of axes. How many units would you have to shift the sine curve (to the left or right) so that it perfectly overlapsy=−cosx ?
Write the equation for each sine or cosine curve below.
Translating Sine and Cosine Functions
Objective
To be able to graph a translated sine or cosine function.
Guidance
Just like other functions, sine and cosine curves can be moved to the left, right, up and down. The general equation for a sine and cosine curve is
Example A
Graph
Solution: This function will be shifted
Example B
Graph
Solution: Because 2 is not written in terms of
Example C
Find the equation of the sine curve below.
Solution: First, we know the amplitude is 1 because the average between 2 and 0 (the maximum and minimum) is 1. Next, we can find the vertical shift. Recall that the maximum is usually 1, in this equation it is 2. That means that the function is shifted up 1 unit
Because
To determine the value of the horizontal shift, you might have to estimate. For example, we estimated that the negative shift was 3 because the maximum value of the parent graph is at
Guided Practice
Graph the following functions from
1.
2.
3. Find the equation of the cosine curve below.
Answers
1. Shift the parent graph down one unit.
2. Shift the parent graph to the left
3. The parent graph is in green. It moves up 3 units and to the right
If you moved the cosine curve backward, then the equation would be
Vocabulary
 Phase Shift
 The horizontal shift of a trigonometric function.
Problem Set
For questions 14, match the equation with its graph.

y=sin(x−π2) 
y=cos(x−π4)+3 
y=cos(x+π4)−2 
y=sin(x−π4)+2  Which graph above represents:

y=cos(x−π) 
y=sin(x+3π4)−2

 Fill in the blanks below.

sinx=cos(x−−−−) 
cosx=sin(x−−−−)

For questions 712, graph the following equations from

y=sin(x+π4) 
y=1+cosx 
y=cos(x+π)−2 
y=sin(x+3)−4 
y=sin(x−π6) 
y=cos(x−1)−3 
Critical Thinking Is there a difference between
y=sinx+1 andy=sin(x+1) ? Explain your answer.
Putting it all Together
Objective
To graph sine and cosine functions where the amplitude is changed and horizontal and vertical shifts.
Guidance
In this concept, we will combine the previous two concepts and change the amplitude, the horizontal shifts, vertical shifts, and reflections.
Example A
Graph
Solution: First, stretch the curve so that the amplitude is 4, making the maximums and minimums 4 and 4. Then, shift the curve
As for the domain, it is all real numbers because the sine curve is periodic and infinite. The range will be from the maximum to the minimum;
Example B
Graph
Solution: The 2 indicates the cosine curve is flipped and stretched so that the amplitude is 2. Then, move the curve up one unit and to the right one unit.
The domain is all real numbers and the range is
Example C
Find the equation of the sine curve to the right.
Solution: First, let’s find the amplitude. The range is from 1 to 5, which is a total distance of 6. Divided by 2, we find that the amplitude is 3. Halfway between 1 and 5 is
Subtracting
Making the equation
Guided Practice
Graph the following functions. State the domain and range. Show two full periods.
1.
2.
3. Write one sine equation and one cosine equation for the curve below.
Answers
1. The domain is all real numbers and the range is
2. The domain is all real numbers and the range is
3. The amplitude and vertical shift is the same, whether the equation is a sine or cosine curve. The vertical shift is 2 because that is the number that is halfway between the maximum and minimum. The difference between the maximum and minimum is 1, so the amplitude is half of that, or
Problem Set
Determine if the following statements are true or false.
 To change a cosine curve into a sine curve, shift the curve
π2 units.  For any given sine or cosine graph, there are infinitely many possible equations that can be written to represent the curve.
 The amplitude is the same as the maximum value of the sine or cosine curve.
 The horizontal shift is always in terms of
π .  The domain of any sine or cosine function is always all real numbers.
Graph the following sine or cosine functions such that

y=sin(x+π4)+1 
y=2−3cosx 
y=34sin(x−2π3) 
y=−5sin(x−3)−2 
y=2cos(x+5π6)−1.5 
y=−2.8cos(x−8)+4
Use the graph below to answer questions 1215.
 Write a sine equation for the function where the amplitude is positive.
 Write a cosine equation for the function where the amplitude is positive.
 How often does a sine or cosine curve repeat itself? How can you use this to help you write different equations for the same graph?
 Write a second sine and cosine equation with different horizontal shifts.
Use the graph below to answer questions 1620.
 Write a sine equation for the function where the amplitude is positive.
 Write a cosine equation for the function where the amplitude is positive.
 Write a sine equation for the function where the amplitude is negative.
 Write a cosine equation for the function where the amplitude is negative.
 Describe the similarities and differences between the four equations from questions 1619.
Changes in the Period of a Sine and Cosine Function
Objective
Here you’ll learn how to change the period of a sine and cosine function.
Guidance
The last thing that we can manipulate on the sine and cosine curve is the period
The normal period of a sine or cosine curve is
To determine the period from an equation, we introduce
Example A
Determine the period of the following sine and cosine functions.
a)
b)
c)
Solution: a) The 6 in the equation tells us that there are 6 repetitions within
b) The
c) The
Example B
Graph part a) from the previous example from
Solution: The amplitude is 3, so it will be stretched and flipped. The period is
The minimums occur at 3 and the
Example C
Find all the solutions from the function in Example B from
Solution: Before this concept, the zeros didn’t change in the frequency because we hadn’t changed the period. Now that the period can be different, we can have a different number of zeros within
0 &=\cos 6x
Now, use the inverse cosine function to determine when the cosine is zero. This occurs at the multiples of
We went much past
Guided Practice
1. Determine the period of the function
2. Find the zeros of the function from #1 from
3. Determine the equation of the sine function with an amplitude of 3 and a period of
Answers
1. The period is
2. The zeros would be when
0 &=\cos \frac{3}{4}x \\
\frac{3}{4}x &=\cos^{1}0=\frac{\pi}{2},\frac{3 \pi}{2} \\
x &=\frac{4}{3}\left(\frac{\pi}{2},\frac{3 \pi}{2}\right) \\
x &=\frac{2\pi}{3},2\pi
3. The general equation of a sine curve is
\frac{2\pi}{8\pi} &=b \\
\frac{1}{4} &=b
The equation of the curve is
Vocabulary
 Period
 The length in which an entire sine or cosine curve is completed.
 Frequency

The number of times a curve is repeated within
2π .
Problem Set
Find the period of the following sine and cosine functions.

y=5sin3x 
y=−2cos4x 
y=−3sin2x 
y=cos34x 
y=12cos2.5x 
y=4sin3x
Use the equation
 Graph the function from
[0,2π] and find the domain and range.  Determine the coordinates of the maximum and minimum values.
 Find all the zeros from
[0,2π] .
Use the equation
 Graph the function from
[0,4π] and find the domain and range.  Determine the coordinates of the maximum and minimum values.
 Find all the zeros from
[0,2π] .
Use the equation
 Graph the function from
[0,2π] and find the domain and range.  Determine the coordinates of the maximum and minimum values.
 Find all the zeros from
[0,2π] .  What is the domain of every sine and cosine function? Can you make a general rule for the range? If so, state it.
Write the equation of the sine function, in the form
 Amplitude: 2 Period:
3π4  Amplitude:
35 Period:5π  Amplitude: 9 Period: 6

Challenge Find all the zeros from
[0,2π] ofy=12sin3(x−π3) .
Graphing Tangent
Objective
Here you’ll learn how to graph a tangent function.
Guidance
The graph of the tangent function is very different from the sine and cosine functions. First, recall that the tangent ratio is
y && \tan \theta && 0 && \frac{\sqrt{3}}{3} && 1 && \sqrt{3} && \text{und.} && \sqrt{3} && 1 && \frac{\sqrt{3}}{3} && 0
After
The red portion of the graph represents the coordinates in the table above. Repeating this portion, we get the entire tangent graph. Notice that there are vertical asymptotes at
The standard form of the equation is
Example A
Graph
Solution: First, the amplitude is 3, which means each
Notice that the vertical asymptotes did not change. The period of this function is still
The domain will be all real numbers, except where the asymptotes occur. Therefore, the domain of this function will be x\in \mathbb{R}, x \notin n\pi \pm \frac{\pi}{2}. The range is all real numbers.
Example B
Graph @$y=\tan 2\pi@$ from @$[0, 2\pi]@$ and state the domain and range. Find all zeros within this domain.
Solution: The period of this tangent function will be @$\frac{\pi}{2}@$ and the curves will be reflected over the @$x@$axis. The domain is all real numbers, @$x \notin \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{\pi}{4}\pm \frac{\pi}{2}n@$ where @$n@$ is any integer. The range is all real numbers. To find the zeros, set @$y = 0@$.
@$$0 &=\tan 2x \\ 0 &=\tan 2x \\ 2x &=\tan^{1}0=0, \pi, 2\pi, 3\pi, 4\pi \\ x &=0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi@$$
Example C
Graph @$y=\frac{1}{4}\tan\frac{1}{4}x@$ from @$[0, 4\pi]@$ and state the domain and range.
Solution: This function has a period of @$\frac{\pi}{\frac{1}{4}}=4\pi@$. The domain is all real numbers, except @$2\pi, 6\pi, 10\pi, 2\pi \pm 4\pi n@$, where @$n@$ is any integer. The range is all real numbers.
Guided Practice
1. Find the period of the function @$y=4\tan \frac{3}{2}x@$.
2. Find the zeros of the function from #1, from @$[0, 2\pi]@$.
3. Find the equation of the tangent function with an amplitude of 8 and a period of @$6\pi@$.
Answers
1. The period is @$\frac{\pi}{\frac{3}{2}}=\pi \cdot \frac{2}{3}=\frac{2\pi}{3}@$.
2. The zeros are where @$y@$ is zero. @$$0 &=4\tan \frac{3}{2}x \\ 0 &=\tan \frac{3}{2}x \\ \frac{3}{2}x &=\tan ^{1}0=0, \pi, 2\pi, 3\pi \\ x &=\frac{2}{3}(0, \pi, 2\pi, 3\pi ) \\ x &=0, \frac{2\pi}{3}, \frac{4\pi}{3}, 2\pi@$$
3. The general equation is @$y=a\tan bx@$. We know that @$a = 8@$. Let’s use the period to solve for the frequency, or @$b@$.
@$$\frac{\pi}{b} &=6\pi \\ b &=\frac{\pi}{6\pi}=\frac{1}{6}@$$
The equation is @$y=8\tan \frac{1}{6}x@$.
Vocabulary
 Tangent Function
 Defined by the coordinates @$(\theta, \tan \theta)@$, where @$\theta@$ is the central angle from the unit circle and tangent is the ratio of the sine and cosine functions.
Problem Set
Graph the following tangent functions over @$[0, 4\pi]@$. Determine the period, domain, and range.
 @$y=2\tan x@$
 @$y=\frac{1}{3}\tan x@$
 @$y=\tan 3x@$
 @$y=4\tan 2x@$
 @$y=\frac{1}{2}\tan 4x@$
 @$y=\tan \frac{1}{2}x@$
 @$y=4+\tan x@$
 @$y=3+\tan 3x@$
 @$y=1+\frac{2}{3}\tan \frac{1}{2}x@$
 Find the zeros of the function from #1.
 Find the zeros of the function from #3.
 Find the zeros of the function from #5.
Write the equation of the tangent function, in the form @$y=a\tan bx@$, with the given amplitude and period.
 Amplitude: 3 Period: @$\frac{3\pi}{2}@$
 Amplitude: @$\frac{1}{4}@$ Period: @$2\pi@$
 Amplitude: 2.5 Period: 8
 Challenge Graph @$y=2\tan \frac{1}{3}\left(x+\frac{\pi}{4}\right)1@$ over @$[0, 6\pi]@$. Determine the domain and period.
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Apr 23, 2013Last Modified:
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