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 where is the central angle. To graph, rewrite the coordinates as where is the central angle, in radians. Below we expanded the sine coordinates for .
Notice that the curve ranges from 1 to -1. The maximum value is 1, which is at . The minimum value is -1 at . This “height” of the sine function is called the amplitude. The amplitude is the absolute value of average between the highest and lowest points on the curve.
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 . After , the sine values repeat. Therefore, the curve above will repeat every units, making the period . The domain is all real numbers.
Similarly, when we expand the cosine curve, , from the unit circle, we have:
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 . If we draw the graph past , it would look like:
Comparing and (below), we see that the curves are almost identical, except that the sine curve starts at and the cosine curve starts at .
If we shift either curve units to the left or right, they will overlap. Any horizontal shift of a trigonometric function is called a phase shift. We will discuss phase shifts more in the upcoming concepts.
Example A
Identify the highlighted points on and below.
,
Solution: For each point, think about what the sine or cosine value is at those values. For point , , therefore the point is . For point , we have to work backwards because it is not exactly on a vertical line, but it is on a horizontal one. When is ? When or . By looking at point ’s location, we know it is the second option. Therefore, the point is .
For the cosine curve, point is the same as point because the sine and cosine for is the same. As for point , we use the same logic as we did for point . When does ? When or . Again, looking at the location of point , we know it is the second option. The point is .
More Guidance
In addition to graphing and , we can stretch the graphs by placing a number in front of the sine or cosine, such as or . is the amplitude of the curve. In the next concept, we will shift the curves up, down, to the left and right.
Example B
Graph over two periods.
Solution: Start with the basic sine curve. Recall that one period of the parent graph, , is . Therefore, two periods will be . The 3 indicates that the range will now be from 3 to -3 and the curve will be stretched so that the maximum is 3 and the minimum is -3. The red curve is .
Notice that the -intercepts are the same as the parent graph. Typically, when we graph a trigonometric function, we always show two full periods of the function to indicate that it does repeat.
Example C
Graph over two periods.
Solution: Now, the amplitude will be and the function will be “smooshed” rather than stretched.
Example D
Graph over two periods.
Solution: The last two examples dealt with changing and was positive. Now, is negative. Just like with other functions, when the leading coefficient is negative, the function is reflected over the -axis. is in red.
Guided Practice
1. Is the point on ? How do you know?
Graph the following functions for two full periods.
2.
3.
4.
Answers
1. Substitute in the point for and and see if the equation holds true.
This is true, so is on the graph.
2. Stretch the cosine curve so that the maximum is 6 and the minimum is -6.
3. The graph is reflected over the -axis and stretched so that the amplitude is 3.
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 plane such that , where is the central angle from the unit circle and is the sine, cosine, or tangent of that angle.
- Amplitude
- The height of a sine or cosine curve. In the equation, or , the amplitude is .
- Periodic
- When a function repeats its -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 or .
- List all the points in the interval where . Use the graph from #1 to help you.
- Draw from from . Find and . Plot these values on the curve.
For questions 4-12, graph the sine or cosine curve over two periods.
- Graph and 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 and 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 ?
Write the equation for each sine or cosine curve below. for both questions.
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 and , respectively. Also, just like in other functions, is the horizontal shift, also called a phase shift, and is the vertical shift. Notice, that because it is in the equation, will always shift in the opposite direction of what is in the equation.
Example A
Graph .
Solution: This function will be shifted units to the right. The easiest way to sketch the curve, is to start with the parent graph and then move it to the right the correct number of units.
Example B
Graph .
Solution: Because -2 is not written in terms of (like the -axis), we need to estimate where it would be on the axis. So, -2 will be shifted not quite to the tic mark. Then, the entire function will be shifted up 3 units. The red graph is the final answer.
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 . The horizontal shift is the hardest to find. Because sine curves are periodic, the horizontal shift can either be positive or negative.
Because is , we can say that “moves back almost units” is -3 units. So, the equation is . If we did the positive horizontal shift, we could say that the equation would be .
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 and the maximum to the left of it didn’t quite make it to (the distance between and is ). Then, to determine the positive shift equation, recall that a period is , which is So, the positive shift would be or .
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 units and down 2 units.
3. The parent graph is in green. It moves up 3 units and to the right units. Therefore, the equation is .
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 1-4, match the equation with its graph.
- Which graph above represents:
- Fill in the blanks below.
For questions 7-12, graph the following equations from .
- Critical Thinking Is there a difference between and ? 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 . Find the domain and range.
Solution: First, stretch the curve so that the amplitude is 4, making the maximums and minimums 4 and -4. Then, shift the curve units to the right.
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 . Find the domain and range.
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 , so that is our vertical shift. Lastly, we need to find the horizontal shift. The easiest way to do this is to superimpose the curve over this curve and determine the movement from one maximum to the closest maximum of this curve.
Subtracting and , we have:
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 . As a sine curve, the function is . As a cosine curve, there will be a shift of .
Problem Set
Determine if the following statements are true or false.
- To change a cosine curve into a sine curve, shift the curve 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 .
- The domain of any sine or cosine function is always all real numbers.
Graph the following sine or cosine functions such that . State the domain and range.
Use the graph below to answer questions 12-15.
- 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 16-20.
- 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 16-19.
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 stretch out the curve, then the period would have to be longer than . Below we have sine curves with a period of and then the second has a period of .
To determine the period from an equation, we introduce into the general equation. So, the equations are and , where is the amplitude, is the frequency, is the phase shift, and is the vertical shift. The frequency is the number of times the sine or cosine curve repeats within . Therefore, the frequency and the period are indirectly related. For the first sine curve, there is half of a sine curve in . Therefore the equation would be . The second sine curve has two curves within , making the equation . To find the period of any sine or cosine function, use , where is the frequency. Using the first graph above, this is a valid formula: .
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 . So, the period is .
b) The in the equation tells us the frequency. The period is .
c) The is the frequency. The period is .
Example B
Graph part a) from the previous example from . Determine where the maximum and minimum values occur. Then, state the domain and range.
Solution: The amplitude is -3, so it will be stretched and flipped. The period is (from above) and the curve should repeat itself 6 times from 0 to . The first maximum value is 3 and occurs at half the period, or and then repeats at Writing this as a formula we start at and add to get the next maximum, so each point would be where is any integer.
The minimums occur at -3 and the -values are multiples of . The points would be , again is any integer. The domain is all real numbers and the range is .
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 . In this case, we will have 6 times the number of zeros that the parent function. To solve this function, set and solve for .
Now, use the inverse cosine function to determine when the cosine is zero. This occurs at the multiples of .
We went much past because when we divide by 6, to get by itself, all of these answers are going to also be divided by 6 and smaller.
so we have found all the zeros in the range.
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 is zero.
3. The general equation of a sine curve is . We know that and that the period is . Let’s use this to find the frequency, or .
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 .
Problem Set
Find the period of the following sine and cosine functions.
Use the equation to answer the following questions.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
Use the equation to answer the following questions.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
Use the equation to answer the following questions.
- Graph the function from and find the domain and range.
- Determine the coordinates of the maximum and minimum values.
- Find all the zeros from .
- 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 , with the given amplitude and period.
- Amplitude: -2 Period:
- Amplitude: Period:
- Amplitude: 9 Period: 6
- Challenge Find all the zeros from of .
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 . In radians, the coordinate for the tangent function would be
After , the -values repeat, making the tangent function periodic with a period of .
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 and . If we were to extend the graph out in either direction, there would continue to be vertical asymptotes at the odd multiples of . Therefore, the domain is all real numbers, , where is an integer. The range would be all real numbers. Just like with sine and cosine functions, you can change the amplitude, phase shift, and vertical shift.
The standard form of the equation is where and are the same as they are for the other trigonometric functions. For simplicity, we will not address phase shifts in this concept.
Example A
Graph from . State the domain and range.
Solution: First, the amplitude is 3, which means each -value will be tripled. Then, we will shift the function up one unit.
Notice that the vertical asymptotes did not change. The period of this function is still . Therefore, if we were to change the period of a tangent function, we would use a different formula than what we used for sine and cosine. To change the period of a tangent function, use the formula .
The domain will be all real numbers, except where the asymptotes occur. Therefore, the domain of this function will be . The range is all real numbers.
Example B
Graph from and state the domain and range. Find all zeros within this domain.
Solution: The period of this tangent function will be and the curves will be reflected over the -axis. The domain is all real numbers, where is any integer. The range is all real numbers. To find the zeros, set .
Example C
Graph from and state the domain and range.
Solution: This function has a period of . The domain is all real numbers, except , where is any integer. The range is all real numbers.
Guided Practice
1. Find the period of the function .
2. Find the zeros of the function from #1, from .
3. Find the equation of the tangent function with an amplitude of 8 and a period of .
Answers
1. The period is .
2. The zeros are where is zero.
3. The general equation is . We know that . Let’s use the period to solve for the frequency, or .
The equation is .
Vocabulary
- Tangent Function
- Defined by the coordinates , where 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 . Determine the period, domain, and range.
- 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 , with the given amplitude and period.
- Amplitude: 3 Period:
- Amplitude: Period:
- Amplitude: -2.5 Period: 8
- Challenge Graph over . Determine the domain and period.