Pre-Calculus For Dummies, 2nd Edition (2012)
Part II. The Essentials of Trigonometry
Chapter 7. Simplifying the Graphing and Transformation of Trig Functions
In This Chapter
Plotting and transforming the sine and cosine parent graphs
Picturing and changing tangent and cotangent
Charting and altering secant and cosecant
“Graph the trig function. . . .” This command sends shivers down the spines of many pre-calc students. But we’re here to say that you have nothing to fear, because graphing functions can be easy. Graphing functions is simply a matter of inserting the x value (the domain) in place of the function’s variable and solving the equation to get the y value (the range). You continue with that calculation until you have enough points to plot. When do you know you have enough? When your graph has a clear line, ray, curve, or what have you.
You’ve dealt with functions before in math, but up until now, the input of a function was typically x. In trig functions, however, the input of the function is typically θ, which is basically just another variable to use. This chapter shows you how to graph trig functions by using various values for θ. We start with the parent graphs — the foundation on which everything else for graphing is built. From there, you can stretch a trig function graph, move it around on the coordinate plane, or flip and shrink it, which we also cover in this chapter.
In Chapter 6, you use two ways of measuring angles: degrees and radians. Lucky for you, you now get to focus solely on radians when graphing trig functions. Mathematicians have always graphed in radians when working with trig functions, and we want to continue that tradition — until someone comes up with a better way, of course.
Drafting the Sine and Cosine Parent Graphs
The trig functions, especially sine and cosine, displayed their usefulness in the last chapter. After putting them under the microscope, you’re now ready to begin graphing them. Just like with the parent functions in Chapter 3, after you discover the basic shape of the sine and cosine graphs, you can begin to graph more complicated versions, using the same transformations you discover in Chapter 3:
Vertical and horizontal transformations
Vertical and horizontal translations
Vertical and horizontal reflections
Knowing how to graph trig functions allows you to measure the movement of objects that move back and forth or up and down in a regular interval, such as pendulums. Sine and cosine as functions are perfect ways of expressing this type of movement, because their graphs are repetitive and they oscillate (like a wave). The following sections illustrate these facts.
Sine graphs move in waves. The waves crest and fall over and over again forever, because you can keep plugging in values for θ for the rest of your life. This section shows you how to construct the parent graph for the sine function, f(θ) = sin θ (for more on parent graphs, refer to Chapter 3).
Because all the values of the sine function come from the unit circle, you should be pretty comfy and cozy with the unit circle before proceeding with this work. If you’re not, we advise that you return to Chapter 6 for a brush-up.
You can graph any trig function in four or five steps. Here are the steps to construct the graph of the parent function f(θ) = sin θ:
1. Find the values for domain and range.
No matter what you put into the sine function, you get an answer as output, because θ can rotate around the unit circle in either direction an infinite amount of times. Therefore, the domain of sine is all real numbers, or (–∞, ∞).
On the unit circle, the y values are your sine values — what you get after plugging the value of θ into the sine function. Because the radius of the unit circle is 1, the y values can’t be more than 1 or less than negative 1 — your range for the sine function. So in the x-direction, the wave (or sinusoid, in math language) goes on forever, and in the y-direction, the sinusoid oscillates only between –1 and 1, including these values. In interval notation, you write this as [–1, 1].
2. Calculate the graph’s x-intercepts.
When you graphed lines in algebra, the x-intercepts occurred when y = 0. In this case, sine is the y value. Find out where the graph crosses the x-axis by finding unit circle values where sine is 0. The graph crosses the x-axis three times: once at 0, once at π, and once at 2π. You now know that three of the coordinate points are (0, 0), (π, 0), and (2π, 0).
3. Calculate the graph’s maximum and minimum points.
To complete this step, use your knowledge of the range from Step 1. You know that the highest value of y is 1. Where does this happen? At π/2. You now have another coordinate point at (π/2, 1). You also can see that the lowest value of y is –1, when x is 3π/2. Hence, you have another coordinate point: (3π/2, –1).
4. Sketch the graph of the function.
Using the five key points as a guide, connect the points with a smooth, round curve. Figure 7-1 approximately shows the parent graph of sine.
Figure 7-1: The parent graph of sine, f(θ) = sin θ.
The parent graph of the sine function has a couple of important characteristics worth noting:
It repeats itself every 2π radians. This repetition occurs because 2π radians is one trip around the unit circle — called the period of the sine graph — and after that, you start to go around again. Usually, you’re asked to draw the graph to show one period of the function, because in this period you capture all possible values for sine before it starts repeating over and over again. The graph of sine is called periodic because of this repeating pattern.
It’s symmetrical about the origin (thus, in math speak, it’s an odd function). The sine function has 180°-point symmetry about the origin. If you look at it upside down, the graph looks exactly the same. The official math definition of an odd function, though, is f(–x) = –f(x) for every value of x in the domain. In other words, if you put in an opposite input, you’ll get an opposite output. For example, sin(π/6) is 1/2, but if you look at sin(–π/6), you get –1/2.
Looking at cosine
The parent graph of cosine looks very similar to the sine function parent graph, but it has its own sparkling personality (like fraternal twins, we suppose). Cosine graphs follow the same basic pattern and have the same basic shape as sine graphs; the difference lies in the location of the maximums and minimums. These extremes occur at different domains, or x values, 1/4 of a period away from each other. Thus, the two graphs are shifts of 1/4 of the period from each other.
Just as with the sine graph, you use the five key points of graphing trig functions to get the parent graph of the cosine function. If necessary, you can refer to the unit circle for the cosine values to start with (see Chapter 6). As you work more with these functions, your dependence on the unit circle should decrease until eventually you don’t need it at all. Here are the steps:
1. Find the values for domain and range.
Like with sine graphs (see the previous section), the domain of cosine is all real numbers, and its range is –1 ≤ y ≤ 1, or [–1, 1].
2. Calculate the graph’s x-intercepts.
Referring to the unit circle, find where the graph crosses the x-axis by finding unit circle values of 0. It crosses the x-axis twice — once at π/2 and once at 3π/2. Those crossings give you two coordinate points: (π/2, 0) and (3π/2, 0).
3. Calculate the graph’s maximum and minimum points.
Using your knowledge of the range for cosine from Step 1, you know the highest value that y can be is 1, which happens twice for cosine — once at 0 and once at 2π (see Figure 7-2), giving you two maximums: (0, 1) and (2π, 1). The minimum value that y can be is –1, which occurs at π. You now have another coordinate pair at (π, –1).
4. Sketch the graph of the function.
Figure 7-2 shows the full parent graph of cosine with the five key points plotted.
Figure 7-2: The parent graph of cosine, f(θ) = cos θ.
The cosine parent graph has a couple of characteristics worth noting:
It repeats every 2π radians. This repetition means it’s a periodic function, so its waves rise and fall in the graph (see the previous section for the full explanation).
It’s symmetrical about the y-axis (in mathematical dialect, it’s an even function). Unlike the sine function, which has 180° symmetry, cosine has y-axis symmetry. In other words, you can fold the graph in half at the y-axis and it matches exactly. The formal definition of an even function is f(x) = f(–x) — if you plug in the opposite input, you’ll get the same output. For example,
Even though the input sign changed, the output sign stayed the same, and it always does for any θ value and its opposite for cosine.
Graphing Tangent and Cotangent
The graphs for the tangent and cotangent functions are quite different from the sine and cosine graphs. The graphs of sine and cosine are very similar to one another in shape and size. However, when you divide one function by the other, the graph you create looks nothing like either of the graphs it came
from. (Tangent is defined as , and cotangent is .)
The graphs of tangent and cotangent can be tough for some students to grasp, but you can master them with practice. The hardest part of graphing tangent and cotangent comes from the fact that they both have asymptotes in their graphs (see Chapter 3), because they’re rational functions.
The tangent graph has an asymptote wherever the cosine is 0, and the cotangent graph has an asymptote wherever the sine is 0. Keeping these asymptotes separate from one another helps you draw your graphs.
The tangent and cotangent functions have parent graphs just like any other function. Using the graphs of these functions, you can make the same types of transformations that apply to the parent graphs of any function. The following sections plot the parent graphs of tangent and cotangent.
The easiest way to remember how to graph the tangent function is to remember
that (see Chapter 6 to review).
Because cos θ = 0 for various values of θ, some interesting things happen to tangent’s graph. When the denominator of a fraction is 0, the fraction is undefined. Therefore, the graph of tangent jumps over an asymptote, which is where the function is undefined, at each of these places.
Table 7-1 presents θ, sin θ, cos θ, and tan θ. It shows the roots (or zeros), the asymptotes (where the function is undefined), and the behavior of the graph in between certain key points on the unit circle.
To plot the parent graph of a tangent function, you start out by finding the vertical asymptotes. Those asymptotes give you some structure from which you can fill in the missing points.
1. Find the vertical asymptotes so you can find the domain.
In order to find the domain of the tangent function, you have to locate the vertical asymptotes. The first asymptote occurs when θ = π/2, and it repeats every π radians (see the unit circle in Chapter 6). (Note: The period of the tangent graph is π radians, which is different from that of sine and cosine.) Tangent, in other words, has asymptotes when θ = π/2 and 3π/2.
The easiest way to write this is
where n is an integer. You write “+ nπ” because the period of tangent is π radians, so if an asymptote is at π/2 and you add or subtract π, you automatically find the next asymptote.
2. Determine values for the range.
Recall that the tangent function can be defined as .
Both of these values can be decimal. The closer you get to the values where cos θ = 0, the smaller the number on the bottom of the fraction gets and the larger the value of the overall fraction gets — in either the positive or negative direction.
The range of tangent has no restrictions; you aren’t stuck between 1 and –1, like with sine and cosine. In fact, the ratios are any and all numbers. The range is (–∞, ∞).
3. Calculate the graph’s x-intercepts.
Tangent’s parent graph has roots (it crosses the x-axis) at 0, π, and 2π.
You can find these values by setting equal to 0 and then solving.
The x-intercepts for the parent graph of tangent are located wherever the sine value is 0.
4. Figure out what’s happening to the graph between the intercepts and the asymptotes.
The first quadrant of tangent is positive and points upward toward the asymptote at π/2, because all sine and cosine values are positive in the first quadrant. Quadrant II is negative because sine is positive and cosine is negative. Quadrant III is positive because both sine and cosine are negative, and quadrant IV is negative because sine is negative and cosine is positive.
Note: A tangent graph has no maximum or minimum points.
Figure 7-3 shows what the parent graph of tangent looks like when you put it all together.
Figure 7-3: The parent graph of tangent, f(θ) = tan θ.
The parent graphs of sine and cosine are very similar because the values are exactly the same; they just occur for different values of θ. Similarly, the parent graphs of tangent and cotangent are comparable because they both have asymptotes and x-intercepts. The only differences you can see are the values of θ where the asymptotes and x-intercepts occur. You can find the parent graph of the cotangent function,
by using the same techniques you use to find the tangent parent graph (see the previous section).
Table 7-2 shows θ, cos θ, sin θ, and cot θ so that you can see both the x-intercepts and the asymptotes in comparison. These points help you find the general shape of your graph so that you have a nice place to start.
To sketch the full parent graph of cotangent, follow these steps:
1. Find the vertical asymptotes so you can find the domain.
Because cotangent is the quotient of cosine divided by sine, and sin θ is sometimes 0, the graph of the cotangent function may have asymptotes, just like with tangent. However, these asymptotes occur whenever the sin θ = 0. The asymptotes of cot θ are at 0, π, and 2π.
The cotangent parent graph repeats every π units. Its domain is based on its vertical asymptotes: The first one comes at 0 and then repeats every π radians. The domain, in other words, is , where n is an integer.
2. Find the values for the range.
Similar to the tangent function, you can define cotangent as
Both of these values can be decimals. The range of cotangent also has no restrictions; the ratios are any and all numbers — (–∞, ∞). The closer you get to the values where sin θ = 0, the smaller the number on the bottom of our fraction is and the larger the value of our overall fraction — in either the positive or negative direction.
3. Determine the x-intercepts.
The roots (or zeros) of cotangent occur wherever the cosine value is 0: at π/2 and 3π/2.
4. Evaluate what happens to the graph between the x-intercepts and the asymptotes.
The positive and negative values in the four quadrants stay the same as in tangent, but the asymptotes change the graph. You can see the full parent graph for cotangent in Figure 7-4.
Figure 7-4: The parent graph of cotangent, cot θ.
Putting Secant and Cosecant in Pictures
As with tangent and cotangent, the graphs of secant and cosecant have asymptotes. They have asymptotes because
Both sine and cosine have values of 0, which causes the denominators to be 0 and the functions to have asymptotes. These considerations are important when plotting the parent graphs, which we do in the sections that follow.
Secant is defined as
You can graph it by using steps similar to those from the tangent and cotangent sections.
The cosine graph crosses the x-axis on the interval [0, 2π] at two places, so the secant graph has two asymptotes, which divide the period interval into three smaller sections. The parent secant graph doesn’t have any x-intercepts (finding them on any transformed graph is hard, so usually you won’t be asked to).
Follow these steps to picture the parent graph of secant:
1. Find the asymptotes of the secant graph.
Because secant is the reciprocal of cosine (see Chapter 6), any place on the cosine graph where the value is 0 creates an asymptote on the secant graph. (And any point with 0 in the denominator is undefined.) Finding these points first helps you define the rest of the graph.
The parent graph of cosine has values of 0 at π/2 and 3π/2. So the graph of secant has asymptotes at those same values. Figure 7-5 shows only the asymptotes.
Figure 7-5: The graph of cosine reveals the asymptotes of secant.
2. Calculate what happens to the graph at the first interval between the asymptotes.
The period of the parent cosine graph starts at 0 and ends at 2π. You need to figure out what the graph does in between the following points:
• Zero and the first asymptote at π/2
• The two asymptotes in the middle
• The second asymptote and the end of the graph at 2π
Start on the interval (0, π/2). The graph of cosine goes from 1, into fractions, and all the way down to 0. Secant takes the reciprocal of all these values and ends on this first interval at the asymptote. The graph gets bigger and bigger rather than smaller, because as the fractions in the cosine function get smaller, their reciprocals in the secant function get bigger.
3. Repeat Step 2 for the second interval (π/2, 3π/2).
If you refer to the cosine graph, you see that halfway between π/2 and 3π/2, from the low point, the line has nowhere to go except closer to 0 in both directions. So secant’s graph (the reciprocal) gets bigger in a negative direction.
4. Repeat Step 2 for the last interval (3π/2, 2π).
This interval is a mirror image of what happens in the first interval.
5. Find the domain and range of the graph.
Its asymptotes are at π/2 and repeat every π, so the domain of secant, where n is an integer, is
The graph exists only for numbers ≥ 1 or ≤ –1. Its range, therefore, is .
You can see the parent graph of secant in Figure 7-6.
Figure 7-6: The parent graph of secant, sec θ.
Checking out cosecant
Cosecant is almost exactly the same as secant because it’s the reciprocal of sine (as opposed to cosine). Anywhere sine has a value of 0, you see an asymptote in the cosecant graph. Because the sine graph crosses the x-axis three times on the interval [0, 2π], you have three asymptotes and two sub-intervals to graph.
The reciprocal of 0 is undefined, and the reciprocal of an undefined value is 0. Because the graph of sine is never undefined, the reciprocal of sine can never be 0. For this reason, the parent graph of the cosecant function has no x-intercepts, so don’t bother looking for them.
The following list explains how to graph cosecant:
1. Find the asymptotes of the graph.
Because cosecant is the reciprocal of sine, any place on sine’s graph where the value is 0 creates an asymptote on cosecant’s graph. The parent graph of sine has values of 0 at 0, π, and 2π. So cosecant has three asymptotes. Figure 7-7 shows these asymptotes.
Figure 7-7: The graph of sine reveals the asymptotes of cosecant.
2. Calculate what happens to the graph at the first interval between 0 and π.
The period of the parent sine graph starts at 0 and ends at 2π. You can figure out what the graph does in between the first asymptote at 0 and the second asymptote at π.
The graph of sine goes from 0 to 1 and then back down again. Cosecant takes the reciprocal of these values, which causes the graph to get bigger.
3. Repeat for the second interval (π, 2π).
If you refer to the sine graph, you see that it goes from 0 down to –1 and then back up again. Because cosecant is the reciprocal, its graph gets bigger in the negative direction.
4. Find the domain and range of the graph.
Cosecant’s asymptotes start at 0 and repeat every π. Its domain is , where n is an integer. The graph also exists for numbers ≥ 1 or ≤ –1. Its range, therefore, is .
You can see the full graph in Figure 7-8.
Figure 7-8: The parent graph of cosecant, csc θ.
Transforming Trig Graphs
The basic parent graphs open the door to many advanced and complicated graphs, which ultimately have more real-world applications. Usually, functions that model real-world situations are stretched, shrunk, or even shifted to an entirely different location on the coordinate plane. The good news is that the transformation of trig functions follows the same set of general guidelines as the transformations you see in Chapter 3.
The rules for graphing complicated trig functions are actually pretty simple. When asked to graph a more-complicated trig function, you can take the parent graph (which you know from the previous sections) and alter it in some way to find the more complex graph. Basically, you can change each parent graph of a trig function in four ways:
Transforming vertically (When dealing with the graph for sine and cosine functions, a vertical transformation changes the graph’s height, also known as the amplitude.)
Transforming horizontally (This transformation makes it move faster or slower, which affects its horizontal length.)
Shifting up, down, left, or right
Reflecting across the x- or y-axis
The following sections cover how to transform the parent trig graphs. However, before you move on to transforming these graphs, make sure you’re comfortable with the parent graphs from the previous sections. Otherwise, you may get confused on nitpicky things.
Screwing with sine and cosine graphs
The sine and cosine graphs look similar to a spring. If you pull the ends of this spring, all the points are farther apart from one another; in other words, the spring is stretched. If you push the ends of the spring together, all the points are closer together; in other words, the spring is shrunk. So the graphs of sine and cosine look and act a lot like a spring, but these springs can be changed both horizontally and vertically; aside from pulling the ends or pushing them together, you can make the spring taller or shorter. Now that’s some spring!
In this section we show you how to alter the parent graphs for sine and cosine using both vertical and horizontal stretches and shrinks. You also see how to move the graph around the coordinate plane using translations (which can be both vertical and horizontal).
Changing the amplitude
Multiplying a trig function by a constant changes the graph of the parent function; specifically, you change the amplitude of the graph. When measuring the height of a graph, you measure the distance between the maximum crest and the minimum wave. Smack dab in the middle of that measurement is a horizontal line called the sinusoidal axis. Amplitude is the measure of the distance from the sinusoidal axis to the maximum or the minimum. Figure 7-9 illustrates this point further.
Figure 7-9: The sinusoidal axis and amplitude of a trig function graph.
By multiplying a trig function by certain values, you can make the graph taller or shorter:
Positive values of amplitudes greater than 1 make the height of the graph taller. Basically, 2 sin θ makes the graph taller than sin θ; 5 sin θ makes it even taller, and so on. For example, if g(θ) = 2 sin θ, you multiply the height of the original sine graph by 2 at every point. Every place on the graph, therefore, is twice as tall as the original.
If you remember that amplifying a sound makes it louder, you may have an easier time remembering that greater amplitudes increase the height.
Fraction values between 0 and 1 make the graph shorter. You can say
that is shorter than sin θ, and is even shorter. For example,
if , you multiply the parent graph’s height by 1/5 at each
point, making it that much shorter.
The change of amplitude affects the range of the function as well, because the maximum and minimum values of the graph change. Before you multiply a sine or cosine function by 2, for instance, its graph oscillated between –1 and 1; now it moves between –2 and 2.
Sometimes you multiply a trigonometric function by a negative number. That negative number doesn’t make the amplitude negative, however! Amplitude is a measure of distance, and distance can’t be negative. You can’t walk –5 feet, for instance, no matter how hard you try. Even if you walk backward, you still walk 5 feet. Similarly, if k(θ) = –5 sin θ, its amplitude is still 5. The negative sign just flips the graph upside down.
Table 7-3 shows a comparison of an original input (θ) and the value of sin θ
with g(θ) = 2 sin θ, , and k(θ) = –5 sin θ.
Don’t worry, you won’t have to re-create Table 7-3 for any pre-calc reasons. We just want you to see the comparison between the parent function and the more complicated functions. Keep in mind that this table displays only values of the sine function and transformations of it; you can easily do the same thing for cosine.
Figure 7-10 illustrates what the graphs of sine look like after the transformations. Figure 7-10a is the graph of f(θ); Figure 7-10b is g(θ); Figure 7-11a is h(θ); and Figure 7-11b is k(θ).
Figure 7-10:The graphs of example of transformations of sine.
Figure 7-11:More examples of transformations of sine.
Altering the period
The period of the parent graphs of sine and cosine is 2π, which is once around the unit circle (see the earlier section “Sketching sine”). Sometimes in trig, the variable θ, not the function, gets multiplied by a constant. This action affects the period of the trig function graph. For example, f(x) = sin 2x makes the graph repeat itself twice in the same amount of time; in other words, the graph moves twice as fast. Think of it like fast-forwarding a DVD. Figure 7-12 shows function graphs with various period changes.
To find the period of f(x) = sin 2x, set 2 · period = 2π (the period of the original sine function) and solve for the period. In this case, the period = π, so the graph finishes its trip at π. Each point along the x-axis also moves at twice the speed.
Figure 7-12:Creating period changes on function graphs.
You can make the graph of a trig function move faster or slower with different constants:
Positive values of period greater than 1 make the graph repeat itself more and more frequently. You see this rule in the example of f(x).
Fraction values between 0 and 1 make the graph repeat itself less
frequently. For example, if , you can find its period by
setting 1/4 · period = 2π. Solving for period gets you 8π. Before, the graph finished at 2π; now it waits to finish at 8π, which slows it down by 1/4.
You can have a negative constant multiplying the period. A negative constant affects how fast the graph moves, but in the opposite direction of the positive constant. For example, say p(x) = sin(3x) and q(x) = sin(–3x). The period of p(x)
is , whereas the period of q(x) is . The graph of p(x) moves to the right
of the y-axis, and the graph of q(x) moves to the left. Figure 7-13 illustrates this point clearly.
Figure 7-13:Graphs with negative periods move to the opposite side of the y-axis.
Don’t confuse amplitude and period when graphing trig functions. For example, f(x) = 2 sin x and g(x) = sin 2x affect the graph differently: f(x) = 2 sin x makes it taller, and g(x) = sin 2x makes it move faster.
Shifting the waves on the coordinate plane
The movement of a parent graph around the coordinate plane is another type of transformation known as a translation or a shift. For this type of transformation, every point on the parent graph is moved somewhere else on the coordinate plane. A translation doesn’t affect the overall shape of the graph; it only changes its location on the plane. In this section, we show you how to take the parent graphs of sine and cosine and shift them both horizontally and vertically.
Did you pick up the rules for shifting a function horizontally and vertically from Chapter 3? If not, go back and check them out, because they’re important for sine and cosine graphs as well.
Most math books write the horizontal and vertical shifts as sin(x – h) + v, or cos(x – h) + v. The variable h represents the horizontal shift of the graph, and v represents the vertical shift of the graph. The sign makes a difference in the direction of the movement. For example,
f(x) = sin(x – 3) moves the parent graph of sine to the right by 3.
g(x) = cos(x + 2) moves the parent graph of cosine to the left by 2.
k(x) = sin x + 4 moves the parent graph of sine up 4.
p(x) = cos x – 4 moves the parent graph of cosine down 4.
For example, if you need to graph y = sin(θ – π/4) + 3, follow these steps:
1. Identify the parent graph.
You’re looking at sine, so draw its parent graph (see the earlier section “Sketching sine”). The starting value for the parent graph of sin θ is at x = 0.
2. Shift the graph horizontally.
To find the new starting place, set what’s inside the parentheses equal to the starting value of the parent graph: θ – π/4 = 0, so θ = π/4 is where this graph starts its period. You move every point on the parent graph to the right by π/4. Figure 7-14 shows what you have so far.
Figure 7-14:Shifting the parent graph of sine to the right by π/4.
3. Move the graph vertically.
The sinusoidal axis of the graph moves up three positions in this function, so shift all the points of the parent graph this direction now. You can see this shift in Figure 7-15.
Figure 7-15:Moving the graph of sine up by three.
4. State the domain and range of the transformed graph, if asked.
The domain and range of a function may be affected by a transformation. When this happens, you may be asked to state the new domain and range. Usually, you can visualize the range of the function easily by looking at the graph. Two factors that change the range are a vertical transformation (stretch or shrink) and a vertical translation.
Keep in mind that the range of the parent sine graph is [–1, 1]. Shifting the parent graph up three units makes the range of y = sin(θ – π/4) + 3 shift up three units also. Therefore, the new range is [2, 4]. The domain of this function isn’t affected; it’s still (–∞, ∞).
Combining transformations in one fell swoop
When you’re asked to graph a trig function with multiple transformations, we suggest that you do them in this order:
1. Change the amplitude.
2. Change the period.
3. Shift the graph horizontally.
4. Shift the graph vertically.
The equations that combine all the transformations into one are as follows:
f(x) = a · sin[p(x – h)] + v
f(x) = a · cos[p(x – h)] + v
The absolute value of the variable a is the amplitude. You take 2π and divide by p to find the period. The variable h is the horizontal shift, and v is the vertical shift.
The most important thing to know is that sometimes a problem is written so that it looks like the period and the horizontal shift are both inside the trig function. For example, f(x) = sin(2x – π) makes it look like the period is twice as fast and the horizontal shift is π, but that isn’t correct. All period shifts must be factored out of the expression to actually be period shifts, which in turn reveals the true horizontal shifts. You need to rewrite f(x) as sin 2(x – π/2). This function tells you that the period is twice as fast but that the horizontal shift is actually π/2 to the right.
Because this concept is so important, we present another example just so you
grasp what we mean. With the following steps, graph :
1. Write the equation in its proper form by factoring out the period constant.
This step gives you .
2. Graph the parent graph.
Graph the original cosine function as you know it (see the earlier section “Looking at cosine”).
3. Change the amplitude.
This graph has an amplitude of 3, but the negative sign turns it upside down, which affects the graph’s range. The range is now [–3, 3]. You can see the amplitude change in Figure 7-16.
Figure 7-16:Changing the amplitude of a function with multiple alterations.
4. Alter the period.
The constant 1/2 affects the period. Solving the equation 1/2 · period = 2π gives you the period of 4π. The graph moves half as fast and finishes at 4π, which you can see in Figure 7-17.
Figure 7-17:Changing the period to 4π.
5. Shift the graph horizontally.
When you factored out the period constant in Step 1, you discovered that the horizontal shift is to the left π/2. This shift is shown in Figure 7-18.
Figure 7-18: A horizontal shift to the left.
6. Shift the graph vertically.
Because of the – 2 you see in Step 1, this graph moves down two positions, which you can see in Figure 7-19.
Figure 7-19:Changing the period to 4π.
7. State the new domain and range.
The functions of sine and cosine are defined for all values of θ. The domain for the cosine function is all real numbers, or (–∞, ∞). The range of the graph in Figure 7-18 has been stretched because of the amplitude change and shifted down.
To find the range of a function that has been shifted vertically, you add or subtract the vertical shift (–2) from the altered range based on the amplitude. For this problem, the range of the transformed cosine function is [–3 – 2, 3 – 2], or [–5, 1].
Tweaking tangent and cotangent graphs
The transformations for sine and cosine work for tangent and cotangent, too (see the earlier transformations section in this chapter for more). Specifically, you can transform the graph vertically, change the period, shift the graph horizontally, or shift it vertically. As always, though, you should take each transformation one step at a time.
For example, to graph , follow these steps:
1. Sketch the parent graph for tangent (see the “Graphing Tangent and Cotangent” section).
2. Shrink or stretch the parent graph.
The vertical shrink is 1/2 for every point on this function, so each point on the tangent parent graph is half as tall.
Seeing vertical changes for tangent and cotangent graphs is harder, but they’re there. Concentrate on the fact that the parent graph has points (π/4, 1) and (–π/4, –1), which in the transformed function become (π/4, 1/2) and (–π/4, –1/2). As you can see in Figure 7-20, the graph really is half as tall!
Figure 7-20:The amplitude is half as tall in the changed graph.
3. Change the period.
The constant 1/2 doesn’t affect the period. Why? Because it sits in front of the tangent function, which only affects vertical, not horizontal, movement.
4. Shift the graph horizontally and vertically.
This graph doesn’t shift horizontally, because no constant is added inside the grouping symbols (parentheses) of the function. So you don’t need to do anything horizontally. The – 1 at the end of the function is a vertical shift that moves the graph down one position. Figure 7-21 shows the shift.
Figure 7-21:The transformed graph of
5. State the transformed function’s domain and range, if asked.
Because the range of the tangent function is all real numbers, transforming its graph doesn’t affect the range, only the domain. The domain of the tangent function isn’t all real numbers because of the asymptotes. The domain of the example function hasn’t been affected by the transformations, however. Where n is an integer,
Now that you’ve graphed the basics, you can graph a function that has a period change, as in the function y = cot(2πx + π/2). You see a lot of π in that one. Relax! You know this graph has a period change because you see a number inside the parentheses that’s multiplied by the variable. This constant changes the period of the function, which in turn changes the distance between the asymptotes. In order for the graph to show this change correctly, you must factor this constant out of the parentheses. Take the transformation one step at a time:
1. Sketch the parent graph for cotangent.
See the information in the “Clarifying cotangent” section to determine how to get the graph of cotangent.
2. Shrink or stretch the parent graph.
No constant is multiplying the outside of the function; therefore, you can apply no amplitude change.
3. Find the period change.
You factor out the 2π, which affects the period. The function now reads y = cot[2π(x + 1/4)].
The period of the parent function for cotangent is π. Therefore, instead of dividing 2π by the period constant to find the period change (like you did for the sine and cosine graphs), you must divide π by the period constant. This step gives you the period for the transformed cotangent function.
Set 2π · period = π and solve for the period; you get a period of 1/2 for the transformed function. The graph of this function starts to repeat at 1/2, which is different from π/2, so be careful when you’re labeling your graph.
Up until now, every trig function you’ve graphed has been a fraction of π (such as π/2), but this period isn’t a fraction of π; it’s just a rational number. When you get a rational number, you must graph it as such. Figure 7-22 shows this step.
Figure 7-22:Graphing of y(x) = cot 2πx shows a period of 1/2.
4. Determine the horizontal and vertical shifts.
Because you’ve already factored the period constant, you can see that the horizontal shift is to the left 1/4. Figure 7-23 shows this transformation on the graph.
No constant is being added to or subtracted from this function on the outside, so the graph doesn’t experience a vertical shift.
Figure 7-23:The transformed graph of y(x) = cot 2π(x + 1/4).
5. State the transformed function’s domain and range, if asked.
The horizontal shift affects the domain of this graph. To find the first asymptote, set 2πx + π/2 = 0 (setting the period shift equal to the original first asymptote). You find that x = –1/4 is your new asymptote. The graph repeats every 1/2 radians because of its period. So the domain is
, where n is an integer. The graph’s range isn’t affected: (–∞, ∞).
Transforming the graphs of secant and cosecant
To graph transformed secant and cosecant graphs, your best bet is to graph their reciprocal functions and transform them first. The reciprocal functions, sine and cosine, are easier to graph because they don’t have as many complex parts (no asymptotes, basically). If you can graph the reciprocals first, you can deal with the more complicated pieces of the secant/cosecant graphs last.
For example, take a look at the graph .
1. Graph the transformed reciprocal function.
Look at the reciprocal function for secant, which is cosine. Pretend just
for a bit that you’re graphing . Follow all the rules for
the cosine graph in order to end up with a graph that looks like the one in Figure 7-24.
2. Sketch the asymptotes of the transformed reciprocal function.
Wherever the transformed graph of cos θ crosses its sinusoidal axis, you have an asymptote in sec θ. You see that cos θ = 0 when θ = π/2 and 3π/2.
Figure 7-24:Graphing the cosine function first.
3. Find out what the graph looks like between each asymptote.
Now that you’ve identified the asymptotes, you simply figure out what happens on the intervals between them, like you do in Steps 2 through 4 in the earlier section “Graphing secant.” The finished graph ends up looking like the one in Figure 7-25.
Figure 7-25:The transformed secant graph
4. State the domain and range of the transformed function.
Because the new transformed function may have different asymptotes than the parent function for secant and it may be shifted up or down, you may be required to state the new domain and range.
This example, , has asymptotes at π/2, 3π/2, and so on,
repeating every π radians. Therefore, the domain is restricted to not
include these values and is written where n is an integer. In
addition, the range of this function changes because the transformed function is shorter than the parent function and has been shifted down 2. The range has two separate intervals, (–∞, –5/4] and [–3/4, ∞).
You can graph a transformation of the cosecant graph by using the same steps you use when graphing the secant function, only this time you use the sine function to guide you.
The shape of the transformed cosecant graph should be very similar to the secant graph, except the asymptotes are in different places. For this reason, be sure you’re graphing with the help of the sine graph (to transform the cosecant graph) and the cosine function (to guide you for the secant graph).
For the last example of this chapter, graph the transformed cosecant graph g(θ) = csc(2x – π) + 1:
1. Graph the transformed reciprocal function.
Look first at the function g(θ) = sin(2x – π) + 1. The rules to transforming a sine function tell you to first factor out the 2 and get g(θ) = sin 2(x – π/2) + 1. It has a horizontal shrink of 2, a horizontal shift of π/2 to the right, and a vertical shift of up 1. Figure 7-26 shows the transformed sine graph.
Figure 7-26: A transformed sine graph.
2. Sketch the asymptotes of the reciprocal function.
The sinusoidal axis that runs through the middle of the sine function is the line y = 1. Therefore, an asymptote of the cosecant graph exists everywhere the transformed sine function crosses this line. The asymptotes of the cosecant graph are at π/2 and π and repeat every π/2 radians.
3. Figure out what happens to the graph between each asymptote.
You can use the transformed graph of the sine function to determine where the cosecant graph is positive and negative. Because the graph of the transformed sine function is positive in between π/2 and π, the cosecant graph is positive as well and extends up when getting closer to the asymptotes. Similarly, because the graph of the transformed sine function is negative in between π and 3π/2, the cosecant is also negative in this interval. The graph alternates between positive and negative in equal intervals for as long as you want to sketch the graph.
Figure 7-27 shows the transformed cosecant graph.
Figure 7-27:The transformed cosecant graph, based on the sine graph.
4. State the new domain and range.
Just as with the transformed graph of the secant function (see the previous list), you may be asked to state the new domain and range for the cosecant function. The domain of the transformed cosecant function is all values of θ except for the values that are asymptotes. From the graph,
you can see that the domain is all values of θ, where , where
n is an integer. The range of the transformed cosecant function is also split up into two intervals: .