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Exercises Trigonometry

Example 3 from Precalculus, 7.6 Modeling with Trigonometric Functions

Graphing Sinusoidal Functions Using Key Points

Graph the function $y = −4 \cos (π x)$ using amplitude, period, and key points.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The amplitude is $| - 4 | = 4.$ The period is $\frac{2 π}{ω} = \frac{2 π}{π} = 2.$ (Recall that we sometimes refer to $B$ as $ω .)$ One cycle of the graph can be drawn over the interval $[0, 2] .$ To find the key points, we divide the period by 4. Make a table similar to Table 2, starting with $x = 0$ and then adding $\frac{1}{2}$ successively to $x$ and calculate $−4$ See the graph in Figure 7.

$x$ $−4$ $\frac{1}{2}$ $1$ $\frac{3}{2}$ $2$

$y = −4 \cos (π x)$ $−4$ $0$ $4$ $0$ $−4$

Table 2

Figure 7

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Graph the function $y = 3 \sin (3 x)$ using the amplitude, period, and five key points.

How did it go?

PreviousFinding the Amplitude and Period of a Function Next Modeling an Equation and Sketching a Sinusoidal Graph to Fit Criteria

From Precalculus (1st edition) by OpenStax (Rice University), section 7.6, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.