Graphing Trigonometric Functions - SAT Subject Test in Math II

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Question

Which of the given functions has the greatest amplitude?

Answer

The amplitude of a function is the amount by which the graph of the function travels above and below its midline. When graphing a sine function, the value of the amplitude is equivalent to the value of the coefficient of the sine. Similarly, the coefficient associated with the x-value is related to the function's period. The largest coefficient associated with the sine in the provided functions is 2; therefore the correct answer is $2\sin(x)$ .

The amplitude is dictated by the coefficient of the trigonometric function. In this case, all of the other functions have a coefficient of one or one-half.

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Question

Which of these functions has a graph with amplitude 4?

Answer

The functions in each of the choices take the form of a cosine function

$f(x)= A \cos Nx$ .

The graph of a cosine function in this form has amplitude . Therefore, for this function to have amplitude 4, . Of the five choices, only

$f(x) = 4 \cos \frac{\pi x}{5}$

matches this description.

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Question

Which of the following sine functions has a graph with period of 7?

Answer

The period of the graph of a sine function $f(x)= A \sin B x$ , is $\frac{2 \pi}{B}$ , or $2 \pi \div B$ .

Therefore, we solve for :

$\frac{2 \pi}{B} = 7$

$\frac{B}{2 \pi} = \frac{1}{7}$

$B= \frac{1}{7} \cdot 2 \pi = \frac{2 \pi}{7}$

The correct choice is therefore $f(x)= \frac{11}{5} \sin \frac{2 \pi x}{7}$ .

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Question

Which of these functions has a graph with amplitude $\frac{4}{7}$ ?

Answer

The functions in each of the choices take the form of a sine function

$f(x)= A \sin Bx$ .

The graph of a sine function in this form has amplitude . Therefore, for this function to have amplitude 4, $A = \frac{4}{7}$ . Of the five choices, only

$f(x)= \frac{4}{7} \sin x$

matches this description.

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Question

Give the amplitude of the graph of the function

$f (x) = 2 \pi \sin 2 \pi x$

Answer

The amplitude of the graph of a sine function $f(x) = A \sin Bx$ is . Here, $A= 2 \pi$ , so this is the amplitude.

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