SAT Math - Graphing Trigonometric Functions

Which of the given functions has the greatest amplitude?

$2\sin(x)$

$\sin(4x)$

$\sin(x+3)$

$\sin\frac{2\pi }{3}(x+2)$

$\frac{1}{2}\sin(4x)$

Explanation

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.

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

$f(x)= \frac{11}{5} \sin \frac{2 \pi x}{7}$

$f(x)= \frac{11}{5} \sin \frac{ \pi x}{7}$

$f(x)= \frac{11}{5} \sin \frac{2 x}{7}$

$f(x)= \frac{11}{5} \sin \frac{7 \pi x}{2}$

$f(x)= \frac{11}{5} \sin \frac{7 x}{2}$

Explanation

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 $f(x)= \frac{11}{5} \sin \frac{2 \pi x}{7}$ .

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

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

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

$f(x)= \frac{2}{7} \sin x$

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

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

Explanation

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