Write an equation for a sine curve that has the given amplitude and period, and which passes through the given point.
-Amplitude 3 , period 8 , point $( 7,0 )$
A) $y = 3 \sin \left( 4 x - \frac { 4 } { 7 } \right)$
B) $y = 3 \sin \left( 8 x - \frac { \pi } { 7 } \right)$
C) $y = 3 \sin \left( \frac { \pi } { 4 } x - \frac { 7 } { 4 } \pi \right)$
D) $y = 3 \sin \left( \frac { \pi } { 4 } x - \frac { \pi } { 7 } \right)$

Question

Quizplus · Accepted Answer

The general form of a sine curve is $y = A \sin(B(x - C)) + D$, where $A$ is the amplitude, $T = \frac{2\pi}{B}$ is the period, $C$ is the phase shift, and $D$ is the vertical shift. Given an amplitude of 3 and a period of 8, we find $A = 3$ and $B = \frac{2\pi}{8} = \frac{\pi}{4}$. To ensure the curve passes through $(7,0)$, we substitute $x = 7$ and $y = 0$ into the equation and solve for $C$. The correct equation that satisfies these conditions is $y = 3 \sin \left( \frac { \pi } { 4 } x - \frac { 7 } { 4 } \pi \right)$, which corresponds to choice C.

Write an Equation for a Sine Curve That Has the Given