Question

g is a trigonometric function of the form g(x) = a sin(bx + c) + d. Below is the graph g(x). The function intersects its midline at 3/4 *pi, 3) and has a maximum point at (pi,7) Find a formula for g(x). Give an exact expression.​

Answer 1

Answer:

$g(x)=4sin(2x+\frac{\pi}{2})+3$

Step-by-step explanation:

Because the function intersects its midline at $(\frac{3\pi}{4},3)$, then the midline is $d=3$.

Additionally, the amplitude is just the positive distance between the maximum/minimum and the midline, so the amplitude is $a=7-3=4$.

Also, given that period is $\frac{2\pi}{b}$ and the fact that the period is $\pi-0=\pi$ from our given maximum, we have the equation $\frac{2\pi}{b}=\pi$ where $b=2$.

Lastly, to find $c$, we know that the phase shift, $-\frac{c}{b}$, is $-\frac{\pi}{4}$ (or $\frac{\pi}{4}$ to the left) since $\pi-\frac{3\pi}{4}=-\frac{\pi}{4}$. Therefore, we have the equation $-\frac{c}{2}=-\frac{\pi}{4}$ where $c=\frac{\pi}{2}$.

Putting it all together, our final equation is $g(x)=4sin(2x+\frac{\pi}{2})+3$