the graph of a sinusoidal function has a maximum point at (0, 10) and then intersects its midline at (pi/4…

Explanation:

Step1: Determine the amplitude $A$

The amplitude is the distance from the mid - line to the maximum or minimum. The maximum is at $y = 10$ and the mid - line value can be found from the mid - point between the maximum and a point on the mid - line. Let the mid - line be $y = k$. Since the maximum is at $(0,10)$ and a point on the mid - line is $(\frac{\pi}{4},4)$, the mid - line $k=\frac{10 + 4}{2}=7$. The amplitude $A=10 - 7 = 3$.

Step2: Determine the vertical shift $k$

As calculated above, the vertical shift $k = 7$.

Step3: Determine the period $T$

The function goes from a maximum at $x = 0$ to the mid - line at $x=\frac{\pi}{4}$. A quarter of the period $T$ is $\frac{\pi}{4}-0=\frac{\pi}{4}$, so $T=\pi$. Using the formula $T=\frac{2\pi}{\omega}$, we get $\omega = 2$.

Step4: Determine the phase shift $\varphi$

The general form of a sinusoidal function is $y = A\sin(\omega(x-\varphi))+k$. Since the maximum occurs at $x = 0$, for a sine function $y = A\sin(\omega(x-\varphi))+k$, when $x = 0$, $\omega(0-\varphi)=\frac{\pi}{2}+2n\pi,n\in\mathbb{Z}$. Substituting $\omega = 2$, we have $- 2\varphi=\frac{\pi}{2}+2n\pi$. Taking $n = 0$, we get $\varphi=-\frac{\pi}{4}$.

Step5: Write the function formula

Substituting $A = 3$, $\omega = 2$, $\varphi=-\frac{\pi}{4}$, and $k = 7$ into $y = A\sin(\omega(x-\varphi))+k$, we get $f(x)=3\sin(2(x+\frac{\pi}{4}))+7$.

Answer:

$f(x)=3\sin(2(x + \frac{\pi}{4}))+7$