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

Explanation:

Step1: Determine the amplitude $A$

The amplitude is the distance from the mid - line to the maximum or minimum. Given a maximum at $(0,10)$ and mid - line passing through $(\frac{\pi}{4},4)$, so $A=10 - 4=6$.

Step2: Determine the vertical shift $D$

The mid - line value gives the vertical shift. The mid - line passes through $(\frac{\pi}{4},4)$, so $D = 4$.

Step3: Determine the phase shift $\varphi$

The function has a maximum at $x = 0$. For a cosine function $y=A\cos(B(x-\varphi))+D$, when $x = 0$ it reaches a maximum. The general form of a sinusoidal function is $y = A\cos(B(x-\varphi))+D$. Since it has a maximum at $x = 0$, $\varphi=0$.

Step4: Determine the frequency $B$

The distance from a maximum to the mid - line is a quarter of the period. The $x$ - value changes from $x = 0$ to $x=\frac{\pi}{4}$. If $\frac{T}{4}=\frac{\pi}{4}$, then the period $T=\pi$. Using the formula $T=\frac{2\pi}{B}$, and since $T = \pi$, we have $\pi=\frac{2\pi}{B}$, so $B = 2$.

Answer:

$f(x)=6\cos(2x)+4$