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

Explanation:

Step1: Determine the amplitude $A$

The amplitude is half the distance between the maximum and minimum values. The maximum is $y = 10$ and the mid - line value is $y = 4$. So $A=10 - 4=6$.

Step2: Determine the vertical shift $D$

The mid - line of the sinusoidal function gives the vertical shift. The mid - line is $y = 4$, so $D = 4$.

Step3: Determine the phase shift $\varphi$

The function has a maximum at $(0,10)$. For a cosine function $y=A\cos(B(x - \varphi))+D$, when $x = 0$, we have a maximum. For the standard cosine function $y=\cos(x)$ which has a maximum at $x = 0$, the phase shift $\varphi=0$.

Step4: Determine the value of $B$

The function goes from a maximum to the mid - line in a quarter of a period. The horizontal distance from the maximum point $(0,10)$ to the mid - line point $(\frac{\pi}{4},4)$ is $\frac{\pi}{4}$. Since a quarter of the period $T$ is $\frac{\pi}{4}$, and $T=\frac{2\pi}{B}$, then $\frac{T}{4}=\frac{\pi}{4}$. Substituting $T=\frac{2\pi}{B}$ into $\frac{T}{4}=\frac{\pi}{4}$, we get $\frac{1}{4}\times\frac{2\pi}{B}=\frac{\pi}{4}$. Cross - multiplying gives $2\pi=B\pi$, so $B = 2$.

The general form of a sinusoidal function is $y = A\cos(B(x-\varphi))+D$. Substituting $A = 6$, $B=2$, $\varphi = 0$ and $D = 4$ into the formula, we get $f(x)=6\cos(2x)+4$.

Answer:

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