the graph of a sinusoidal function intersects its midline at (0, -7) and then has a minimum point at (π/4…

Explanation:

Step1: Determine the vertical shift

The mid - line is at $y = - 7$, so the vertical shift $D=-7$.

Step2: Determine the amplitude

The mid - line value is $y=-7$ and the minimum is at $y = - 9$. The amplitude $A$ is the distance from the mid - line to the maximum or minimum. So $A=\vert-7-(-9)\vert = 2$. Since it is a minimum point after the mid - line intersection, $A=-2$.

Step3: Determine the period

The distance from the mid - line intersection to the minimum is $\frac{\pi}{4}$ units. For a sinusoidal function, this is $\frac{1}{4}$ of the period. Let the period be $T$. Then $\frac{T}{4}=\frac{\pi}{4}$, so $T=\pi$. Using the formula $T=\frac{2\pi}{B}$, we get $\pi=\frac{2\pi}{B}$, and solving for $B$ gives $B = 2$.

Step4: Determine the phase shift

The function intersects the mid - line at $x = 0$. For a cosine function $y=A\cos(Bx - C)+D$, when $x = 0$, we can assume $C = 0$ (since there is no horizontal shift indicated by the given points). The general form of a sinusoidal function is $y = A\cos(Bx - C)+D$. Substituting $A=-2$, $B = 2$, $C = 0$, and $D=-7$ into the formula, we get $f(x)=-2\cos(2x)-7$.

Answer:

$f(x)=-2\cos(2x)-7$