Task B-3.4. Determine the function $f(x)=A \sin (b x+c)+d$ if its minimum is at the point $(3,-1)$, and the first maximum after that point is at the point $(5,7)$.
Solution. The amplitude is $A=\frac{7-(-1)}{2}=4$

The vertical shift is $d=\frac{7+(-1)}{2}=3$

The half period is the distance between $x_{m}=3$ and $x_{M}=5$, so the period $T=4$. From this, we get $\frac{2 \pi}{b}=4$ so $b=\frac{\pi}{2}$.

(1 point)

It remains to determine the horizontal shift.

The equilibrium position is at the midpoint between the minimum and maximum, so for $x=4$, we have $-\frac{c}{\frac{\pi}{2}}=4$, or $c=-2 \pi$.

The desired function is

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

which can also be written as

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

Note: If the student has correctly calculated all four parameters $a, b, c, d$, but has not written the function, deduct 1 point. For the parameter $c$, accept any integer multiple of $2 \pi$.