Question Type 2: Given the period and amplitude, finding the trigonometric functions Exercises

Find the function $y=a\sin(bx)$ that has amplitude $9$ and period $\pi$.

A sine wave has amplitude $9$, period $\pi$, and midline $y=1$. The first maximum of the function for $x > 0$ occurs at $x=\frac{\pi}{4}$.

Find the equation of the sine wave in the form $y= a\sin\bigl(b(x+c)\bigr)+d$, where $a, b, d > 0$.

Find the equation of the sine curve $y=a\sin\bigl(b(x+c)\bigr)+d$, where $a, b > 0$, that has amplitude $9$, period $\pi$, a maximum at $x=\frac{\pi}{12}$ and a minimum at $x=\frac{7\pi}{12}$. Assume the curve oscillates about the $x$-axis.

Write down $y=a\sin(bx)+d$ with amplitude $9$, period $\pi$, and a vertical shift of $5$ units up.

Determine the equation $y=a\sin\bigl(b(x+c)\bigr)+d$ for a wave with amplitude $9$, period $\pi$, midline $y=-3$, and rising through the midline at $x=\frac{\pi}{8}$.

Determine the equation of the function $y=a\sin\bigl(b(x+c)\bigr)+d$ with amplitude $9$, period $\pi$, a vertical shift $2$, and intersecting its midline decreasing at $x=\frac{\pi}{6}$.

Find the equation $y=a\sin\bigl(b(x+c)\bigr)$ with amplitude $9$, period $\pi$, and a phase shift of $\frac{\pi}{3}$ to the right.

Find $y= a\sin\bigl(b(x+c)\bigr)+d$ with amplitude $9$, period $\pi$, a phase shift of $\frac{\pi}{6}$ to the right, and a vertical shift of $-2$.

Find the equation of the sine function $y=a\sin\bigl(b(x+c)\bigr)+d$ with amplitude $6$, period $\pi$, that passes through the points $(\frac{\pi}{6},4)$ and $(\frac{\pi}{3},-2)$.

Find the equation of the function $y=a\sin\bigl(b(x+c)\bigr)+d$ given: amplitude $9$, period $π$, midline $y=0$, minimum at $x=0$, and it passes through the midline going up at $x=\frac{π}{4}$.

Determine $y=a\sin\bigl(b(x+c)\bigr)+d$ given amplitude $9$, period $\pi$, vertical shift $-1$, and its first positive peak at $x=\tfrac{\pi}{8}$.

Determine $y=a\sin\bigl(b(x+c)\bigr)$ with amplitude $9$, period $\pi$, and a phase shift of $\frac{\pi}{4}$ to the left.