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

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

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.

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

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 $-2$.

Given that a sine wave has amplitude $9$, period $\pi$, and midline $y=1$, with its maximum occurring at $x=\tfrac{\pi}{4}$, find its equation in the form $y= a\sin\bigl(b(x+c)\bigr)+d$.

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

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

Find the sine function $y=a\sin\bigl(b(x+c)\bigr)+d$ with amplitude $9$, period $\pi$, that passes through the points $(\tfrac{\pi}{6},4)$ and $(\tfrac{\pi}{3},-2)$.

Construct $y=a\sin\bigl(b(x+c)\bigr)+d$ with amplitude $9$, period $\pi$, a phase shift $\tfrac{\pi}{12}$ to the left, a vertical shift $2$, and intersecting its midline decreasing at $x=\tfrac{\pi}{6}$.

Find the sine curve $y=a\sin\bigl(b(x+c)\bigr)+d$ that has amplitude $9$, period $\pi$, maxima at $x=\tfrac{\pi}{12}$ and minima at $x=\tfrac{5\pi}{12}$.

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