Question Type 3: Applying and describing transformations applied to trigonometric function Exercises

Find an equation of the sinusoid obtained from $y=4\sin(x)$ that has amplitude $5$, period $\pi$, phase shift $\pi$ to the right, vertical shift $+5$ and is reflected in the $x$-axis.

Describe the sequence of transformations that maps $f(x)=4\sin(x)$ to $g(x)=-5\sin(x-\pi)+5$.

Determine the x-intercepts of $y=-5\sin(x-\pi)+5$ in the interval $[0,2\pi]$.

Determine the domain and range of $y=-5\sin(x-\pi)+5$.

Find the coordinates of the maximum and minimum points of one period of $y=-5\sin(x-\pi)+5$ in the interval $[0,2\pi]$.

Write an equation for the function obtained from $y=4\sin(x)$ by reflecting in the $y$-axis, applying a vertical stretch by factor $3$, shifting right by $\frac{π}{2}$ and down by $2$.

Describe the sequence of transformations that maps $f(x)=4\sin(x)$ to $h(x)=-5\sin\left(2\left(x-\frac{\pi}{2}\right)\right)+5$.

Compute the average value of $y=5-5\sin(x-\pi)$ over one period.

Solve the equation $-5\sin(x-\pi)+5=0$ for $x$.

Find the derivative of $g(x)=-5\sin(x-\pi)+5$ and determine its critical points in $[0,2\pi]$.

For $g(x)=-5\sin(x-\pi)+5$, calculate $g\left(\frac{\pi}{2}\right)$.

State the amplitude, period, phase shift and vertical shift of $y=-5\sin(x-\pi)+5$.