Question Type 4: Creating a description explaining function transformations Exercises

For the function $y=5\sin(2x-6),$ determine the $x$-coordinate of the first local minimum after $x=3$.

Find the derivative $\frac{dy}{dx}$ of $y=5\sin(2x-6)$ and determine the $x$-values where this derivative is zero for $3 \le x \le 3+\pi$.

What is the equation of the midline of the function $y = 5 \sin(2x - 6)?$

Solve the equation $5\sin(2x-6)=3$ for $0 \le x \le 2\pi$.

Find the range of the function $y=5\sin(2x-6).$

Express the function $y=5\sin(2x-6)$ in the form $y=A\sin\bigl(B(x-C)\bigr)+D$ by identifying $A$, $B$, $C$, and $D$.

Find the $x$-intercepts of $y=5\sin(2x-6)$ within the interval $[0,2\pi]$.

Describe in words how the graph of $y=\sin(x)$ changes when it is transformed into $y=5\sin(2x-6)$.

Starting from $y=\sin(x)$, write the equation of the function after applying a horizontal compression by a factor of $\frac{1}{2}$, a horizontal shift right by 3 units, and a vertical stretch by a factor of 5.

Find the equation of the function obtained by transforming the graph of $y=\sin(x)$ such that it has an amplitude of 5, a period of $\pi$, a phase shift of 3 units to the right, and no vertical shift.

Determine the smallest positive period of $y = 5\sin(2x - 6).$

A point moves according to the law $y = 5\sin(2x - 6).$ Find the smallest positive $x$ such that $y = -5$.

Sketch one period of $y=5\sin(2x-6)$ and label all maximum, minimum, and $x$-intercept points.

Describe the sequence of transformations that maps the graph of $y = \sin(x)$ to the graph of $y = 5\sin(2x - 6)$.

Determine all $x$-values (general form) for which $y = 5\sin(2x - 6)$ attains its maximum value.

State the amplitude, period, phase shift, and vertical shift of the function $y=5\sin(2x-6).$