Question Type 4: Creating a description explaining function transformations Exercises

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

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

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 $D$.

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

Starting from $y=\sin(x)$, write the equation of the function after applying a horizontal compression by a factor of $\tfrac12$, a horizontal shift right by 3 units, and a vertical stretch by a factor of 5.

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

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

Find the equation of the transform of $y=\sin(x)$ that has amplitude 5, period $\pi$, phase shift of 3 units to the right, and no vertical shift.

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

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

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

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

Solve the equation $5\sin(2x-6)=3$ for $x$ in the interval $[0,2\pi]$.

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 within one period.