Question Type 3: Given a description of transformations, finding the appropriate transformed function Exercises

If $f(x)=|x|$, apply a reflection in the $x$-axis followed by a vertical compression by a factor of $\tfrac12$. Write the equation of the transformed function.

Describe the changes from $y=\sin x$ to $y=\tfrac{1}{4}\sin(3x)$.

Let $f(x)=e^x$. Translate it 3 units left, then reflect in the $y$-axis. Determine the resulting function.

Let $f(x)=\tfrac{3}{x}$. Perform the transformations: translate up 2 units, stretch horizontally by a factor of 3, then shift 4 units to the left. Express the resulting function.

Given $f(x)=\sqrt{x}$, reflect in the $y$-axis, then stretch horizontally by a factor of 4, then translate up by 5. Determine the new function.

Given $g(x)=\log(x)$, apply in order: vertical stretch by 2, translate right 1 unit, reflect in the $y$-axis. Write the resulting function.

Describe the transformations mapping $y=\sin x$ to $y=5\sin(2x-6)$.

Given the function $f(x)=x^2$, apply the following transformations in order: stretch vertically by a factor of 3, translate 2 units to the right, then reflect in the $x$-axis. Write the equation of the transformed function.

Describe the transformations from $y=\sin x$ to $y=2\sin\bigl(-\tfrac{x}{2}\bigr)-1$.

Given $f(x)=7(x+5)^2$, apply a vertical stretch by 4, then translate right 6 units, then stretch horizontally by factor 2. Find the resulting function.

Describe the transformations mapping $y=\sqrt{x}$ to $y=\sqrt{-2x+8}+3$.

Describe the sequence of transformations to obtain $y=-3\cos\bigl(4(x+\tfrac{\pi}{3})\bigr)+2$ from $y=\cos x$.