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

Given the function $f(x) = \sqrt{x}$, the following transformations are applied in order:

• a reflection in the $y$-axis;
• a horizontal stretch by a factor of 4;
• a translation up by 5 units.

Determine the equation of the resulting function.

Given $g(x) = \log(x)$, apply the following transformations in the order shown:

1. a vertical stretch by a factor of 2
2. a translation 1 unit to the right
3. a reflection in the $y$-axis

Write down an expression for the resulting function.

Given $f(x) = |x|$, apply a reflection in the $x$-axis followed by a vertical compression by a factor of $\frac{1}{2}$.

Write the equation of the transformed function.

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

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

Describe the transformations that map the graph of $y = \sin x$ onto the graph of $y = \frac{1}{4}\sin(3x)$.

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

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

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.

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