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

Given $f(x)=\frac{1}{x}$, reflect in the $y$-axis, translate up 4 units, then apply a vertical stretch with scale factor $\frac{1}{2}$. Write the resulting function.

Given $f(x) = \sin x$, the graph is horizontally stretched by a factor of 2, then translated up 1 unit, and finally reflected in the $y$-axis. Find an expression for the transformed function.

Given $f(x)=|x+1|$, apply a horizontal compression by factor 2, then shift right 3 units, and finally apply a vertical stretch by factor 2. Write the new function.

Given $f(x)=e^x$, translate left 3 units, then apply a vertical stretch by 5, and then translate down 2 units. Find the new function.

Given $f(x)= \sqrt{x}$, apply a reflection in the $x$-axis, followed by a translation by the vector $\begin{pmatrix} 5 \\ 0 \end{pmatrix}$, and finally a vertical stretch with scale factor $2$. Find an expression for the new function $g(x)$.

Given $f(x)=\sqrt{x-2}$, apply a vertical stretch by scale factor $\frac{1}{3}$, then translate up 6 units, and then translate left 1 unit. Find the transformed function.

Given $f(x)=x^2+1$, reflect the graph in the $x$-axis, then translate it 2 units to the right, and then translate it 4 units up. Determine the resulting function.

Given $f(x)=\ln x$, reflect in the $y$-axis, then shift right 2 units, and then apply a vertical stretch by factor 3. Write the transformed function.

Given $f(x)=\frac{1}{x+2}$, reflect in the $x$-axis, then apply a horizontal stretch with scale factor $\frac{1}{4}$, translate $1$ unit to the right, and translate $3$ units up. Find the resulting function.

Given $f(x)=x^2$, apply a vertical stretch by a factor of 3, then translate up 2 units, and then shift left 4 units. Write the equation of the transformed function.

Given $f(x)=|x|$, shift down 3 units, then apply a horizontal stretch by factor 2, and then shift left 1 unit. Find the transformed function.

Given $f(x)=x^3$, apply a vertical stretch by factor 4, then a horizontal compression by factor 3, and then translate down 2 units. Find the final function.