Graphing Functions Using Vertical and Horizontal Shift Practice

The graph of function ‍$h$ is the graph of function ‍$f$ shifted ‍$12$ units to the right and down $4$ units.

To shift the graph of $y=f(x)$ right ‍$12$ units, we can replace ‍$x$ with $x-12$ in the equation.

To shift the graph of $y=f(x-12)$ down ‍$4$ units, we can replace ‍$f(x-12)$ with $f(x-12)-4$ in the equation.

Notice that this is the graph of function $h$, and so $h(x)=f(x-12)-4$.

The answer:

$h(x)=f(x-12)-4$

The graph of ‍$y=x^2$ can be transformed to get the graph of ‍‍$y=(x+4)^2+2$.

• Replacing ‍$x$ with ‍‍$x+4$ shifts the graph of ‍‍$y=x^2$ to the left by ‍$4$ units.
• Adding $2$ to the function shifts the graph of ‍‍$y=x^2$ up by ‍$2$ units.

Therefore, the graph of ‍$y=(x+4)^2+2$ is obtained by shifting the graph of ‍‍$y=x^2$ to the left by ‍‍$4$ units and up by‍ $2$ units.

The graph that corresponds with this transformation is graph B

The graph of ‍$y=f(x)$ can be transformed to get the graph of ‍‍$y=f(x-4)+2$.

• Replacing ‍$x$ with ‍‍$x-4$ shifts the graph of ‍‍$y=f(x)$ to the right by ‍$4$ units.

• Adding $2$ to the function shifts the graph of ‍‍$y=f(x)$ up by ‍$2$ units.

Therefore, the graph of ‍$y=f(x-4)+2$ is obtained by shifting the graph of ‍‍$y=f(x)$ to the right by ‍‍$4$ units and up by‍ $2$ units.

The graph that corresponds with this transformation is graph D

The graph of function ‍$g$ is the graph of function ‍$f$ shifted ‍left $2$ units and up $5$ units.

To shift the graph of $y=f(x)$ left $2$ units, we can replace ‍$x$ with $x+2$ in the equation.

$\begin{aligned}f({x})&=\sqrt{{x}+4}-2\\\\f({x+2})&=\sqrt{{x+2}+4}-2\\\\&=\sqrt{x+6}-2\end{aligned}$

To shift this graph up $5$, we can add $5$ to the function value:

$\begin{aligned}f({x+2})&=\sqrt{{x+6}}-2\\\\f(x+2)+{5}&=\sqrt{{x+6}}-2 + 5\\\\&=\sqrt{x+6}+3\end{aligned}$

The answer:

$g(x)=\sqrt{x+6}+3$