Graphing Functions Using Vertical and Horizontal Shift Practice

1. The graph of ‍$y=h(x)$ is a transformation of the graph of ‍‍$y=f(x)$.
   

   
   

   Write a formula for function ‍‍$h$ in terms of function ‍‍$f$.
   

   ‍$h(x)=$

2. The graph of ‍$y=x^2$ is shown on the grid below.

   
   

   Which of the following is the graph of ‍‍$y=(x+4)^2+2$?

   Choose 1 answer:
   

   1. 
   2. 
   3. 

   4. 

3. The graph of ‍$y=f(x)$ is shown on the grid below.

   
   

   Which of the following is the graph of ‍‍$y=f(x-4)+2$?

   Choose 1 answer:
   

   1. 
   2. 
   3. 

   4. 

4. The graph of $y=g(x)$ is a transformation of the graph of $y=f(x)$.

   
   
   

   Given that $f(x)=\sqrt{x+4}-2$, write an expression for $g(x)$ in terms of $x$.
   

Source: Khan Academy, https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations/x2ec2f6f830c9fb89:shift/e/shift-functions
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. 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:

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

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2. 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

   
   

   ---

3. 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
   

   
   

   ---

4. 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:
   
   $g(x)=\sqrt{x+6}+3$