Graphing Functions Using Reflections Practice

1. Functions ‍$f$ (solid) and ‍$g$ (dashed) are graphed.

   
   

   $f$ is defined as ‍$f(x)=-2\log_2(x+1)+3$.

   What is the equation of ‍$g$?

   Choose 1 answer:

   1. $g(x)=-2\log_2(x+1)+3$
   2. $g(x)=2\log_2(x+1)-3$
   3. $g(x)=-2\log_2(1-x)+3$

   4. $g(x)=2\log_2(1-x)-3$

2. This is the graph of function $f(x)=\sqrt[3]{x-2}$.

   
   

   What is the graph of $g(x)=-\sqrt[3]{x-2}$.
   

   Choose 1 answer:

   1. 

   2. 
      
   3. 
      

   4. 

3. Functions ‍$f$ (solid) and ‍$g$ (dashed) are graphed.

   
   

   What is the equation of ‍$g$ in terms of $f$?

   Choose 1 answer:

   1. $g(x)=f(x)$
   2. $g(x)=-f(x)$
   3. $g(x)=f(-x)$

   4. $g(x)=-f(-x)$

4. This is the graph of function $f$.

   
   

   Function $g$ is defined as $g(x)=f(-x)$.
   

   What is the graph of $g$.
   

   Choose 1 answer:

   1. 

   2. 
      
   3. 
      
   4. 
      

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

1. Function $g$ is a reflection of function $f$. There are different kinds of function reflections:

   Expression	Type	Meaning
   $-f(x)$	Vertical (reflection across the $x$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(x)$ with the same ‍$x$-value but the opposite‍ $y$-value.
   $f(-x)$	Horizontal (reflection across the $y$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=f(-x)$ with the same ‍$y$-value but the opposite $x$-value.
   $-f(-x)$	Vertical and horizontal	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(-x)$ with the opposite ‍$x$- and $y$-values.

   
   

   We need to determine which reflection to apply on $f$ to get ‍$g$, and then we need to find the expression for ‍$g$.

   $g$ is a horizontal reflection of $f$.
   

   So $g(x)=f(-x)$.

   Now recall that $f(x)=-2\log_2(x+1)+3$. Let's see what expression we get for $f(-x)$:

   $\begin{aligned}f(-x)&=-2\log_2\big((-x)+1\big)+3\\\\&=-2\log_2(1-x)+3\end{aligned}$
   

   In conclusion, $g(x)=-2\log_2(1-x)+3$.

   ---

2. Function $g$ is a reflection of function $f$. There are different kinds of function reflections:

   Expression	Type	Meaning
   $-f(x)$	Vertical (reflection across the $x$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(x)$ with the same ‍$x$-value but the opposite‍ $y$-value.
   $f(-x)$	Horizontal (reflection across the $y$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=f(-x)$ with the same ‍$y$-value but the opposite $x$-value.
   $-f(-x)$	Vertical and horizontal	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(-x)$ with the opposite ‍$x$- and $y$-values.

   
   

   First, we need to determine which kind of reflection is ‍$g(x)=-\sqrt[3]{x-2}$.

   Recall that $f(x)=\sqrt[3]{x-2}$ Let's see what expression we get for  $-f(x)$:

   $-f(x)=-\sqrt[3]{x-2}$
   

   So $g(x)=-f(x)$.

   Since $g(x)=-f(x)$, it is a vertical reflection of ‍$f$.

   So the correct answer is B.
   

   
   

   ---

3. Function $g$ is a reflection of function $f$. There are different kinds of function reflections:

   Expression	Type	Meaning
   $-f(x)$	Vertical (reflection across the $x$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(x)$ with the same ‍$x$-value but the opposite‍ $y$-value.
   $f(-x)$	Horizontal (reflection across the $y$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=f(-x)$ with the same ‍$y$-value but the opposite $x$-value.
   $-f(-x)$	Vertical and horizontal	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(-x)$ with the opposite ‍$x$- and $y$-values.

   
   

   $g$ is a horizontal reflection of $f$.

   So $g(x)=f(-x)$.

   ---

4. Function $g$ is a reflection of function $f$. There are different kinds of function reflections:

   Expression	Type	Meaning
   $-f(x)$	Vertical (reflection across the $x$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(x)$ with the same ‍$x$-value but the opposite‍ $y$-value.
   $f(-x)$	Horizontal (reflection across the $y$-axis)	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=f(-x)$ with the same ‍$y$-value but the opposite $x$-value.
   $-f(-x)$	Vertical and horizontal	Every point on the graph of $y=f(x)$ corresponds to a point on the graph of ‍$y=-f(-x)$ with the opposite ‍$x$- and $y$-values.

   
   

   Since $g(x)=f(-x)$, it is a horizontal reflection of ‍$f$.

   So the correct answer is C.