About Reflecting Graphs of Functions:

In addition to horizontal and vertical stretches and compressions, we also need to know how to reflect a graph across an axis (x-axis, y-axis). When we reflect a graph across the x-axis, the same x-values will now correspond to the negative of the old y-values. Similarly, when we reflect a graph across the y-axis, the same y-values will now correspond to the negative of the old x-values. Additionally, we will learn how to stretch or shrink a function when a reflection is involved.


Test Objectives
  • Demonstrate an understanding of horizontal and vertical stretches and compressions
  • Demonstrate the ability to reflect a graph across the x-axis
  • Demonstrate the ability to reflect a graph across the y-axis
Reflecting Graphs of Functions Practice Test:

#1:

Instructions: describe the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=-\frac{3}{x}$$

$$b)\hspace{.2em}$$ $$f(x)=[x]$$ $$g(x)=-3[-x]$$


#2:

Instructions: describe the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=-|3x|$$

$$b)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=-\left(\frac{1}{2}\right)^2$$


#3:

Instructions: describe the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=-\frac{1}{3x}$$

$$b)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=-\left|\frac{1}{2}x\right|$$


#4:

Instructions: describe the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x}$$ $$g(x)=-\frac{1}{2}\sqrt{-x}$$

$$b)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=-|2x|$$


#5:

Instructions: describe the transformation from f(x) to g(x).

$$a)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=-(2x)^3$$

$$b)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=-3x^3$$


Written Solutions:

#1:

Solutions:

a) vertically stretched by a factor of 3, reflected across the x-axis

b) vertically stretched by a factor of 3, reflected across the x-axis and the y-axis


#2:

Solutions:

a) horizontally compressed by a factor of 3, reflected across the x-axis

b) horizontally stretched by a factor of 2, reflected across the x-axis


#3:

Solutions:

a) horizontally compressed by a factor of 3, reflected across the x-axis

b) horizontally stretched by a factor of 2, reflected across the x-axis


#4:

Solutions:

a) vertically compressed by a factor of 2, reflected across the x-axis and y-axis

b) horizontally compressed by a factor of 2, reflected across the x-axis


#5:

Solutions:

a) horizontally compressed by a factor of 2, reflected across the x-axis

b) vertically expanded by a factor of 3, reflected across the x-axis