### 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 practice stretching or compressing 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

#1:

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

$$a)\hspace{.2em}$$ $$f(x)=x^2$$ $$g(x)=-3x^2$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=\sqrt{x}$$ $$g(x)=\sqrt{-2x}$$ Desmos Link for More Detail

Watch the Step by Step Video Solution View the Written Solution

#2:

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

$$a)\hspace{.2em}$$ $$f(x)=|x|$$ $$g(x)=-|5x|$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=\sqrt[3]{x}$$ $$g(x)=\sqrt[3]{-8x}$$ Desmos Link for More Detail

Watch the Step by Step Video Solution View the Written Solution

#3:

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

$$a)\hspace{.2em}$$ $$f(x)=\frac{1}{x}$$ $$g(x)=-\frac{5}{x}$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=x^3 - 2x^2 - x$$ $$g(x)=-\frac{1}{2}\left(x^3 - 2x^2 - x\right)$$ Desmos Link for More Detail

Watch the Step by Step Video Solution View the Written Solution

#4:

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

$$a)\hspace{.2em}$$ $$f(x)=\sqrt{x}$$ $$g(x)=-\frac{1}{2}\sqrt{-x}$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=x^3 - x$$ $$g(x)=f\left(-\frac{1}{4}x\right)$$ Desmos Link for More Detail

Watch the Step by Step Video Solution View the Written Solution

#5:

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

$$a)\hspace{.2em}$$ $$f(x)=x^3$$ $$g(x)=(-2x)^3$$ Desmos Link for More Detail

$$b)\hspace{.2em}$$ $$f(x)=x^3 - x$$ $$g(x)=-2f\left(\frac{1}{2}x\right)$$ Desmos Link for More Detail

Watch the Step by Step Video Solution View the Written Solution

Written Solutions:

#1:

Solutions:

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

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

Watch the Step by Step Video Solution

#2:

Solutions:

a) horizontally compressed by a factor of 5, reflected across the x-axis

b) horizontally compressed by a factor of 8, reflected across the y-axis

Watch the Step by Step Video Solution

#3:

Solutions:

a) vertically stretched by a factor of 5, reflected across the x-axis

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

Watch the Step by Step Video Solution

#4:

Solutions:

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

b) horizontally stretched by a factor of 4, reflected across the y-axis

Watch the Step by Step Video Solution

#5:

Solutions:

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

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