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 
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#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 
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#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 
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#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 
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#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 
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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
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#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
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#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
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#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
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#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
