Graphing Calculator Drawing: Absolute Value

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Setup. Navigate to https://www.desmos.com/calculator. Be sure to sign in so that you can save your graph.

Demonstration - Absolute Value. Observe the graph as you type each of the following inputs. In general, an absolute value graph $y=m \vert x \vert$ makes a “V” shape, with the magnitude of $m$ controlling the slope of the V, and the sign of $m$ controlling whether the V opens upward or downward.

$\begin{align*} y&=5|x| \\[10pt] y&=1|x| \\[10pt] y&=0.1|x| \\[10pt] y&=-0.1|x| \\[10pt] y&=-1|x| \\[10pt] y&=-5|x| \end{align*}$


Demonstration - Shifts. Observe the graph as you type each of the following inputs. In general, the graph of $y=m \vert x-a \vert +b$ shifts the absolute value graph $y=m \vert x \vert$ so that the pointy part of the “V” occurs at the point $(a,b).$

$\begin{align*} y&=|x-1|+2 \\[10pt] y&=-2|x-1|-3 \\[10pt] y&=-0.5|x+3|-1 \\[10pt] y&=10|x+2|+1 \end{align*}$


Exercise. Draw the two absolute value functions shown below. (Hint: Remember that you can limit the domain and range of your functions with parentheses, e.g. $y= \vert x \vert \lbrace -1 < x < 1 \rbrace$ or $y=\vert x \vert \lbrace y < 3 \rbrace.$)

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Exercise. Draw more absolute value functions to create a person! (The person’s back will be a vertical line, but everything else can be made out of absolute value functions.)

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Challenge. Try to draw yourself, or your friend! You can include hair, shoes, ears, hands, clothes, etc.)

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