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Notes
Parent Function
\(f(x)=|x|\)
can be expressed as a piecewise function
\(f(x) = \begin{cases}
-x & \text{if }x\lt 0 \\
x & \text{if }x\geq 0
\end{cases}\)
Translated Function
\(f(x)=|x-h|+k\)
can be expressed as a piecewise function
\(f(x) = \begin{cases}
-x+(k+h) & \text{if }x\lt h \\
x+(k-h) & \text{if }x\geq h
\end{cases}\)
Transformed Function
\(f(x)=a|x-h|+k\)
can be expressed as a piecewise function
\(f(x) = \begin{cases}
-ax+(k+ah) & \text{if }x\lt h \\
ax+(k-ah) & \text{if }x\geq h
\end{cases}\)
Practice Problems
Express as a piecewise function
\(\small{\textbf{1)}}\) \( f(x)=|x+3|-1 \)
\(\small{\textbf{2)}}\) \( f(x)=|x-5|+2 \)
\(\small{\textbf{3)}}\) \( f(x)=|x-4|-1 \)
\(\small{\textbf{4)}}\) \( f(x)=|x+5|+2 \)
\(\small{\textbf{5)}}\) \( f(x)=2|x+4|-2 \)
\(\small{\textbf{6)}}\) \( f(x)=-3|x+2|-5 \)
\(\small{\textbf{7)}}\) \( f(x)=4|x-2|+3 \)
See Related Pages\(\)
\(\bullet\text{ Graphing Piecewise Functions}\)
\(\,\,\,\,\,\,\,\,\)
In Summary
The parent function for absolute value functions is \(f(x)=|x|\). It can be thought of as representing the distance between the number \(x\) and zero on the number line. The shape of absolute value functions is a “v”. This means that each absolute value function can be thought of as two separate lines. This means that piecewise functions are a great way to express absolute value functions. One linear function for when the number inside the absolute value is positive and another linear function when negative.
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