2.2 Absolute Value Functions - Introduction to Calculus

Absolute Value Functions

A very special and common piecewise function is the absolute value function.

y=|x|=\begin{cases}x&x\ge0\\[.5em]-x&x<0\end{cases}

Domain:

x\in\mathbb{R}

Range:

y\ge0

Adjust the scale and shifts, then move the input to see which side of the vertex you are using.

\displaystyle y=|x|

\displaystyle y=1|x-(0)|+(0)

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Vertex: (0, 0) · x − h ≥ 0: use x − h

At the initial settings, the graph is y = |x|. Negative a reflects it downward, while h and k move the vertex.

QUESTION

Which value equals

\displaystyle |-3|

THEOREM

\begin{aligned} |ab|&=|a|\cdot|b|\\[.65em] \left|\dfrac ab\right|&=\dfrac{|a|}{|b|}\qquad(b\ne0)\\[.65em] \text{If }|a|\le b,&\quad\text{then }-b\le a\le b\\[.65em] \text{If }|a|\ge b,&\quad\text{then }a\ge b\text{ or }a\le-b\\[.65em] |a+b|&\le|a|+|b|\quad\text{(Triangle Inequality)} \end{aligned}

Here

a

and

b

are real numbers; the quotient rule requires

b\ne0

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Absolute Value Functions

A very special and common piecewise function is the absolute value function.

y=|x|=\begin{cases}x&x\ge0\\[.5em]-x&x<0\end{cases}

Domain:

x\in\mathbb{R}

Range:

y\ge0

Adjust the scale and shifts, then move the input to see which side of the vertex you are using.

\displaystyle y=|x|

\displaystyle y=1|x-(0)|+(0)

a · vertical scale1h · horizontal shift0k · vertical shift0Input x1

\displaystyle f(1)=1

Vertex: (0, 0) · x − h ≥ 0: use x − h

At the initial settings, the graph is y = |x|. Negative a reflects it downward, while h and k move the vertex.

QUESTION

Which value equals

\displaystyle |-3|

THEOREM

\begin{aligned} |ab|&=|a|\cdot|b|\\[.65em] \left|\dfrac ab\right|&=\dfrac{|a|}{|b|}\qquad(b\ne0)\\[.65em] \text{If }|a|\le b,&\quad\text{then }-b\le a\le b\\[.65em] \text{If }|a|\ge b,&\quad\text{then }a\ge b\text{ or }a\le-b\\[.65em] |a+b|&\le|a|+|b|\quad\text{(Triangle Inequality)} \end{aligned}

Here

a

and

b

are real numbers; the quotient rule requires

b\ne0

Absolute Value Functions