Question

1. what does absolute value mean?

please make a function table and graph for the following functions. use graph paper.

1. f(x) = 4|x|
2. f(x) = -3|x|
3. f(x) = |x| - 2

graph the following piecewise function. use graph paper.
f(x) = x - 2, x ≤ 1
-2x + 3, x > 1

Explanation:

1. The absolute value of a real number is its non-negative distance from zero on the number line, regardless of direction.
2. For each absolute value function, generate input values (positive, negative, zero) to calculate corresponding output values, then plot the points. For the piecewise function, evaluate each sub-function over its defined domain, plot the segments, and note the endpoint behavior (closed/open circles).

Answer:

1. The absolute value of a number, denoted as $|x|$, is the non-negative value of $x$ without regard to its sign. Mathematically, $|x| =

$$\begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

$. It represents the distance of the number from 0 on the number line.

1. ### Function Table for $f(x)=4|x|$:

$x$	$f(x)=4	x	$
-1	$4	-1	=4$
0	$4	0	=0$
1	$4	1	=4$
2	$4	2	=8$

Graph: V-shaped with vertex at (0,0), opening upward, steeper than $|x|$ (slope $\pm4$).

1. ### Function Table for $f(x)=-3|x|$:

$x$	$f(x)=-3	x	$
-1	$-3	-1	=-3$
0	$-3	0	=0$
1	$-3	1	=-3$
2	$-3	2	=-6$

Graph: V-shaped with vertex at (0,0), opening downward, slope $\pm(-3)$.

1. ### Function Table for $f(x)=|x|-2$:

$x$	$f(x)=	x	-2$
-1	$	-1	-2=-1$
0	$	0	-2=-2$
1	$	1	-2=-1$
2	$	2	-2=0$

Graph: V-shaped with vertex at (0,-2), opening upward, slope $\pm1$.

1. ### Function Table for piecewise $f(x)=

$$\begin{cases}x-2, & x\leq1 \\ -2x+3, & x>1\end{cases}$$

$:

$x$	$f(x)$
0	$0-2=-2$
1	$1-2=-1$ (closed circle)
2	$-2(2)+3=-1$
3	$-2(3)+3=-3$

Graph:

• For $x\leq1$: Line with slope 1, y-intercept -2, ending at closed circle (1,-1).
• For $x>1$: Line with slope -2, passing through (2,-1) (open circle at (1,1) since $-2(1)+3=1$, not included), extending downward to the right.