Question

graphing absolute value functions (tables) name date:
graph the absolute value function. be sure to first identify the vertex.

1. graph $f(x) = |x - 4|$
2. graph $f(x) = |x| - 5$

Explanation:

Step1: Analyze \( f(x) = |x - 4| \)

The general form of an absolute - value function is \( f(x)=|x - h|+k \), where the vertex is at \( (h,k) \). For \( f(x)=|x - 4| \), we can rewrite it as \( f(x)=|x - 4|+0 \), so the vertex is at \( (4,0) \).

To create a table, we choose values of \( x \) around \( x = 4 \) (since the vertex is at \( x = 4 \)). Let's choose \( x=2,3,4,5,6 \).

• When \( x = 2 \): \( f(2)=|2 - 4|=|-2| = 2 \)
• When \( x = 3 \): \( f(3)=|3 - 4|=|-1| = 1 \)
• When \( x = 4 \): \( f(4)=|4 - 4|=|0| = 0 \)
• When \( x = 5 \): \( f(5)=|5 - 4|=|1| = 1 \)
• When \( x = 6 \): \( f(6)=|6 - 4|=|2| = 2 \)

So the table for \( f(x)=|x - 4| \) is:

\( x \)	\( f(x) \)
3	1
4	0
5	1
6	2

To graph, we plot the points \((2,2)\), \((3,1)\), \((4,0)\), \((5,1)\), \((6,2)\) and draw a V - shaped graph with the vertex at \((4,0)\).

Step2: Analyze \( f(x)=|x|-5 \)

The general form of an absolute - value function is \( f(x)=|x - h|+k \). For \( f(x)=|x|-5 \), we can rewrite it as \( f(x)=|x - 0|-5 \), so the vertex is at \((0,-5)\).

To create a table, we choose values of \( x \) around \( x = 0 \). Let's choose \( x=-2,-1,0,1,2 \).

• When \( x=-2 \): \( f(-2)=|-2|-5 = 2 - 5=-3 \)
• When \( x=-1 \): \( f(-1)=|-1|-5 = 1 - 5=-4 \)
• When \( x = 0 \): \( f(0)=|0|-5 = 0 - 5=-5 \)
• When \( x = 1 \): \( f(1)=|1|-5 = 1 - 5=-4 \)
• When \( x = 2 \): \( f(2)=|2|-5 = 2 - 5=-3 \)

So the table for \( f(x)=|x|-5 \) is:

\( x \)	\( f(x) \)
-1	-4
0	-5
1	-4
2	-3

To graph, we plot the points \((-2,-3)\), \((-1,-4)\), \((0,-5)\), \((1,-4)\), \((2,-3)\) and draw a V - shaped graph with the vertex at \((0,-5)\).

Answer:

For \( f(x)=|x - 4| \):

• Vertex: \((4,0)\)
• Table:

\( x \)	\( f(x) \)
3	1
4	0
5	1
6	2

• Graph: V - shaped graph with vertex at \((4,0)\) and passing through the points \((2,2)\), \((3,1)\), \((5,1)\), \((6,2)\)

For \( f(x)=|x|-5 \):

• Vertex: \((0,-5)\)
• Table:

\( x \)	\( f(x) \)
-1	-4
0	-5
1	-4
2	-3

• Graph: V - shaped graph with vertex at \((0,-5)\) and passing through the points \((-2,-3)\), \((-1,-4)\), \((1,-4)\), \((2,-3)\)