Question

kuta software - infinite algebra 1
graphing absolute value functions
graph each equation.

1. $y = |x - 2| - 4$
2. $y = |x + 1|$
3. $y = |x| + 1$
4. $y = |x| + 2$
5. $y = |x + 2|$
6. $y = |x + 1| + 3$

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Explanation:

For each absolute value function, we use the parent function $y=|x|$ (vertex at $(0,0)$) and apply horizontal/vertical shifts:

• Horizontal shift: $y=|x-h|$ shifts $h$ units right if $h>0$, left if $h<0$
• Vertical shift: $y=|x|+k$ shifts $k$ units up if $k>0$, down if $k<0$

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1) $y=|x-2|-4$

Step1: Identify vertex

Vertex: $(2, -4)$

Step2: Find 2 additional points

For $x=0$: $y=|0-2|-4=2-4=-2$ → $(0,-2)$
For $x=4$: $y=|4-2|-4=2-4=-2$ → $(4,-2)$

Step3: Plot & connect points

Draw a V-shape through $(2,-4)$, $(0,-2)$, $(4,-2)$

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2) $y=|x+1|$

Step1: Identify vertex

Rewrite as $y=|x-(-1)|$, vertex: $(-1, 0)$

Step2: Find 2 additional points

For $x=0$: $y=|0+1|=1$ → $(0,1)$
For $x=-2$: $y=|-2+1|=1$ → $(-2,1)$

Step3: Plot & connect points

Draw a V-shape through $(-1,0)$, $(0,1)$, $(-2,1)$

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3) $y=|x|+1$

Step1: Identify vertex

Vertex: $(0, 1)$

Step2: Find 2 additional points

For $x=1$: $y=|1|+1=2$ → $(1,2)$
For $x=-1$: $y=|-1|+1=2$ → $(-1,2)$

Step3: Plot & connect points

Draw a V-shape through $(0,1)$, $(1,2)$, $(-1,2)$

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4) $y=|x|+2$

Step1: Identify vertex

Vertex: $(0, 2)$

Step2: Find 2 additional points

For $x=1$: $y=|1|+2=3$ → $(1,3)$
For $x=-1$: $y=|-1|+2=3$ → $(-1,3)$

Step3: Plot & connect points

Draw a V-shape through $(0,2)$, $(1,3)$, $(-1,3)$

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5) $y=|x+2|$

Step1: Identify vertex

Rewrite as $y=|x-(-2)|$, vertex: $(-2, 0)$

Step2: Find 2 additional points

For $x=0$: $y=|0+2|=2$ → $(0,2)$
For $x=-4$: $y=|-4+2|=2$ → $(-4,2)$

Step3: Plot & connect points

Draw a V-shape through $(-2,0)$, $(0,2)$, $(-4,2)$

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6) $y=|x+1|+3$

Step1: Identify vertex

Rewrite as $y=|x-(-1)|+3$, vertex: $(-1, 3)$

Step2: Find 2 additional points

For $x=0$: $y=|0+1|+3=4$ → $(0,4)$
For $x=-2$: $y=|-2+1|+3=4$ → $(-2,4)$

Step3: Plot & connect points

Draw a V-shape through $(-1,3)$, $(0,4)$, $(-2,4)$

Answer:

1. Graph with vertex at $(2, -4)$, passing through $(0,-2)$ and $(4,-2)$
2. Graph with vertex at $(-1, 0)$, passing through $(0,1)$ and $(-2,1)$
3. Graph with vertex at $(0, 1)$, passing through $(1,2)$ and $(-1,2)$
4. Graph with vertex at $(0, 2)$, passing through $(1,3)$ and $(-1,3)$
5. Graph with vertex at $(-2, 0)$, passing through $(0,2)$ and $(-4,2)$
6. Graph with vertex at $(-1, 3)$, passing through $(0,4)$ and $(-2,4)$