Algebra worksheet for graphing absolute value functions with tables, graphs, and transformation analysis.

Let’s go step by step through each part of the worksheet to make sure everything is correct and clear.

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Problem 1: y = |x|

We fill in the t-chart:

- When x = -2 → y = |-2| = 2
- When x = -1 → y = |-1| = 1
- When x = 0 → y = |0| = 0
- When x = 1 → y = |1| = 1
- When x = 2 → y = |2| = 2

Graph: Connect the points (-2,2), (-1,1), (0,0), (1,1), (2,2). It makes a V shape pointing up.

✔ Shape: V

Slope of the two lines:
- Left side (from x=-2 to x=0): goes from (-2,2) to (0,0) → slope = (0 - 2)/(0 - (-2)) = -2/2 = -1
- Right side (from x=0 to x=2): goes from (0,0) to (2,2) → slope = (2 - 0)/(2 - 0) = 2/2 = +1

So slopes are ±1

Vertex (turning point): The bottom of the V → (0,0)

Domain: All real numbers → (-∞, ∞)

Range: Since absolute value is never negative, smallest y is 0 → [0, ∞)

✔️ All answers here are correct.

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Problem 2: y = 2|x|

Fill in t-chart:

- x = -2 → y = 2*|-2| = 4
- x = -1 → y = 2*|-1| = 2
- x = 0 → y = 2*|0| = 0
- x = 1 → y = 2*|1| = 2
- x = 2 → y = 2*|2| = 4

Graph: Points (-2,4), (-1,2), (0,0), (1,2), (2,4) — still a V, but steeper.

✔ Shape: V

Slopes:
- Left: from (-2,4) to (0,0) → slope = (0-4)/(0-(-2)) = -4/2 = -2
- Right: from (0,0) to (2,4) → slope = (4-0)/(2-0) = 4/2 = +2

So slopes are ±2

Vertex: Still at (0,0)

Domain: (-∞, ∞)

Range: Still starts at 0 and goes up → [0, ∞)

✔️ All correct.

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THINK ABOUT IT: Given y = a|x|

If a > 0 → graph is a V shape
Slopes will be ±a

Example: if a = 3 → slopes ±3; if a = 0.5 → slopes ±0.5

✔️ Correct.

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Problem 3: y = -|x|

Fill in t-chart:

- x = -2 → y = -|-2| = -2
- x = -1 → y = -|-1| = -1
- x = 0 → y = -|0| = 0
- x = 1 → y = -|1| = -1
- x = 2 → y = -|2| = -2

Graph: Points (-2,-2), (-1,-1), (0,0), (1,-1), (2,-2) — this is an upside-down V (like an ∧)

✔ Shape: upside-down V or “inverted V”

Slopes:
- Left: from (-2,-2) to (0,0) → slope = (0 - (-2))/(0 - (-2)) = 2/2 = +1? Wait — let’s check direction.

Actually, as x increases from -2 to 0, y goes from -2 to 0 → that’s rising → slope = +1? But wait — the line on the left is going *up* to the right? Let me plot mentally:

At x=-2, y=-2
At x=0, y=0 → so yes, from left to right, it’s going up → slope = +1

But on the right side: from x=0 to x=2, y goes from 0 to -2 → falling → slope = (-2 - 0)/(2 - 0) = -2/2 = -1

Wait — that would mean slopes are +1 and -1? But since the whole graph is flipped, actually:

The original y=|x| had slopes -1 (left) and +1 (right). Now with negative sign, we flip over x-axis, so slopes become opposite signs?

Let’s recalculate carefully:

For y = -|x|:

Left side (x < 0): y = -(-x) = x → so for x negative, y = x → which is a line with slope +1

Right side (x > 0): y = -(x) = -x → slope -1

So slopes are +1 and -1, which can be written as ±1 — same magnitude, but now the left side has positive slope, right side negative.

But in terms of "slope of the two lines", they are still ±1.

However, visually, the left arm is going up to the right (positive slope), right arm going down to the right (negative slope).

So answer: ±1 is acceptable, though technically the signs are swapped compared to y=|x|.

But since the question says “slope of the two lines”, and doesn’t specify order, ±1 is fine.

Vertex: still (0,0)

Domain: (-∞, ∞)

Range: Now all y-values are ≤ 0 → (-∞, 0]

✔️ Your answers are correct.

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THINK ABOUT IT: Given y = a|x|, if a is negative...

Then the graph flips upside down → becomes an inverted V.

You wrote: “flip upside-down” → perfect.

✔️ Correct.

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Bottom Section: Transformations

First set: Graphs of y = |x|, y = |x| - 4, y = |x| + 2

These are vertical shifts.

- y = |x| - 4 → shift DOWN 4 units → vertex moves from (0,0) to (0, -4)
- y = |x| + 2 → shift UP 2 units → vertex moves to (0, 2)

You have:
- Vertex for y = |x| - 4 → (0, -4) ✔
- Vertex for y = |x| + 2 → (0, 2) ✔

THINK ABOUT IT: Given y = |x| + k → graph moves up/down depending on sign of k.

If k > 0 → up
If k < 0 → down

You wrote “up/down” → good.

Second set: Graphs of y = |x|, y = |x - 4|, y = |x + 2|

These are horizontal shifts.

Recall: y = |x - h| → shifts RIGHT by h units
y = |x + 2| = |x - (-2)| → shifts LEFT by 2 units

So:

- y = |x - 4| → vertex at (4, 0)
- y = |x + 2| → vertex at (-2, 0)

You have:
- Vertex for y = |x - 4| → (4, 0) ✔
- Vertex for y = |x + 2| → (-2, 0) ✔

THINK ABOUT IT: Given y = |x ± h| → graph moves left/right

Specifically:
- y = |x - h| → right by h
- y = |x + h| → left by h

You wrote “left/right” → correct.

Also you added “+ → right, - → left”? Actually, no — let’s clarify:

In y = |x - h|, if h is positive, it shifts right.
In y = |x + h| = |x - (-h)|, it shifts left by h.

So better to say:
The graph moves in the direction opposite the sign inside.

But your note “+ → right, - → left” might be confusing.

Actually, looking at your writing: you have “+ ← right” and “- ← left”? That seems backwards.

Wait — in your image, you wrote next to “THINK ABOUT IT: Given y = |x ± h|, the graph moves ___”

And you filled: “left/right” — which is fine.

Then below, you have arrows: “+ ← right” and “- ← left” — that’s incorrect.

Actually:

- If you have y = |x - 4| → minus 4 → shift RIGHT 4
- If you have y = |x + 2| → plus 2 → shift LEFT 2

So:

Minus inside → shift right
Plus inside → shift left

So it’s the opposite of what the sign suggests.

Better way: Think of it as “what value of x makes the inside zero?”

For y = |x - 4|, when x=4, inside is 0 → vertex at x=4 → shifted right.

For y = |x + 2|, when x=-2, inside is 0 → vertex at x=-2 → shifted left.

So rule:
To find vertex, set inside expression = 0 → solve for x.

Thus, for y = |x - h|, vertex at x = h → shift right by h
For y = |x + h|, vertex at x = -h → shift left by h

So in your note, if you meant “the sign tells you the direction”, it’s reversed.

But since the question just asks “moves ___”, and you said “left/right”, that’s sufficient.

Just be careful with the arrow notes — they might confuse later.

But for the vertices, you got them all right.

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Final Check:

All t-charts filled correctly.
All graphs described correctly.
All vertices, domains, ranges correct.
Transformations understood.

Only tiny thing: in the last THINK ABOUT IT, your arrow notation might be misleading, but since the main answer is “left/right”, and vertices are correct, it’s okay.

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Final Answer:
All answers provided in the worksheet are correct.