The image shows the
graph of the absolute value function $ f(x) = |x| $. Let's analyze and explain the graph and its properties.
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🔍 Understanding the Graph
The graph is a
V-shaped curve with the vertex at the origin $(0, 0)$.
- For $ x \geq 0 $: $ f(x) = x $
- For $ x < 0 $: $ f(x) = -x $
This means:
- On the
right side (positive $x$), the graph is a straight line with slope $+1$: $ y = x $
- On the
left side (negative $x$), the graph is a straight line with slope $-1$: $ y = -x $
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✔ Key Features of the Graph
1.
Vertex: The lowest point of the graph is at $ (0, 0) $
2.
Symmetry: The graph is symmetric about the
y-axis
3.
Domain: All real numbers $ (-\infty, \infty) $
4.
Range: $ [0, \infty) $ — since absolute value is always non-negative
5.
Increasing/Decreasing:
- Decreasing on $ (-\infty, 0) $
- Increasing on $ (0, \infty) $
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📌 Points Plotted on the Graph
Let’s verify a few points:
| $ x $ | $ f(x) = |x| $ | Point |
|--------|------------------|-------|
| -5 | 5 | (-5, 5)
✔ |
| -4 | 4 | (-4, 4)
✔ |
| -3 | 3 | (-3, 3)
✔ |
| -2 | 2 | (-2, 2)
✔ |
| -1 | 1 | (-1, 1)
✔ |
| 0 | 0 | (0, 0)
✔ |
| 1 | 1 | (1, 1)
✔ |
| 2 | 2 | (2, 2)
✔ |
| 3 | 3 | (3, 3)
✔ |
| 4 | 4 | (4, 4)
✔ |
| 5 | 5 | (5, 5)
✔ |
All points match the definition of $ f(x) = |x| $, so the graph is correct.
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💡 What Can We Learn from This Graph?
- The absolute value function outputs the
distance of a number from zero on the number line.
- It "folds" negative values to positive: $ |-a| = a $
- The graph has a sharp corner at $ x = 0 $, which means it is
not differentiable at that point.
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✔ Conclusion
The graph correctly represents the
absolute value function $ f(x) = |x| $. It shows:
- A V-shape with vertex at $ (0, 0) $
- Symmetry about the y-axis
- Linear behavior with slopes $-1$ and $+1$
- Correctly plotted points
✔ Answer: The graph is accurate and illustrates $ f(x) = |x| $ perfectly.
Let me know if you'd like to transform this function (e.g., shift, stretch) or solve an equation involving it!
Parent Tip: Review the logic above to help your child master the concept of graphing absolute value functions worksheet answers.
