Identifying the Constant Function: Graph Interpretation Challenge

Q: Why does the graph look like a V-shape?

Absolute value functions create V-shapes because they reflect negative outputs to positive. When x 5, it increases away from the vertex.

Q: What's the difference between |x-5| and |x+5|?

$ |x-5| $ has vertex at (5, 0) while $ |x+5| $ has vertex at (-5, 0). The sign inside the absolute value bars determines the horizontal shift direction.

Q: How can I tell which function matches the graph?

Look at the vertex location first! The graph shows vertex at (5, 0), so you need the function where the inside expression equals zero when x = 5. Only $ |x-5| $ works.

Absolute Value Functions with Vertex Identification

The graph corresponds to

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Understand the problem

The graph corresponds to

Step-by-step solution

To solve this problem, we'll conduct the following steps:

Step 1: Identify the vertex of the graph shown.
Step 2: Compare the vertex to each function given in the choices.
Step 3: Select the function whose vertex matches the graph.

Now, let’s work through these steps:

Step 1: The graph shows a vertex at the point (5, 0). This suggests that the graph is the absolute value function centered at $x = 5$ .

Step 2: We compare this with each available function:
- $|x-5|$ : This corresponds to a vertex at $(5, 0)$ .
- $|x-3|$ would give a vertex at $(3, 0)$ .
- $|x|$ would give a vertex at $(0, 0)$ .
- $|5x|$ involves a similar transformation but with varied scaling and zero vertex.

Step 3: Therefore, the function $|x-5|$ perfectly matches the vertex (5, 0) of the graph plotted.

Our analysis confirms that the absolute value function corresponding to the graph is $|x-5|$ .

Final Answer

$|x-5|$

Key Points to Remember

Essential concepts to master this topic

Vertex Location: Identify where the V-shape touches its lowest point
Form Recognition: $|x-h|$ has vertex at (h, 0), so vertex (5, 0) means h = 5
Verification: Check that $|5-5| = 0$ gives the minimum value ✓

Common Mistakes

Avoid these frequent errors

Confusing the sign in the absolute value expression
Don't think $|x+5|$ has vertex at (5, 0) = wrong vertex location! The expression $|x+5|$ actually has vertex at (-5, 0). Always remember that $|x-h|$ has vertex at (h, 0), so subtract to find h.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

How do I find the vertex of an absolute value function?

The vertex is where the expression inside the absolute value bars equals zero. For $|x-5|$ , set x - 5 = 0, so x = 5. The vertex is at (5, 0).

Why does the graph look like a V-shape?

Absolute value functions create V-shapes because they reflect negative outputs to positive. When x < 5, the function decreases toward the vertex. When x > 5, it increases away from the vertex.

What's the difference between |x-5| and |x+5|?

$|x-5|$ has vertex at (5, 0) while $|x+5|$ has vertex at (-5, 0). The sign inside the absolute value bars determines the horizontal shift direction.

How can I tell which function matches the graph?

Look at the vertex location first! The graph shows vertex at (5, 0), so you need the function where the inside expression equals zero when x = 5. Only $|x-5|$ works.

Do I need to check multiple points on the graph?

Not usually! For basic absolute value functions, finding the vertex is enough. You can verify by checking one point on each side: when x = 4, $|4-5| = 1$ ✓

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