Analyzing the Graph: Understanding the Corresponding Function

Q: How do I know when a graph involves absolute value?

Look for a V-shaped pattern where parts that would normally be below the x-axis are reflected upward. Regular parabolas can dip below the x-axis, but absolute value graphs never go below y = 0!

Q: Why does the graph have a vertex at (0, 3) instead of (0, -3)?

The function $ x^2 - 3 $ would have vertex at (0, -3), but $ |x^2 - 3| $ flips that negative value to positive, creating a vertex at (0, 3).

Q: What happens to the x-intercepts in absolute value functions?

The x-intercepts stay the same! For $ |x^2 - 3| $, when $ x^2 - 3 = 0 $, we get $ x = ±\sqrt{3} $. These points don't change because absolute value of zero is still zero.

Q: How can I tell the difference between |x² - 3| and |x² + 3|?

$ |x^2 + 3| $ is always positive since $ x^2 + 3 \geq 3 $ for all x. It looks like a regular upward parabola shifted up 3 units, with no reflection needed!

Q: Why does the graph look like a W shape?

The W shape comes from reflecting the negative portion of $ x^2 - 3 $ upward. Between $ x = -\sqrt{3} $ and $ x = \sqrt{3} $, the parabola dips below the x-axis, but absolute value flips it up, creating that distinctive double-valley pattern.

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Step-by-step written solution

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1

Understand the problem

The graph corresponds to

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Analyze the graph provided. The graph appears to have a distinct 'V' shape, indicating that at least part of the graph is reflected about the x-axis.
Step 2: Recognize that a standard quadratic function $f(x) = x^2 - 3$ would have a vertex at (0, -3) if unaltered. Any sections of this below the x-axis being reflected upwards suggests the graph of an absolute value function.
Step 3: Apply the absolute value function: When the function is $f(x) = |x^2 - 3|$ , all values of $f(x)$ which are below the x-axis due to $x^2$ terms less than 3 are reflected above the x-axis.

We can eliminate any function that does not involve an absolute value or shift consistent with reflected x-intercepts. Comparing with each choice:

Choice 1 $f(x) = x^2 - 3$ is incorrect because the graph wouldn't show the V-like reflection above the x-axis.
Choice 2 $f(x) = |x^2 - 3|$ matches perfectly since this transformation reflects all negative portions of the $x^2 - 3$ graph above the x-axis as observed.
Choice 3 $f(x) = |x^2 + 3|$ is incorrect as this has no crossing or vertex shifts indicating a reflection starting point below the x-axis.
Choice 4 $f(x) = x^2$ doesn't reflect along y-values shifted down from 3.

Therefore, the function that corresponds to the graph is $f(x) = |x^2 - 3|$ .

3

Final Answer

$f(x)=|x^2-3|$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value reflects negative portions above the x-axis
Technique: When |x² - 3|, values below x-axis flip upward
Check: Vertex at (0,3) and symmetric V-shape confirms |x² - 3| ✓

Common Mistakes

Avoid these frequent errors

Confusing regular parabola with absolute value graph
Don't assume x² - 3 matches the V-shaped graph = missing the reflection! The downward portion below the x-axis would stay negative, not flip up. Always identify the absolute value bars that reflect negative y-values upward.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

How do I know when a graph involves absolute value?

+

Look for a V-shaped pattern where parts that would normally be below the x-axis are reflected upward. Regular parabolas can dip below the x-axis, but absolute value graphs never go below y = 0!

Why does the graph have a vertex at (0, 3) instead of (0, -3)?

+

The function $x^2 - 3$ would have vertex at (0, -3), but $|x^2 - 3|$ flips that negative value to positive, creating a vertex at (0, 3).

What happens to the x-intercepts in absolute value functions?

+

The x-intercepts stay the same! For $|x^2 - 3|$ , when $x^2 - 3 = 0$ , we get $x = ±\sqrt{3}$ . These points don't change because absolute value of zero is still zero.

How can I tell the difference between |x² - 3| and |x² + 3|?

+

$|x^2 + 3|$ is always positive since $x^2 + 3 \geq 3$ for all x. It looks like a regular upward parabola shifted up 3 units, with no reflection needed!

Why does the graph look like a W shape?

+

The W shape comes from reflecting the negative portion of $x^2 - 3$ upward. Between $x = -\sqrt{3}$ and $x = \sqrt{3}$ , the parabola dips below the x-axis, but absolute value flips it up, creating that distinctive double-valley pattern.

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Analyzing the Graph: Understanding the Corresponding Function

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