The graph corresponds to(3, 2)(3, 2)(3, 2)(7, 2)(7, 2)(7, 2)25

$ \begin{cases} f(x)=\lvert x-3| \\ g(x)=2 \end{cases} $

$ \begin{cases} f(x)=\lvert x| \\ g(x)=2 \end{cases} $

$ \begin{cases} f(x)=\lvert x| \\ g(x)=3 \end{cases} $

Graph Analysis: Identifying Coordinates and Points of Interception

Q: How do I find the vertex of an absolute value function from a graph?

Look for the bottom point of the V-shape where the graph changes from going down to going up. This is where the expression inside the absolute value bars equals zero.

Q: Why is the red line g(x) = 2 and not something else?

The red line is horizontal and passes through y = 2 on the coordinate system. Horizontal lines are always constant functions of the form g(x) = c where c is the y-value.

Q: How can I verify these functions are correct?

Substitute the marked points into both functions:At x = 3: $ |3-5| = 2 $ ✓At x = 7: $ |7-5| = 2 $ ✓Both equal g(x) = 2 ✓

Q: What if I chose f(x) = |x-3| instead?

Test it at x = 7: $ |7-3| = 4 $, but the graph shows the point (7,2). Since 4 ≠ 2, this function is incorrect.

Q: Do absolute value functions always make V-shapes?

Yes! The graph of $ f(x) = |x - h| $ always creates a V-shape with the vertex at (h, 0). The absolute value ensures all y-values are non-negative.

Absolute Value Functions with Horizontal Lines

The graph corresponds to

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Understand the problem

The graph corresponds to

Step-by-step solution

We will determine the functions represented by the provided graphs. There are two distinct graphs: a V-shaped graph and a horizontal line. Let's identify each:

1. Observing the V-shaped blue graph, we interpret it as an absolute value function. The shape of such graphs is $f(x) = |x-a|$ , where the vertex occurs at $x = a$ . The vertex sits at the lowest/highest point, most likely along the line $x = 5$ . This suggests $f(x) = |x-5|$ .

2. The horizontal red line is a constant function where $g(x) = c$ is consistent across all values of $x$ . This line crosses at $y = 2$ , hence, $g(x) = 2$ .

Thus, the functions are:

$f(x) = |x-5|$
$g(x) = 2$

Upon evaluation, these equations best fit the graph representations.

The solution to the problem is $f(x) = |x-5|$ and $g(x) = 2$ .

Final Answer

$\begin{cases} f(x)=|x-5| \\ g(x)=2 \end{cases}$

Key Points to Remember

Essential concepts to master this topic

Vertex Identification: The V-shaped graph's lowest point determines the absolute value function
Function Form: $f(x) = |x - 5|$ has vertex at x = 5
Verification: Check that both graphs pass through intersection points (3,2) and (7,2) ✓

Common Mistakes

Avoid these frequent errors

Misidentifying the vertex of absolute value functions
Don't assume the vertex is at x = 3 just because (3,2) is marked = wrong function! The vertex occurs where the graph changes direction from decreasing to increasing. Always locate the bottom point of the V-shape at x = 5.

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 from a graph?

Look for the bottom point of the V-shape where the graph changes from going down to going up. This is where the expression inside the absolute value bars equals zero.

Why is the red line g(x) = 2 and not something else?

The red line is horizontal and passes through y = 2 on the coordinate system. Horizontal lines are always constant functions of the form g(x) = c where c is the y-value.

How can I verify these functions are correct?

Substitute the marked points into both functions:

At x = 3: $|3-5| = 2$ ✓
At x = 7: $|7-5| = 2$ ✓
Both equal g(x) = 2 ✓

What if I chose f(x) = |x-3| instead?

Test it at x = 7: $|7-3| = 4$ , but the graph shows the point (7,2). Since 4 ≠ 2, this function is incorrect.

Do absolute value functions always make V-shapes?

Yes! The graph of $f(x) = |x - h|$ always creates a V-shape with the vertex at (h, 0). The absolute value ensures all y-values are non-negative.

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