The graph corresponds to333-6

$ \begin{cases} f(x)= |x-3| \\ g(x)=6 \end{cases} $

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

Deciphering the Graph: Identifying Horizontal Line and Key Point at (994, 431)

Q: How do I find the absolute value function from the vertex?

Look at the vertex coordinates! If the vertex is at $(-6, 0)$, then the inside expression equals zero when $x = -6$. This gives us $x + 6 = 0$, so the function is $f(x) = |x + 6|$.

Q: Why is the horizontal line g(x) = 3 and not y = 3?

Both are correct! $g(x) = 3$ is function notation showing that for any input x, the output is always 3. The line $y = 3$ is the same thing written differently.

Q: What does the point (994, 431) represent?

This appears to be where the two functions intersect! At $x = 994$, both $f(x) = |x + 6|$ and $g(x) = 3$ have the same y-value, creating an intersection point.

Q: Why does the graph show two intersection points?

Absolute value functions create V-shapes that can intersect horizontal lines at two points (one on each side of the vertex). This is normal behavior when the horizontal line is above the vertex.

Absolute Value Functions with Horizontal Line Intersections

The graph corresponds to

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Understand the problem

The graph corresponds to

Step-by-step solution

First, we identify the absolute value function associated with the V-shaped graph. The vertex of this V-shape is at the point $(-6, 0)$ , indicating an expression of $f(x) = |x + 6|$ .

Next, observe the red horizontal line, which consistently passes through $y = 3$ . This confirms that the horizontal line function is $g(x) = 3$ .

Having identified both functions, we conclude that they correspond to:

$\begin{cases} f(x)= |x+6| \\ g(x)=3 \end{cases}$

Final Answer

$\begin{cases} f(x)= |x+6| \\ g(x)=3 \end{cases}$

Key Points to Remember

Essential concepts to master this topic

Vertex identification: V-shaped graph vertex determines absolute value expression inside
Horizontal line: Red line at y = 3 represents constant function g(x) = 3
Verification: Check vertex (-6, 0) gives |(-6) + 6| = |0| = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing vertex location with absolute value expression
Don't write |x - 6| when vertex is at (-6, 0) = wrong function! The vertex at (-6, 0) means the expression inside equals zero when x = -6. Always use |x + 6| when vertex is at (-6, 0).

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 absolute value function from the vertex?

Look at the vertex coordinates! If the vertex is at $(-6, 0)$ , then the inside expression equals zero when $x = -6$ . This gives us $x + 6 = 0$ , so the function is $f(x) = |x + 6|$ .

Why is the horizontal line g(x) = 3 and not y = 3?

Both are correct! $g(x) = 3$ is function notation showing that for any input x, the output is always 3. The line $y = 3$ is the same thing written differently.

What does the point (994, 431) represent?

This appears to be where the two functions intersect! At $x = 994$ , both $f(x) = |x + 6|$ and $g(x) = 3$ have the same y-value, creating an intersection point.

How can I tell if my absolute value function is correct?

Check the vertex! Substitute the x-coordinate of the vertex into your function. For $f(x) = |x + 6|$ , when $x = -6$ , we get $f(-6) = |-6 + 6| = |0| = 0$ . Perfect! ✓

Why does the graph show two intersection points?

Absolute value functions create V-shapes that can intersect horizontal lines at two points (one on each side of the vertex). This is normal behavior when the horizontal line is above the vertex.

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