Sure, let's work through the steps to graph the function [tex]\( f(x) = -|x - 5| \)[/tex].
### Step-by-Step Solution
1. Understand the Function:
The given function [tex]\( f(x) = -|x - 5| \)[/tex] involves an absolute value and a negative sign. The absolute value function [tex]\( |x - 5| \)[/tex] creates a "V" shape graph centered at [tex]\( x = 5 \)[/tex]. The outer negative sign flips this "V" shape upside down.
2. Identify Key Points:
The absolute value function [tex]\( |x - 5| \)[/tex] has its vertex at [tex]\( x = 5 \)[/tex]. The value of [tex]\( f(x) \)[/tex] at this point is:
[tex]\[
f(5) = -|5 - 5| = 0
\][/tex]
Thus, the vertex of the function [tex]\( f(x) = -|x - 5| \)[/tex] is at [tex]\( (5, 0) \)[/tex].
3. Behavior of the Function:
For [tex]\( x > 5 \)[/tex]:
[tex]\[
f(x) = -|x - 5| = -(x - 5)
\][/tex]
This is a linear function with a negative slope:
[tex]\[
f(x) = -x + 5
\][/tex]
For [tex]\( x < 5 \)[/tex]:
[tex]\[
f(x) = -|x - 5| = -(-(x - 5)) = x - 5
\][/tex]
This is a linear function with a positive slope:
[tex]\[
f(x) = x - 5
\][/tex]
4. Plot Key Points:
Let's plot some points on either side of the vertex [tex]\( (5, 0) \)[/tex].
- For [tex]\( x > 5 \)[/tex] (e.g., [tex]\( x = 6 \)[/tex]):
[tex]\[
f(6) = -|6 - 5| = -(6 - 5) = -1
\][/tex]
Point: [tex]\( (6, -1) \)[/tex]
- For [tex]\( x < 5 \)[/tex] (e.g., [tex]\( x = 4 \)[/tex]):
[tex]\[
f(4) = -|4 - 5| = -(4 - 5) = -(-1) = -1
\][/tex]
Point: [tex]\( (4, -1) \)[/tex]
5. Draw the Rays:
- From the vertex [tex]\( (5, 0) \)[/tex], draw a ray extending to the right with a slope of -1, passing through points like [tex]\( (6, -1) \)[/tex], [tex]\( (7, -2) \)[/tex], etc.
- From the vertex [tex]\( (5, 0) \)[/tex], draw a ray extending to the left with a slope of 1, passing through points like [tex]\( (4, -1) \)[/tex], [tex]\( (3, -2) \)[/tex], etc.
### Final Graph:
- Plot the vertex at [tex]\( (5, 0) \)[/tex].
- Draw the right ray downwards starting from [tex]\( (5, 0) \)[/tex] and passing through points like [tex]\( (6, -1) \)[/tex], [tex]\( (7, -2) \)[/tex], ...
- Draw the left ray downwards starting from [tex]\( (5, 0) \)[/tex] and passing through points like [tex]\( (4, -1) \)[/tex], [tex]\( (3, -2) \)[/tex], ...
This step-by-step process will help you accurately draw the graph of the function [tex]\( f(x) = -|x - 5| \)[/tex].
