Sure, let's graph the function [tex]\( f(x) = -|x-3| - 6 \)[/tex] step by step.
1. Understand the function:
- The function [tex]\( f(x) = -|x-3| - 6 \)[/tex] combines an absolute value function [tex]\( |x-3| \)[/tex] with a vertical asymptote.
- Recall that [tex]\( |x-3| \)[/tex] shifts the standard absolute value [tex]\( |x| \)[/tex] to the right by 3 units.
- The negative sign in front of the absolute value reflects the function across the x-axis.
- The -6 at the end translates the entire function down by 6 units.
2. Identify key points and vertex:
- The vertex of [tex]\( |x-3| \)[/tex] is at [tex]\( x=3 \)[/tex].
- Thus, the vertex of [tex]\( f(x) = -|x-3| - 6 \)[/tex] is at [tex]\( (3, -6) \)[/tex].
3. Evaluate the function at a few points:
- When [tex]\( x = 3 \)[/tex]:
[tex]\[
f(3) = -|3-3| - 6 = 0 - 6 = -6
\][/tex]
So we have a point [tex]\( (3, -6) \)[/tex].
- When [tex]\( x = 4 \)[/tex]:
[tex]\[
f(4) = -|4-3| - 6 = -1 - 6 = -7
\][/tex]
So we have a point [tex]\( (4, -7) \)[/tex].
- When [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = -|2-3| - 6 = -1 - 6 = -7
\][/tex]
So we have a point [tex]\( (2, -7) \)[/tex].
4. Draw the rays:
- The absolute value function [tex]\( |x-3| \)[/tex] creates a V shape. Reflecting this and shifting, it appears as an upside-down V, starting at [tex]\( (3, -6) \)[/tex] and opening downwards.
5. Graph the left ray:
- Start at the vertex [tex]\( (3, -6) \)[/tex].
- The ray that goes leftwards would have points like [tex]\( (2, -7) \)[/tex].
6. Graph the right ray:
- Again, start at the vertex [tex]\( (3, -6) \)[/tex].
- The ray that goes rightwards would have points like [tex]\( (4, -7) \)[/tex].
Putting it all together, the graph of [tex]\( f(x) = -|x-3| - 6 \)[/tex] is a V shape opening downwards with its vertex at [tex]\( (3, -6) \)[/tex].
Here is a step-by-step description of how you would draw the graph:
1. Plot the vertex at [tex]\( (3, -6) \)[/tex].
2. From the vertex, draw a line going left downwards through points such as [tex]\( (2, -7) \)[/tex].
3. From the vertex, draw a line going right downwards through points such as [tex]\( (4, -7) \)[/tex].
4. Ensure that the function continues to follow the V-shaped nature of the absolute value function reflected across the x-axis and shifted downwards.
And that gives you the graph of the function [tex]\( f(x) = -|x-3| - 6 \)[/tex].
