To determine the graph of the function [tex]\( y = -(x-2)^3 - 5 \)[/tex], let's analyze it step-by-step:
1. Basic Form:
The basic form is [tex]\( y = (x-2)^3 \)[/tex]. This is a cubic function with a horizontal shift to the right by 2 units. The graph typically looks like a stretched "S" shape through the origin.
However, we have modifications to this basic form.
2. Horizontal Shift:
The function [tex]\( y = (x-2)^3 \)[/tex] shifts the standard cubic graph [tex]\( y = x^3 \)[/tex] to the right by 2 units. Therefore, the inflection point (previously at the origin) moves to the point [tex]\((2, 0)\)[/tex].
3. Vertical Reflection:
The negative sign in front of the cubic function, making it [tex]\( y = -(x-2)^3 \)[/tex], reflects the graph across the x-axis. Essentially, this flips the "S" shape upside down. Now, instead of rising from left to right, it falls from left to right.
4. Vertical Shift:
The term [tex]\(-5\)[/tex] at the end of the function, [tex]\( y = -(x-2)^3 - 5 \)[/tex], vertically shifts the entire graph downwards by 5 units. So the inflection point, which was at [tex]\((2, 0)\)[/tex], moves to the new point [tex]\((2, -5)\)[/tex].
Let's summarize key points to plot the graph:
- The basic shape is a cubic function.
- There is a horizontal shift 2 units to the right.
- There's an upside-down reflection due to the negative sign.
- The graph is shifted downward by 5 units.
### Characteristics of the Graph
- Inflection Point: Located at [tex]\( (2, -5) \)[/tex].
- End Behavior: As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex]; and as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
- General Shape: It will appear as an inverted "S" shape, passing through the inflection point at [tex]\( (2, -5) \)[/tex].
### Plotting Points:
- Start from the inflection point [tex]\((2, -5)\)[/tex].
- As [tex]\( x \)[/tex] increases past 2, [tex]\( y \)[/tex] decreases sharply downward.
- As [tex]\( x \)[/tex] decreases past 2, [tex]\( y \)[/tex] increases sharply upward.
Given this analysis, the graph should look like an upside-down stretched-out "S" that crosses the line [tex]\( y = -5 \)[/tex] when [tex]\( x = 2 \)[/tex].
Here is a rough sketch:
[tex]\[
\begin{array}{c}
\ \ \ \ \ \ | \\
\ \ \ \ \ \ | \\
\ \ \ \ \ \ | \\
\ \ \ \ \ \ | \quad *\ \ \ \ \ \__ \dots \dots \dots \displaystyle{\near{-5}{}} \\
\ \ \ \ \ \ \ \ \ \ * \ | \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ | \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | \
*\ \ \ \ \ \ \ \ \ \ \ \ \ \ | \
(-) \ \ (-) \ \ (-) \ \ \ \ (-8)\ (6) (10)\\
\end{array}
\][/tex]
In this sketch, notice that the graph is a smooth curve passing through the [tex]\( y \)[/tex]-axis, with the key shifting and flipping properties described.
