To identify which graph represents the function [tex]\( y = -(x-2)^3 - 5 \)[/tex], we need to analyze the key characteristics of this cubic function and compute specific values to determine its behavior.
### Step-by-Step Solution
1. Understand the Function Form:
- The function is [tex]\( y = -(x-2)^3 - 5 \)[/tex].
- This is a cubic function that has been shifted and reflected.
2. Points of Interest:
- Let's calculate the y-values at three specific x-values to get a sense of the shape and behavior of the graph around key points.
3. Calculations:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0-2)^3 - 5 \][/tex]
[tex]\[ y = -(-2)^3 - 5 \][/tex]
[tex]\[ y = -(-8) - 5 \][/tex]
[tex]\[ y = 8 - 5 \][/tex]
[tex]\[ y = 3 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2-2)^3 - 5 \][/tex]
[tex]\[ y = -(0)^3 - 5 \][/tex]
[tex]\[ y = 0 - 5 \][/tex]
[tex]\[ y = -5 \][/tex]
- For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -(4-2)^3 - 5 \][/tex]
[tex]\[ y = -(2)^3 - 5 \][/tex]
[tex]\[ y = -(8) - 5 \][/tex]
[tex]\[ y = -8 - 5 \][/tex]
[tex]\[ y = -13 \][/tex]
4. Key Points on the Graph:
- The function passes through the points:
- [tex]\( (0, 3) \)[/tex]
- [tex]\( (2, -5) \)[/tex]
- [tex]\( (4, -13) \)[/tex]
5. Behavior of the Function:
- The graph is a vertically shifted and reflected version of the basic cubic function [tex]\( y = x^3 \)[/tex].
- The negative sign in front of the cubic term indicates the graph is reflected over the x-axis.
- The term [tex]\((x-2)\)[/tex] moves the graph 2 units to the right.
- The [tex]\(-5\)[/tex] at the end moves the graph 5 units down.
6. Plotting the Graph:
- Plot the points [tex]\((0, 3)\)[/tex], [tex]\((2, -5)\)[/tex], and [tex]\((4, -13)\)[/tex] on a coordinate plane.
- Sketch a smooth curve through these points, keeping in mind the nature of cubic functions (they have an S-shape due to their inflection point).
### Conclusion
Look for the graph that:
- Passes through the points [tex]\((0, 3)\)[/tex], [tex]\((2, -5)\)[/tex], and [tex]\((4, -13)\)[/tex].
- Has the overall shape of a cubic function, but reflected about the x-axis.
- Is shifted 2 units to the right and 5 units down.
This detailed analysis should confirm which graph correctly represents [tex]\( y = -(x-2)^3 - 5 \)[/tex] based on the provided key characteristics and points.
