https://file.aiquickdraw.com/mathbot/file/1727301258735f027en8r.jpg

We are given the curve equation \( y = x^3 - 3x^2 - 9x + 7 \), and it's mentioned that there are two turning points: one at \( x = 3 \) and another at \( (-1, 12) \). We are tasked with selecting the graph that matches this behavior.

### Let's analyze the equation:

1. **Degree of the Polynomial:**
   - The equation is cubic \( y = x^3 - 3x^2 - 9x + 7 \), meaning the graph will have an "S" shape (inflection point).
   
2. **Turning Points:**
   - The first turning point is given at \( x = 3 \). To determine if it's a local maximum or minimum, we can evaluate the second derivative, but we already know from the problem that this is a turning point.
   - The second turning point is \( (-1, 12) \). This means that when \( x = -1 \), \( y = 12 \), and this should show up in the graph with an upward peak or valley near \( x = -1 \).
   
3. **End Behavior:**
   - Since the highest-degree term is \( x^3 \), as \( x \to \infty \), \( y \to \infty \), and as \( x \to -\infty \), \( y \to -\infty \).

### Analyzing the options:

- **Option A:** This curve has a correct "S" shape, but the turning points are not consistent with \( (-1, 12) \) and \( x = 3 \).
  
- **Option B:** This curve shows a turning point near \( x = 3 \) and another near \( (-1, 12) \), matching the given conditions. The end behavior also matches, with the curve going upwards as \( x \to \infty \) and downwards as \( x \to -\infty \).
  
- **Option C, D, and E:** These options do not match the behavior at \( x = 3 \) or do not have turning points at \( (-1, 12) \).

### Conclusion:
The correct graph is **Option B**, as it satisfies both the turning point locations and the end behavior.

Would you like further details on this analysis or have any other questions? 

Here are 5 related questions to expand your understanding:
1. How do you find the turning points of a cubic function algebraically?
2. What role does the second derivative play in identifying turning points?
3. How does the leading term of a polynomial affect the shape of its graph?
4. What is the significance of the first derivative in understanding the behavior of the curve?
5. How can the inflection point of a cubic function be determined?

**Tip:** For cubic functions, always check the end behavior based on the leading term. This helps in quickly eliminating incorrect graphs.

The curve y = x^3 - 3x^2 - 9x + 7 has two turning points, one of which is where x = 3, and the other of which is at (-1, 12). Sketch the curve.

Analyze the cubic function y = x^3 - 3x^2 - 9x + 7, determine its turning points, and sketch the graph. Includes key concepts like cubic functions, turning points at (3, y) and (-1, 12), and end behavior. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation