Answer
To sketch the graph of \( y = (x + 1)(x - 4)^3(x - 2)^2 \), plot the x-intercepts at \( x = -1 \), \( x = 2 \), and \( x = 4 \). The graph crosses the x-axis at \( x = -1 \) and \( x = 4 \), and touches it at \( x = 2 \). The y-intercept is at \( (0, -256) \). The graph starts from \( +\infty \) as \( x \to -\infty \), crosses at \( x = -1 \), passes through \( (0, -256) \), touches at \( x = 2 \), crosses at \( x = 4 \), and ends at \( +\infty \) as \( x \to +\infty \).
Solution
To sketch the graph of the function \( y = (x + 1)(x - 4)^3(x - 2)^2 \), follow these steps:
### 1. **Identify the Zeros and Their Multiplicities**
- **\( x = -1 \)**:
- **Multiplicity:** 1 (odd)
- **Behavior:** The graph crosses the x-axis at this point.
- **\( x = 2 \)**:
- **Multiplicity:** 2 (even)
- **Behavior:** The graph touches the x-axis and bounces off (does not cross).
- **\( x = 4 \)**:
- **Multiplicity:** 3 (odd)
- **Behavior:** The graph crosses the x-axis at this point.
### 2. **Determine the End Behavior**
- **Degree of the Polynomial:** \( 1 + 3 + 2 = 6 \) (even)
- **Leading Coefficient:** Positive (since all factors have positive leading coefficients when expanded)
- **Behavior:**
- As \( x \to -\infty \), \( y \to +\infty \).
- As \( x \to +\infty \), \( y \to +\infty \).
### 3. **Plot Key Points**
- **Y-intercept:**
- Set \( x = 0 \):
\[
y = (0 + 1)(0 - 4)^3(0 - 2)^2 = 1 \times (-64) \times 4 = -256
\]
- Point: \( (0, -256) \)
- **X-intercepts:**
- \( (-1, 0) \)
- \( (2, 0) \)
- \( (4, 0) \)
### 4. **Analyze the Behavior Around Zeros**
- **At \( x = -1 \):**
- The graph crosses from positive to negative because the multiplicity is odd.
- **At \( x = 2 \):**
- The graph touches the x-axis and turns around (does not cross) because the multiplicity is even.
- **At \( x = 4 \):**
- The graph crosses the x-axis, similar to \( x = -1 \).
### 5. **Sketch the Graph**
- **Starting from \( x \to -\infty \):**
- The graph comes from \( +\infty \).
- **Crosses at \( x = -1 \):**
- Moves from positive to negative.
- **Passes through \( (0, -256) \):**
- Shows a deep dip due to the y-intercept.
- **Touches and bounces at \( x = 2 \):**
- The graph remains non-positive around this point.
- **Crosses at \( x = 4 \):**
- Moves from negative to positive.
- **Ends at \( x \to +\infty \):**
- The graph heads to \( +\infty \).
### 6. **Final Graph Features**
- **Symmetry:** None.
- **Turning Points:** At least two between the zeros due to the multiplicities.
- **Shape:** Based on the multiplicities and end behavior, the graph will have a "W" shape with specific behaviors at each zero.
### **Graph Overview:**
1. **From left to right:**
- Starts high (\( +\infty \)), decreases, crosses the x-axis at \( x = -1 \).
2. **Between \( x = -1 \) and \( x = 2 \):**
- Continues decreasing to the y-intercept at \( (0, -256) \).
3. **At \( x = 2 \):**
- Bounces off the x-axis without crossing.
4. **Between \( x = 2 \) and \( x = 4 \):**
- Decreases to a local minimum.
5. **At \( x = 4 \):**
- Crosses the x-axis and rises towards \( +\infty \).
### **Sample Sketch:**
While I can't provide a visual sketch here, you can plot the points and behaviors described to create an accurate graph. Here's a rough outline to guide you:
```
y
|
| *
| *
| *
| * *
| | |
|--------*--------*--------*-------- x
-1 2 4
```
- The asterisks (*) represent the curve passing through or bouncing at the x-intercepts.
- The exact curvature near each intercept should follow the multiplicity rules explained above.
**Remember:** The critical points and exact curvature can be further refined by taking derivatives, but for a sketch (especially worth 4 marks), the above analysis should suffice.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
