VIPSolutionsgive answer in 2 step with explanation at the end of each step and final answer at the end: sketch the following. show appropriate work, including end behaviourv 10 2x-4 b === 5 marks ) py [5 marks] 8 6 4 2 -10.-8 6-4-2 2 4 5 810 2 in hd 8 -10
Question:
give answer in 2 step with explanation at the end of each step and final answer at the end:
sketch the following. show appropriate work, including end behaviour
v
10
2x-4
b === 5 marks
) py [5 marks] 8
6
4
2
-10.-8 6-4-2 2 4 5 810
2
in
hd
8
-10
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Asked by: Pagal
Created at: 2025-06-17 17:59:21
Asked by: Pagal
Created at: 2025-06-17 17:59:21
Final Answer:
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Let's answer this in **2 steps with explanations**:
---
### **Step 1: Analyze the key features (asymptotes, intercepts, end behavior)**
**a) Vertical Asymptote:**
Set the denominator equal to zero:
\( x + 3 = 0 \implies x = -3 \)
So, there is a **vertical asymptote** at \( x = -3 \).
**b) Horizontal Asymptote (End Behavior):**
For large \(|x|\), look at the leading coefficients:
\[
y = \frac{2x - 4}{x + 3} \approx \frac{2x}{x} = 2
\]
So, the **horizontal asymptote** is \( y = 2 \).
**c) x-intercept:**
Set \( y = 0 \):
\[
0 = \frac{2x - 4}{x + 3} \implies 2x - 4 = 0 \implies x = 2
\]
So, the **x-intercept** is at \( (2, 0) \).
**d) y-intercept:**
Set \( x = 0 \):
\[
y = \frac{2(0) - 4}{0 + 3} = \frac{-4}{3}
\]
So, the **y-intercept** is at \( (0, -\frac{4}{3}) \).
**Explanation:**
This step helps us identify the main features needed to sketch the rational function: the vertical and horizontal asymptotes, and the intercepts.
---
### **Step 2: Sketch the function using these features**
1. **Draw the vertical asymptote** at \( x = -3 \) (dashed vertical line).
2. **Draw the horizontal asymptote** at \( y = 2 \) (dashed horizontal line).
3. **Mark the intercepts:**
- \( x \)-intercept at \( (2, 0) \)
- \( y \)-intercept at \( (0, -\frac{4}{3}) \)
4. **Sketch the curve:**
- To the left of \( x = -3 \), the function approaches \( y = 2 \) as \( x \to -\infty \) and drops to \( -\infty \) as \( x \to -3^- \).
- To the right of \( x = -3 \), the function rises from \( +\infty \) as \( x \to -3^+ \) and approaches \( y = 2 \) as \( x \to +\infty \).
- The curve passes through the intercepts found.
**Explanation:**
This step puts together all the information and shows how the graph behaves near the asymptotes and intercepts.
---
## **Final Answer:**
- **Vertical asymptote:** \( x = -3 \)
- **Horizontal asymptote:** \( y = 2 \)
- **x-intercept:** \( (2, 0) \)
- **y-intercept:** \( (0, -\frac{4}{3}) \)
**The sketch should show:**
- Dashed lines at \( x = -3 \) and \( y = 2 \)
- The curve passing through \( (2, 0) \) and \( (0, -\frac{4}{3}) \)
- Correct end behavior as described above
Below is a quick sketch for visualization:
```
|
10 -| . . . .
| /
8 -| . . . /
| /
6 -| . . /
| /
4 -| . . /
| /
2 -|------//---o-----/-------> (asymptote y=2)
| . / /
0 -| . / o (2,0)
| / /
-2 -| / /
| / /
-4 -| o (0,-4/3)
| /
-6 -| /
| /
-10 -|/ . . . . . . . . . . . . .
-10 -8 -6 -4 -2 0 2 4 6 8 10
(asymptote at x=-3)
```
(For a neater sketch, plot the points and asymptotes on the provided grid.)
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