QUESTION 3
Whe equation of a hyperbola is given by .
Write down the equation of the new function that is formed when is transformed as follow
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asymptotes at and

Ask by Rodriguez Todd. in South Africa

Dec 28,2024

Question

QUESTION 3
Whe equation of a hyperbola is given by .
Write down the equation of the new function that is formed when is transformed as follow
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asymptotes at and

Ask by Rodriguez Todd. in South Africa

Dec 28,2024

UpStudy AI · Accepted Answer

To determine the equation of the transformed hyperbola and verify the new asymptotes, let's break down the transformations step by step.

### Original Function:
$$ f(x) = \frac{3}{x - 7} - 4 $$

**Asymptotes of the Original Function:**
- **Vertical Asymptote:** $$ x = 7 $$
- **Horizontal Asymptote:** $$ y = -4 $$

### Transformations:

1. **Shift Two Units to the Left:**
   - Replace $$ x $$ with $$ x + 2 $$.
   $$ f_1(x) = \frac{3}{(x + 2) - 7} - 4 = \frac{3}{x - 5} - 4 $$
   
   **New Asymptotes After Shift:**
   - **Vertical:** $$ x = 5 $$
   - **Horizontal:** $$ y = -4 $$

2. **Shift Three Units Up:**
   - Add 3 to the entire function.
   $$ f_2(x) = \frac{3}{x - 7} - 4 + 3 = \frac{3}{x - 7} - 1 $$
   
   **New Asymptotes After Shift:**
   - **Vertical:** $$ x = 7 $$
   - **Horizontal:** $$ y = -1 $$

3. **Shift One Unit Right and Two Units Down:**
   - Replace $$ x $$ with $$ x - 1 $$ (right shift) and subtract 2.
   $$ f_3(x) = \frac{3}{(x - 1) - 7} - 4 - 2 = \frac{3}{x - 8} - 6 $$
   
   **New Asymptotes After Shift:**
   - **Vertical:** $$ x = 8 $$
   - **Horizontal:** $$ y = -6 $$

4. **Final Transformation Resulting in Asymptotes at $$ x = -4 $$ and $$ y = -1 $$:**
   
   To achieve the desired asymptotes:
   - **Vertical Asymptote at $$ x = -4 $$:** Replace $$ x $$ with $$ x + 4 $$.
   - **Horizontal Asymptote at $$ y = -1 $$:** Adjust the constant term to $$-1$$.
   
   $$ f_{\text{final}}(x) = \frac{3}{x + 4} - 1 $$
   
   **Verification of Asymptotes:**
   - **Vertical Asymptote:** $$ x + 4 = 0 \Rightarrow x = -4 $$
   - **Horizontal Asymptote:** $$ y = -1 $$

### Final Equation of the Transformed Hyperbola:
$$ \boxed{y = \frac{3}{\,x\,+\,4\,} - 1} $$
The transformed hyperbola equation is $$ y = \frac{3}{x + 4} - 1 $$ with asymptotes at $$ x = -4 $$ and $$ y = -1 $$.

QUESTION 3
Whe equation of a hyperbola is given by .
Write down the equation of the new function that is formed when is transformed as follow
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asymptotes at and

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QUESTION 3 Whe equation of a hyperbola is given by . Write down the equation of the new function that is formed when is transformed as follow Shift two units to the left Shift 3 units up Shift 1 unit right and 2 units down The equation of the new hyperbola has new asymptotes at and

Upstudy AI Solution

Answer

Solution

Beyond the Answer

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Latest Pre Algebra Questions

QUESTION 3
Whe equation of a hyperbola is given by .
Write down the equation of the new function that is formed when is transformed as follow
Shift two units to the left
Shift 3 units up
Shift 1 unit right and 2 units down
The equation of the new hyperbola has new asymptotes at and