The equation of a hyperbola is given by $$ f(x)=\frac{3}{x-7}-4 $$. Write down the equation of the new function that is lormed when $$ f $$ is transformed as follows: $$ \begin{array}{ll}3.1 & ext { Shift two units to the left } \ 3.2 & ext { Shift } 3 ext { units up } \ 3.3 & ext { Shift I unit right and } 2 ext { units down } \ 3.4 & ext { The equation of the new hyperbola has new asymptotes at } x=-4 ext { and } y=-1\end{array} $$

Question

The equation of a hyperbola is given by $$ f(x)=\frac{3}{x-7}-4 $$. Write down the equation of the new function that is lormed when $$ f $$ is transformed as follows: $$ \begin{array}{ll}3.1 & 	ext { Shift two units to the left } \ 3.2 & 	ext { Shift } 3 	ext { units up } \ 3.3 & 	ext { Shift I unit right and } 2 	ext { units down } \ 3.4 & 	ext { The equation of the new hyperbola has new asymptotes at } x=-4 	ext { and } y=-1\end{array} $$

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Let's analyze the transformations applied to the original hyperbola $$ f(x) = \frac{3}{x - 7} - 4 $$.

### Understanding the Original Function
- **Vertical Asymptote:** $$ x = 7 $$
- **Horizontal Asymptote:** $$ y = -4 $$

### Transformation Rules
1. **Horizontal Shifts:** To shift the graph horizontally, replace $$ x $$ with $$ x \pm h $$:
   - $$ x - h $$ shifts right by $$ h $$ units.
   - $$ x + h $$ shifts left by $$ h $$ units.
   
2. **Vertical Shifts:** To shift the graph vertically, add or subtract a constant:
   - $$ f(x) + k $$ shifts up by $$ k $$ units.
   - $$ f(x) - k $$ shifts down by $$ k $$ units.

Now, let's apply these transformations step-by-step.

---

### 3.1 Shift Two Units to the Left
**Transformation:** Replace $$ x $$ with $$ x + 2 $$.

$$
\begin{align*}
f_{\text{new}}(x) &= \frac{3}{(x + 2) - 7} - 4 \
&= \frac{3}{x - 5} - 4
\end{align*}
$$

**Equation of the New Function:**
$$
y = \frac{3}{x - 5} - 4
$$

---

### 3.2 Shift 3 Units Up
**Transformation:** Add 3 to the original function.

$$
\begin{align*}
f_{\text{new}}(x) &= \frac{3}{x - 7} - 4 + 3 \
&= \frac{3}{x - 7} - 1
\end{align*}
$$

**Equation of the New Function:**
$$
y = \frac{3}{x - 7} - 1
$$

---

### 3.3 Shift 1 Unit Right and 2 Units Down
**Transformation:** 
- Shift right by replacing $$ x $$ with $$ x - 1 $$.
- Shift down by subtracting 2.

$$
\begin{align*}
f_{\text{new}}(x) &= \frac{3}{(x - 1) - 7} - 4 - 2 \
&= \frac{3}{x - 8} - 6
\end{align*}
$$

**Equation of the New Function:**
$$
y = \frac{3}{x - 8} - 6
$$

---

### 3.4 New Hyperbola with Asymptotes at $$ x = -4 $$ and $$ y = -1 $$
**Transformation:**
- To set the vertical asymptote to $$ x = -4 $$, replace $$ x $$ with $$ x + 4 $$.
- To set the horizontal asymptote to $$ y = -1 $$, set the vertical shift to $$ -1 $$.

**Using the same coefficient $$ 3 $$ as in the original function:**
$$
y = \frac{3}{x + 4} - 1
$$

**Equation of the New Function:**
$$
y = \frac{3}{x + 4} - 1
$$

---

### Summary of All Transformations

3.1. **Shift two units to the left:**
$$
y = \frac{3}{x - 5} - 4
$$

3.2. **Shift 3 units up:**
$$
y = \frac{3}{x - 7} - 1
$$

3.3. **Shift 1 unit right and 2 units down:**
$$
y = \frac{3}{x - 8} - 6
$$

3.4. **New hyperbola with asymptotes at $$ x = -4 $$ and $$ y = -1 $$:**
$$
y = \frac{3}{x + 4} - 1
$$
Here are the equations after each transformation: 1. **Shift two units to the left:** $$ y = \frac{3}{x - 5} - 4 $$ 2. **Shift 3 units up:** $$ y = \frac{3}{x - 7} - 1 $$ 3. **Shift 1 unit right and 2 units down:** $$ y = \frac{3}{x - 8} - 6 $$ 4. **New hyperbola with asymptotes at $$ x = -4 $$ and $$ y = -1 $$:** $$ y = \frac{3}{x + 4} - 1 $$

The equation of a hyperbola is given by .
Write down the equation of the new function that is lormed when is transformed as follows:

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Solución

Understanding the Original Function

Transformation Rules

3.1 Shift Two Units to the Left

3.2 Shift 3 Units Up

3.3 Shift 1 Unit Right and 2 Units Down

3.4 New Hyperbola with Asymptotes at and

Summary of All Transformations

Bonus Knowledge

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The equation of a hyperbola is given by . Write down the equation of the new function that is lormed when is transformed as follows: