QUESTION 3
The equation of a hyperbola is given by .
Write down the equation of the new function that is forme
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 asyr
QUESTION 4
Sketch on the same set of axes the graphs of .
Clearly indicate all intercepts with the axes, turning point

Ask by Burton Vargas. in South Africa

Jan 08,2025

Question

QUESTION 3
The equation of a hyperbola is given by .
Write down the equation of the new function that is forme
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 asyr
QUESTION 4
Sketch on the same set of axes the graphs of .
Clearly indicate all intercepts with the axes, turning point

Ask by Burton Vargas. in South Africa

Jan 08,2025

UpStudy AI · Accepted Answer

### **QUESTION 3**

Given the hyperbola $$ f(x) = \frac{3}{x - 7} - 4 $$, we'll apply various transformations as specified.

#### **3.1 Shift Two Units to the Left**

**Transformation Steps:**
- **Horizontal Shift:** Shifting the graph two units to the **left** involves replacing $$ x $$ with $$ x + 2 $$.

**New Function:**
$$
f_{\text{new}}(x) = \frac{3}{(x + 2) - 7} - 4 = \frac{3}{x - 5} - 4
$$

**Final Equation:**
$$
f(x) = \frac{3}{x - 5} - 4
$$

---

#### **3.2 Shift Three Units Up**

**Transformation Steps:**
- **Vertical Shift:** Shifting the graph three units **up** involves adding 3 to the function.

**New Function:**
$$
f_{\text{new}}(x) = \frac{3}{x - 7} - 4 + 3 = \frac{3}{x - 7} - 1
$$

**Final Equation:**
$$
f(x) = \frac{3}{x - 7} - 1
$$

---

#### **3.3 Shift One Unit Right and Two Units Down**

**Transformation Steps:**
- **Horizontal Shift:** Shifting the graph one unit to the **right** involves replacing $$ x $$ with $$ x - 1 $$.
- **Vertical Shift:** Shifting the graph two units **down** involves subtracting 2 from the function.

**New Function:**
$$
f_{\text{new}}(x) = \frac{3}{(x - 1) - 7} - 4 - 2 = \frac{3}{x - 8} - 6
$$

**Final Equation:**
$$
f(x) = \frac{3}{x - 8} - 6
$$

---

#### **3.4 New Asymptotes of the Transformed Hyperbola**

For the transformed hyperbolas obtained in parts 3.1 to 3.3, the asymptotes change based on the transformations applied.

1. **Original Function $$ f(x) = \frac{3}{x - 7} - 4 $$:**
   - **Vertical Asymptote:** $$ x = 7 $$
   - **Horizontal Asymptote:** $$ y = -4 $$

2. **After 3.1 $$ f(x) = \frac{3}{x - 5} - 4 $$:**
   - **Vertical Asymptote:** $$ x = 5 $$
   - **Horizontal Asymptote:** $$ y = -4 $$

3. **After 3.2 $$ f(x) = \frac{3}{x - 7} - 1 $$:**
   - **Vertical Asymptote:** $$ x = 7 $$
   - **Horizontal Asymptote:** $$ y = -1 $$

4. **After 3.3 $$ f(x) = \frac{3}{x - 8} - 6 $$:**
   - **Vertical Asymptote:** $$ x = 8 $$
   - **Horizontal Asymptote:** $$ y = -6 $$

---

### **QUESTION 4**

**Sketching the Graph of $$ f(x) = -2x^{2} $$:**

Since a sketch cannot be drawn here, I'll describe the key features of the graph:

1. **Shape of the Graph:**
   - The equation $$ f(x) = -2x^{2} $$ represents a **downward-opening parabola** because the coefficient of $$ x^{2} $$ is negative.

2. **Vertex (Turning Point):**
   - The vertex of the parabola is at the origin.
   $$
   \text{Vertex: } (0, 0)
   $$

3. **Intercepts:**
   - **Y-intercept:** Set $$ x = 0 $$.
     $$
     f(0) = -2(0)^2 = 0 \Rightarrow (0, 0)
     $$
     The y-intercept is at the origin.

- **X-intercepts:** Set $$ f(x) = 0 $$.
     $$
     -2x^{2} = 0 \Rightarrow x = 0
     $$
     The only x-intercept is also at the origin.

4. **Additional Points:**
   - For $$ x = 1 $$:
     $$
     f(1) = -2(1)^2 = -2 \Rightarrow (1, -2)
     $$
   - For $$ x = -1 $$:
     $$
     f(-1) = -2(-1)^2 = -2 \Rightarrow (-1, -2)
     $$
   - These points help in sketching the symmetry of the parabola about the y-axis.

5. **Symmetry:**
   - The parabola is symmetric with respect to the **y-axis**.

6. **Behavior:**
   - As $$ |x| $$ increases, $$ f(x) $$ decreases without bound due to the negative coefficient.

**Summary of Key Features:**
- **Vertex:** (0, 0)
- **Y-intercept:** (0, 0)
- **X-intercept:** (0, 0)
- **Opening Direction:** Downward
- **Symmetry:** About the y-axis

By plotting these points and understanding the symmetry, one can sketch an accurate graph of the parabola $$ f(x) = -2x^{2} $$.
**QUESTION 3** 1. **Shift Two Units to the Left:** $$ f(x) = \frac{3}{x - 5} - 4 $$ 2. **Shift Three Units Up:** $$ f(x) = \frac{3}{x - 7} - 1 $$ 3. **Shift One Unit Right and Two Units Down:** $$ f(x) = \frac{3}{x - 8} - 6 $$ 4. **Asymptotes After Transformations:** - **After 3.1:** Vertical asymptote at $$ x = 5 $$, horizontal asymptote at $$ y = -4 $$ - **After 3.2:** Vertical asymptote at $$ x = 7 $$, horizontal asymptote at $$ y = -1 $$ - **After 3.3:** Vertical asymptote at $$ x = 8 $$, horizontal asymptote at $$ y = -6 $$ **QUESTION 4** - **Graph of $$ f(x) = -2x^{2} $$:** - **Vertex:** (0, 0) - **Y-intercept:** (0, 0) - **X-intercept:** (0, 0) - **Symmetric about the y-axis** - **Opens downward** - **Additional Points:** (1, -2) and (-1, -2) By plotting these points and understanding the symmetry, the graph of $$ f(x) = -2x^{2} $$ can be accurately sketched.

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Solución

QUESTION 3

3.1 Shift Two Units to the Left

3.2 Shift Three Units Up

3.3 Shift One Unit Right and Two Units Down

3.4 New Asymptotes of the Transformed Hyperbola

QUESTION 4

Beyond the Answer

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