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 formed when $$ f $$ is transformed as follow: $$ 3.1 \quad $$ Shift two units to the left $$ 3.2 \quad $$ Shift 3 units up $$ 3.3 \quad $$ Shift 1 unit right and 2 units down $$ 3.4 \quad $$ The equation of the new hyperbola has new asymptotes at $$ x=-4 $$ and $$ y=-1 $$ QUESTION 4 Sketch on the same set of axes the graphs of $$ f(x)=-2 x^{2}-4 x+6 $$ and $$ g(x)=-2 \cdot 2^{x-1}+1 $$. Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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### Problem 3: Transformations of the Hyperbola $$ f(x) = \frac{3}{x-7} - 4 $$

Let's analyze each transformation step-by-step.

#### 3.1 Shift Two Units to the Left

**Transformation:** Shifting a graph to the **left** by 2 units involves replacing $$ x $$ with $$ x + 2 $$.

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

#### 3.2 Shift Three Units Up

**Transformation:** Shifting a graph **up** by 3 units involves adding 3 to the entire function.

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

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

**Transformation:** 
- Shifting **right** by 1 unit involves replacing $$ x $$ with $$ x - 1 $$.
- Shifting **down** by 2 units involves subtracting 2 from the entire function.

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

#### 3.4 New Hyperbola with Asymptotes at $$ x = -4 $$ and $$ y = -1 $$

**Understanding Asymptotes:**
- The vertical asymptote of $$ f(x) = \frac{a}{x - h} + k $$ is $$ x = h $$.
- The horizontal asymptote is $$ y = k $$.

**Given:**
- Vertical asymptote: $$ x = -4 $$ $$ \Rightarrow h = -4 $$
- Horizontal asymptote: $$ y = -1 $$ $$ \Rightarrow k = -1 $$

Assuming the same coefficient $$ a = 3 $$ as in the original function,

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

---

### Problem 4: Graphing $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$

Since I can't provide an actual sketch, I'll detail the key features of both functions to help you plot them accurately.

#### 4.1 Graph of $$ f(x) = -2x^{2} - 4x + 6 $$ (Quadratic Function)

**a. Standard Form:** 
$$
f(x) = -2x^{2} - 4x + 6
$$
This is a downward-opening parabola because the coefficient of $$ x^{2} $$ is negative.

**b. Intercepts:**

- **y-intercept:** Set $$ x = 0 $$
  $$
  f(0) = -2(0)^2 - 4(0) + 6 = 6
  $$
  Point: $$ (0, 6) $$

- **x-intercepts:** Set $$ f(x) = 0 $$
  $$
  -2x^{2} - 4x + 6 = 0 \
  \Rightarrow x^{2} + 2x - 3 = 0 \quad (\text{Divide by } -2) \
  \Rightarrow x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
  $$
  $$
  x = 1 \quad \text{and} \quad x = -3
  $$
  Points: $$ (1, 0) $$ and $$ (-3, 0) $$

**c. Vertex (Turning Point):**

Use the vertex formula for a parabola $$ ax^{2} + bx + c $$:
$$
x = -\frac{b}{2a} = -\frac{-4}{2 \times -2} = -\frac{-4}{-4} = -1
$$
$$
f(-1) = -2(-1)^2 - 4(-1) + 6 = -2(1) + 4 + 6 = -2 + 4 + 6 = 8
$$
Vertex: $$ (-1, 8) $$

**d. Axis of Symmetry:** $$ x = -1 $$

**e. Summary of Key Points:**
- Vertex: $$ (-1, 8) $$
- y-intercept: $$ (0, 6) $$
- x-intercepts: $$ (-3, 0) $$ and $$ (1, 0) $$
- Axis of Symmetry: $$ x = -1 $$

#### 4.2 Graph of $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ (Exponential Function)

**a. Standard Form:**
$$
g(x) = -2 \cdot 2^{x-1} + 1
$$
This is an exponential decay function reflected over the x-axis due to the negative coefficient.

**b. Intercepts:**

- **y-intercept:** Set $$ x = 0 $$
  $$
  g(0) = -2 \cdot 2^{-1} + 1 = -2 \cdot \frac{1}{2} + 1 = -1 + 1 = 0
  $$
  Point: $$ (0, 0) $$

- **x-intercept:** Set $$ g(x) = 0 $$
  $$
  -2 \cdot 2^{x-1} + 1 = 0 \
  \Rightarrow -2 \cdot 2^{x-1} = -1 \
  \Rightarrow 2^{x-1} = \frac{1}{2} \
  \Rightarrow x - 1 = -1 \quad (\text{since } 2^{-1} = \frac{1}{2}) \
  \Rightarrow x = 0
  $$
  Point: $$ (0, 0) $$ \ (Same as y-intercept)

**c. Asymptote:**
$$
y = 1
$$
The horizontal asymptote is $$ y = 1 $$ because the function approaches this value as $$ x \to \infty $$ or $$ x \to -\infty $$.

**d. Additional Points:**

To better sketch the graph, calculate a few more points:

- **When $$ x = 1 $$:**
  $$
  g(1) = -2 \cdot 2^{0} + 1 = -2 \cdot 1 + 1 = -2 + 1 = -1
  $$
  Point: $$ (1, -1) $$

- **When $$ x = 2 $$:**
  $$
  g(2) = -2 \cdot 2^{1} + 1 = -2 \cdot 2 + 1 = -4 + 1 = -3
  $$
  Point: $$ (2, -3) $$

- **When $$ x = -1 $$:**
  $$
  g(-1) = -2 \cdot 2^{-2} + 1 = -2 \cdot \frac{1}{4} + 1 = -\frac{1}{2} + 1 = \frac{1}{2}
  $$
  Point: $$ (-1, 0.5) $$

**e. Summary of Key Points:**
- y-intercept and x-intercept: $$ (0, 0) $$
- Additional points: $$ (-1, 0.5) $$, $$ (1, -1) $$, $$ (2, -3) $$
- Asymptote: $$ y = 1 $$

#### 4.3 Plotting Guidelines

- **Quadratic Function $$ f(x) $$:**
  - Plot the vertex at $$ (-1, 8) $$.
  - Plot the intercepts at $$ (-3, 0) $$ and $$ (1, 0) $$.
  - Draw the axis of symmetry $$ x = -1 $$.
  - Sketch the downward-opening parabola passing through these points.

- **Exponential Function $$ g(x) $$:**
  - Plot the intercept at $$ (0, 0) $$.
  - Plot additional points $$ (-1, 0.5) $$, $$ (1, -1) $$, and $$ (2, -3) $$.
  - Draw the horizontal asymptote $$ y = 1 $$.
  - Sketch the curve approaching the asymptote as $$ x $$ moves away from the plotted points.

By accurately plotting these key points and understanding their behavior, you can effectively sketch both functions on the same set of axes.
**Problem 3: Transformations of the Hyperbola** 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. **New Hyperbola with Asymptotes at $$ x = -4 $$ and $$ y = -1 $$:** $$ f(x) = \frac{3}{x + 4} - 1 $$ **Problem 4: Graphing $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$** - **Quadratic Function $$ f(x) $$:** - **Intercepts:** $$ (0, 6) $$, $$ (-3, 0) $$, $$ (1, 0) $$ - **Vertex:** $$ (-1, 8) $$ - **Axis of Symmetry:** $$ x = -1 $$ - **Exponential Function $$ g(x) $$:** - **Intercept:** $$ (0, 0) $$ - **Asymptote:** $$ y = 1 $$ - **Additional Points:** $$ (-1, 0.5) $$, $$ (1, -1) $$, $$ (2, -3) $$ **Graphing Guidelines:** - Plot the key points for both functions. - Draw the axis of symmetry and asymptotes. - Sketch the parabola and the exponential curve based on the plotted points and their behaviors.

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