Sketch on the same set of axes the graphs of $$ f(x)=-2 x^{2}-4 x+6 $$ and $$ g(x)=-2 \cdot 2^{x} $$ Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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Sure! Let's analyze and sketch the graphs of the two functions $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x} $$ on the same set of axes. We'll identify their intercepts, turning points, and asymptotes.

## 1. Function $$ f(x) = -2x^{2} - 4x + 6 $$

### **Type of Function**
- **Quadratic Function:** Represents a parabola.
- **Direction:** Opens **downward** since the coefficient of $$ x^2 $$ is negative.

### **Key Features**

#### **a. Y-Intercept**
- Set $$ x = 0 $$:
  $$
  f(0) = -2(0)^2 - 4(0) + 6 = 6
  $$
- **Y-Intercept:** $$ (0, 6) $$

#### **b. X-Intercepts (Roots)**
- Solve $$ f(x) = 0 $$:
  $$
  -2x^{2} - 4x + 6 = 0 \
  2x^{2} + 4x - 6 = 0 \quad \text{(Multiplying both sides by -1)} \
  x^{2} + 2x - 3 = 0
  $$
- Using the quadratic formula:
  $$
  x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
  $$
  - $$ x = 1 $$
  - $$ x = -3 $$
- **X-Intercepts:** $$ (-3, 0) $$ and $$ (1, 0) $$

#### **c. Vertex (Turning Point)**
- Formula for vertex $$ x $$-coordinate: $$ x = -\frac{b}{2a} $$
  $$
  a = -2,\ b = -4 \
  x = -\frac{-4}{2 \times -2} = \frac{4}{-4} = -1
  $$
- Calculate $$ f(-1) $$:
  $$
  f(-1) = -2(-1)^2 - 4(-1) + 6 = -2(1) + 4 + 6 = 8
  $$
- **Vertex:** $$ (-1, 8) $$

### **Summary for $$ f(x) $$:**
- **Y-Intercept:** $$ (0, 6) $$
- **X-Intercepts:** $$ (-3, 0) $$ and $$ (1, 0) $$
- **Vertex:** $$ (-1, 8) $$
- **Direction:** Opens downward

## 2. Function $$ g(x) = -2 \cdot 2^{x} $$

### **Type of Function**
- **Exponential Function:** Represents exponential decay (scaled by a factor of -2).

### **Key Features**

#### **a. Y-Intercept**
- Set $$ x = 0 $$:
  $$
  g(0) = -2 \cdot 2^{0} = -2 \cdot 1 = -2
  $$
- **Y-Intercept:** $$ (0, -2) $$

#### **b. X-Intercept**
- Set $$ g(x) = 0 $$:
  $$
  -2 \cdot 2^{x} = 0 \implies 2^{x} = 0
  $$
- **No X-Intercept:** $$ 2^{x} $$ is always positive, so $$ g(x) $$ never crosses the x-axis.

#### **c. Asymptote**
- **Horizontal Asymptote:** $$ y = 0 $$ (since as $$ x \to -\infty $$, $$ 2^{x} \to 0 $$, so $$ g(x) \to 0 $$)

#### **d. Additional Points for Sketching**
| $$ x $$ | $$ g(x) = -2 \cdot 2^{x} $$ |
|---------|-----------------------------|
| -2      | $$ -2 \cdot \frac{1}{4} = -0.5 $$ |
| -1      | $$ -2 \cdot \frac{1}{2} = -1 $$   |
| 1       | $$ -2 \cdot 2 = -4 $$            |
| 2       | $$ -2 \cdot 4 = -8 $$            |

### **Summary for $$ g(x) $$:**
- **Y-Intercept:** $$ (0, -2) $$
- **No X-Intercept**
- **Horizontal Asymptote:** $$ y = 0 $$
- **Behavior:**
  - As $$ x \to -\infty $$: $$ g(x) \to 0 $$ (approaches asymptote from below)
  - As $$ x \to \infty $$: $$ g(x) \to -\infty $$

## 3. Combined Sketch

Below is a conceptual sketch of both functions on the same axes. Due to text limitations, this is a simplified representation.

```
Y
|
10|             *
  |             *
  |             *
  |             *       f(x) = -2x² -4x +6
8 |             *       (Vertex at (-1,8))
  |             *
6 |*            * 
  | \           *
4 |  \          *
  |   \         *
2 |    \        *
  |     \       *
0 |-----*-------*----------- X
    -3   -1    0    1    2   
        f(x): (-3,0),(1,0); (0,6)
        
Additional Exponential Function g(x):
- Starts near y=0 for large negative x
- Passes through (0,-2), (-1,-1), (1,-4), etc.
- Approaches y=0 as x decreases

For clarity, here's a more detailed description:

- **Parabola $$ f(x) $$**:
  - **Vertex** at (-1, 8)
  - **X-intercepts** at (-3, 0) and (1, 0)
  - **Y-intercept** at (0, 6)
  - Opens downward

- **Exponential Curve $$ g(x) $$**:
  - **Y-intercept** at (0, -2)
  - **No X-intercept**
  - **Horizontal Asymptote** at y = 0
  - Passes through points like (-2, -0.5), (-1, -1), (1, -4), (2, -8)

## 4. Description of Graph Behavior

- **Intersection Points**:
  - Check if the graphs intersect by solving $$ f(x) = g(x) $$:
    $$
    -2x^{2} -4x +6 = -2 \cdot 2^{x}
    $$
    This equation is transcendental and cannot be solved algebraically. Graphically, they may intersect at certain points, but exact solutions require numerical methods.

- **Overall Layout**:
  - The parabola $$ f(x) $$ peaks at (-1,8) and descends on both sides, crossing the y-axis at (0,6) and the x-axis at (-3,0) and (1,0).
  - The exponential function $$ g(x) $$ decreases rapidly as $$ x $$ increases, passing through (0,-2) and moving towards negative infinity, while approaching the horizontal asymptote y = 0 as $$ x $$ decreases.

## 5. Visual Representation

While I can't provide a graphical plot directly, here's a rough ASCII representation to help visualize the relative positions of the two functions:

```
Y
|
10|           
  |             
  |             
  |             
8 |      * (Vertex f)
  |             
6 |* (0,6)        
  |             
4 |               
  |               
2 |               
  |             
0 |----*-----*----*-----*----------- X
    -3    -1    0     1    2   
       (x-intercepts f)
       
Additional Points for g(x):
- At x=0: (0,-2)
- At x=1: (1,-4)
- At x=-1: (-1,-1)
- As x approaches -∞: approaches y=0
```

**Legend:**
- `*` represents key points on $$ f(x) $$
- The exponential curve $$ g(x) $$ would descend from near y=0 (left side), pass through (0,-2), (-1,-1), (1,-4), etc., moving downward as x increases.

For a precise and accurate graph, consider using graphing software or tools like Desmos, GeoGebra, or a graphing calculator. These tools will allow you to input both functions and visualize their intersections, intercepts, and overall behavior clearly.

## 6. Summary of Features

| Feature           | $$ f(x) = -2x^{2} -4x +6 $$           | $$ g(x) = -2 \cdot 2^{x} $$    |
|-------------------|--------------------------------------|--------------------------------|
| **Type**          | Quadratic (Parabola)                 | Exponential                    |
| **Y-Intercept**   | (0, 6)                               | (0, -2)                         |
| **X-Intercepts**  | (-3, 0) and (1, 0)                   | None                           |
| **Vertex**        | (-1, 8)                              | N/A                            |
| **Asymptote**     | None                                 | Horizontal asymptote at y = 0   |
| **Direction**     | Opens downward                       | Decreasing (approaches -∞)      |

Feel free to use graphing tools to get a precise visualization based on these key points and descriptions!
To sketch the graphs of $$ f(x) = -2x^{2} -4x +6 $$ and $$ g(x) = -2 \cdot 2^{x} $$ on the same axes: 1. **Function $$ f(x) $$:** - **Y-Intercept:** (0, 6) - **X-Intercepts:** (-3, 0) and (1, 0) - **Vertex (Turning Point):** (-1, 8) - **Direction:** Opens downward 2. **Function $$ g(x) $$:** - **Y-Intercept:** (0, -2) - **No X-Intercept** - **Horizontal Asymptote:** $$ y = 0 $$ - **Behavior:** Decreases as $$ x $$ increases, approaching $$ y = 0 $$ as $$ x $$ decreases **Key Points to Plot:** - For $$ f(x) $$: Plot the vertex at (-1,8), intercepts at (-3,0) and (1,0), and the y-intercept at (0,6). - For $$ g(x) $$: Plot the y-intercept at (0,-2) and additional points like (-2, -0.5), (-1, -1), (1, -4), etc. **Intersection Points:** Solve $$ -2x^{2} -4x +6 = -2 \cdot 2^{x} $$ to find where the graphs intersect. This may require numerical methods for exact solutions. **Asymptotes and Behavior:** - $$ f(x) $$ is a downward-opening parabola with a maximum point at the vertex. - $$ g(x) $$ is an exponential decay function approaching the horizontal asymptote $$ y = 0 $$ as $$ x $$ increases. Use graphing tools for a precise visualization of these features.

Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x} \) Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

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