Sketch on the same set of axes the graphs of and
Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

1. Function

Type of Function

• Quadratic Function: Represents a parabola.
• Direction: Opens downward since the coefficient of is negative.

Key Features

a. Y-Intercept

• Set :
• Y-Intercept:

b. X-Intercepts (Roots)

• Solve :
• Using the quadratic formula:
• X-Intercepts: and

c. Vertex (Turning Point)

• Formula for vertex -coordinate:
• Calculate :
• Vertex:

Summary for :

• Y-Intercept:
• X-Intercepts: and
• Vertex:
• Direction: Opens downward

2. Function

Type of Function

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

Key Features

a. Y-Intercept

• Set :
• Y-Intercept:

b. X-Intercept

• Set :
• No X-Intercept: is always positive, so never crosses the x-axis.

c. Asymptote

• Horizontal Asymptote: (since as , , so )

d. Additional Points for Sketching

Summary for :

• Y-Intercept:
• No X-Intercept
• Horizontal Asymptote:
• Behavior:
  • As : (approaches asymptote from below)
  • As :

3. Combined Sketch

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:

• 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!