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

Ask by Deleon Reid. in South Africa

Jan 13,2025

Question

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).

UpStudy AI · Accepted Answer

Certainly! Let's analyze and describe how to sketch the graphs of the functions $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same set of axes. We'll identify all intercepts, turning points, and asymptotes for each function.

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

**Type of Function:** Quadratic (parabola)  
**Orientation:** Opens downward (since the coefficient of $$ x^2 $$ is negative)

**Key Features:**

- **Vertex (Turning Point):**
  - **Formula:** $$ x = -\frac{b}{2a} $$
  - $$ a = -2 $$, $$ b = -4 $$
  - $$ x = -\frac{-4}{2 \cdot -2} = -1 $$
  - $$ f(-1) = -2(-1)^2 - 4(-1) + 6 = -2 + 4 + 6 = 8 $$
  - **Vertex:** $$ (-1, 8) $$

- **Intercepts:**
  - **Y-intercept:** Set $$ x = 0 $$
    - $$ f(0) = -2(0)^2 - 4(0) + 6 = 6 $$
    - **Point:** $$ (0, 6) $$
  
  - **X-intercepts:** Set $$ f(x) = 0 $$
    $$
    -2x^{2} - 4x + 6 = 0 \
    2x^{2} + 4x - 6 = 0 \quad \text{(Multiply both sides by -1)} \
    x^{2} + 2x - 3 = 0 \
    x = \frac{-2 \pm \sqrt{4 + 12}}{2} = \frac{-2 \pm \sqrt{16}}{2} = \frac{-2 \pm 4}{2}
    $$
    - $$ x = 1 $$ and $$ x = -3 $$
    - **Points:** $$ (1, 0) $$ and $$ (-3, 0) $$

**Sketching $$ f(x) $$:**

1. Plot the vertex at $$ (-1, 8) $$.
2. Plot the intercepts at $$ (0, 6) $$, $$ (1, 0) $$, and $$ (-3, 0) $$.
3. Draw a downward-opening parabola passing through these points.

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

**Type of Function:** Exponential  
**Transformation:** 
- **Base:** $$ 2 $$ (since $$ 2^{x-1} $$)
- **Vertical Stretch/Compression:** Multiplied by $$ -2 $$ (reflects over the x-axis and stretches vertically by a factor of 2)
- **Horizontal Shift:** Right by 1 unit (due to $$ x-1 $$)
- **Vertical Shift:** Up by 1 unit

**Key Features:**

- **Horizontal Asymptote:**
  - $$ y = 1 $$ (since $$ \lim_{x \to \pm\infty} g(x) = 1 $$ from below as the exponential term decays or grows)

- **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:** Same as Y-intercept (since $$ g(0) = 0 $$)
    - **Point:** $$ (0, 0) $$

- **Behavior:**
  - As $$ x \to -\infty $$: $$ 2^{x} \to 0 $$, so $$ g(x) \to 1 $$
  - As $$ x \to \infty $$: $$ 2^{x} \to \infty $$, so $$ g(x) \to -\infty $$
  - **Monotonicity:** Strictly decreasing

**Sketching $$ g(x) $$:**

1. Draw the horizontal asymptote at $$ y = 1 $$.
2. Plot the intercept at $$ (0, 0) $$.
3. Choose additional points for clarity, e.g.,
   - $$ x = 1 $$: $$ g(1) = -2 \cdot 2^{0} + 1 = -2 + 1 = -1 $$ ⇒ $$ (1, -1) $$
   - $$ x = -1 $$: $$ g(-1) = -2 \cdot 2^{-2} + 1 = -2 \cdot \frac{1}{4} + 1 = -0.5 + 1 = 0.5 $$ ⇒ $$ (-1, 0.5) $$
4. Draw a smooth curve passing through these points, approaching $$ y = 1 $$ as $$ x \to -\infty $$ and decreasing towards $$ -\infty $$ as $$ x \to \infty $$.

### 3. Combined Sketch Description

- **Axes:** Draw standard x and y axes.
  
- **Plot $$ f(x) $$:**
  - Vertex at $$ (-1, 8) $$
  - X-intercepts at $$ (-3, 0) $$ and $$ (1, 0) $$
  - Y-intercept at $$ (0, 6) $$
  - Draw a downward-opening parabola through these points.

- **Plot $$ g(x) $$:**
  - Horizontal asymptote at $$ y = 1 $$ (draw a dashed line)
  - Only intercept at $$ (0, 0) $$
  - Additional points at $$ (1, -1) $$ and $$ (-1, 0.5) $$
  - Draw a decreasing exponential curve passing through these points, approaching $$ y = 1 $$ as $$ x \to -\infty $$ and descending without bound as $$ x \to \infty $$.

### 4. Summary of Key Features

| Feature          | $$ f(x) = -2x^{2} - 4x + 6 $$ | $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ |
|------------------|-------------------------------|-----------------------------------|
| **Vertex**       | $$ (-1, 8) $$                  | None                              |
| **Y-intercept**  | $$ (0, 6) $$                   | $$ (0, 0) $$                      |
| **X-intercepts** | $$ (-3, 0) $$ and $$ (1, 0) $$| $$ (0, 0) $$                      |
| **Asymptote**    | None                           | Horizontal at $$ y = 1 $$         |
| **Behavior**     | Downward parabola              | Decreasing exponential            |

### 5. Visual Representation

While I can't provide an actual plot here, below is a rough sketch representation using text:

```
y
|
10|                  *
  |                 / \
  |                /   \
  |               /     \
 8|             *       \
  |            /         \
 6|           *           \
  |          /             \
 4|         /               \
  |        /                 \
 2|       /                   \
  |      /                     \
 0|-----*----------*------------*-----> x
     -3    -1     0     1      2
```

- The asterisks (*) represent key points:
  - For $$ f(x) $$: $$ (-3,0) $$, $$ (-1,8) $$, $$ (0,6) $$, $$ (1,0) $$
  - For $$ g(x) $$: $$ (0,0) $$
- The dashed horizontal line at $$ y = 1 $$ represents the asymptote for $$ g(x) $$.

**Note:** For an accurate and precise graph, it's recommended to use graphing software or graph paper to plot the points and draw the curves based on the described features.
To sketch the graphs of $$ f(x) = -2x^{2} - 4x + 6 $$ and $$ g(x) = -2 \cdot 2^{x-1} + 1 $$ on the same axes: 1. **For $$ f(x) $$:** - **Vertex:** At $$ (-1, 8) $$ - **Intercepts:** $$ (0, 6) $$, $$ (-3, 0) $$, and $$ (1, 0) $$ - **Asymptote:** None - **Behavior:** Downward-opening parabola 2. **For $$ g(x) $$:** - **Intercept:** $$ (0, 0) $$ - **Asymptote:** Horizontal line at $$ y = 1 $$ - **Behavior:** Decreasing exponential curve approaching $$ y = 1 $$ as $$ x \to -\infty $$ and descending without bound as $$ x \to \infty $$ **Combined Sketch:** - Plot the vertex and intercepts for $$ f(x) $$ and the intercept for $$ g(x) $$. - Draw the horizontal asymptote for $$ g(x) $$. - Sketch the parabola for $$ f(x) $$ and the exponential curve for $$ g(x) $$ based on the key features. This will give a clear visual representation of both functions on the same set of axes, highlighting their intercepts, turning points, and asymptotes.

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Graph of

2. Graph of

3. Combined Sketch Description

4. Summary of Key Features

5. Visual Representation

Extra Insights

For :

For :

Sketching:

On the Same Set of Axes:

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Latest Pre Calculus Questions

Feature
Vertex		None
Y-intercept
X-intercepts	and
Asymptote	None	Horizontal at
Behavior	Downward parabola	Decreasing exponential

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

Solución de inteligencia artificial de Upstudy

Responder

Solución

1. Graph of

2. Graph of

3. Combined Sketch Description

4. Summary of Key Features

5. Visual Representation

Extra Insights

For :

For :

Sketching:

On the Same Set of Axes:

preguntas relacionadas

Latest Pre Calculus Questions

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