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

Ask by Ward Maxwell. in South Africa

Jan 13,2025

Question

Sketch on the same set of axes the graphs of $$ g(x)=-2 \cdot 2^{x-1}+1 $$ Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).

UpStudy AI · Accepted Answer

Certainly! Let's analyze and describe the graph of the function $$ g(x) = -2 \cdot 2^{x-1} + 1 $$. We'll break down its key features, including intercepts, asymptotes, and turning points.

### 1. **Simplifying the Function**

First, let's simplify the given function for easier analysis:

$$
g(x) = -2 \cdot 2^{x-1} + 1 = -2 \cdot \left(\frac{2^x}{2}\right) + 1 = -2^{x} + 1
$$

So, the function simplifies to:

$$
g(x) = -2^x + 1
$$

### 2. **Domain and Range**

- **Domain:** All real numbers ($$ \mathbb{R} $$)
- **Range:** $$ y < 1 $$ (since $$ -2^x $$ can take any negative value, and adding 1 shifts it upward)

### 3. **Intercepts**

#### a. **Y-Intercept**

The y-intercept occurs where $$ x = 0 $$:

$$
g(0) = -2^0 + 1 = -1 + 1 = 0
$$

- **Y-Intercept:** $$ (0, 0) $$

#### b. **X-Intercept**

The x-intercept occurs where $$ g(x) = 0 $$:

$$
-2^x + 1 = 0 \implies 2^x = 1 \implies x = 0
$$

- **X-Intercept:** $$ (0, 0) $$

*Note:* The function intersects both axes at the same point, $$ (0, 0) $$.

### 4. **Asymptote**

The horizontal asymptote is determined by the behavior of the function as $$ x \to -\infty $$:

$$
\lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} (-2^x + 1) = 1
$$

- **Horizontal Asymptote:** $$ y = 1 $$

### 5. **Behavior and Turning Points**

- **Monotonicity:** The function $$ g(x) = -2^x + 1 $$ is **strictly decreasing** for all real $$ x $$. This is because the exponential term $$ -2^x $$ decreases as $$ x $$ increases.
  
- **Turning Points:** Since the function is strictly decreasing and continuous, it has **no turning points**.

### 6. **Additional Points for Sketching**

To better illustrate the graph, let's calculate a few more points:

| $$ x $$ | $$ g(x) = -2^x + 1 $$ |
|---------|-----------------------|
| -2      | $$ -2^{-2} + 1 = -\frac{1}{4} + 1 = \frac{3}{4} $$ |
| -1      | $$ -2^{-1} + 1 = -\frac{1}{2} + 1 = \frac{1}{2} $$ |
| 1       | $$ -2^{1} + 1 = -2 + 1 = -1 $$ |
| 2       | $$ -2^{2} + 1 = -4 + 1 = -3 $$ |
| 3       | $$ -2^{3} + 1 = -8 + 1 = -7 $$ |

### 7. **Graph Description**

- **Starting Point:** As $$ x $$ approaches $$ -\infty $$, $$ g(x) $$ approaches the horizontal asymptote $$ y = 1 $$ from below.
- **Y-Intercept:** The graph passes through $$ (0, 0) $$.
- **Descending Path:** The graph decreases continuously, passing through points like $$ (-1, 0.5) $$, $$ (1, -1) $$, $$ (2, -3) $$, and so on.
- **Behavior at $$ x \to +\infty $$:** As $$ x $$ increases, $$ g(x) $$ decreases without bound, tending toward $$ -\infty $$.

### 8. **Summary of Key Features**

- **Function:** $$ g(x) = -2^x + 1 $$
- **Y-Intercept:** $$ (0, 0) $$
- **X-Intercept:** $$ (0, 0) $$
- **Horizontal Asymptote:** $$ y = 1 $$
- **Turning Points:** None (strictly decreasing)
- **Behavior:** Decreases from $$ y = 1 $$ (as $$ x \to -\infty $$) to $$ y \to -\infty $$ (as $$ x \to +\infty $$).

### 9. **Sketch of the Graph**

While I can't provide a visual sketch here, you can plot the key points and features as follows:

1. **Plot the Horizontal Asymptote:** Draw a dashed horizontal line at $$ y = 1 $$.
2. **Plot the Intercept:** Mark the origin $$ (0, 0) $$.
3. **Plot Additional Points:** Plot points like $$ (-2, 0.75) $$, $$ (-1, 0.5) $$, $$ (1, -1) $$, $$ (2, -3) $$, and $$ (3, -7) $$.
4. **Draw the Curve:** Connect the points with a smooth, continuously decreasing exponential curve that approaches the asymptote $$ y = 1 $$ as $$ x $$ moves to the left and falls steeply as $$ x $$ moves to the right.

This description should help you visualize and sketch the graph accurately.
The graph of $$ g(x) = -2^x + 1 $$ has a y-intercept at $$ (0, 0) $$, a horizontal asymptote at $$ y = 1 $$, and is strictly decreasing with no turning points.

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

Solución de inteligencia artificial de Upstudy

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Solución

Mind Expander

Intercepts:

Asymptote:

Turning Points:

Summary:

Sketch:

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