Question
How to graph y = 2^x and describe its behavior as x increases
Answers
GPT-4o mini
To graph the function \( y = 2^x \) and describe its behavior as \( x \) increases, follow these steps:
### Steps to Graph \( y = 2^x \):
1. **Create a Table of Values**: Calculate \( y \) for several values of \( x \):
- For \( x = -3 \): \( y = 2^{-3} = \frac{1}{8} = 0.125 \)
- For \( x = -2 \): \( y = 2^{-2} = \frac{1}{4} = 0.25 \)
- For \( x = -1 \): \( y = 2^{-1} = \frac{1}{2} = 0.5 \)
- For \( x = 0 \): \( y = 2^0 = 1 \)
- For \( x = 1 \): \( y = 2^1 = 2 \)
- For \( x = 2 \): \( y = 2^2 = 4 \)
- For \( x = 3 \): \( y = 2^3 = 8 \)
This gives us the points:
\[
(-3, 0.125), (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8)
\]
2. **Plot the Points**: On a Cartesian plane, plot these points.
3. **Draw the Curve**: Connect the points with a smooth, continuous curve that reflects the exponential growth.
### Description of the Behavior as \( x \) Increases:
- **Initial Behavior**: For negative values of \( x \), \( y = 2^x \) approaches zero but never actually reaches zero. This indicates that the function has a horizontal asymptote at \( y = 0 \).
- **Growth**: As \( x \) increases, \( y \) increases rapidly. The function doubles for each increase of 1 in \( x \) (e.g., \( y(1) = 2 \) and \( y(2) = 4 \) shows that \( y \) doubles).
- **Exponential Growth**: The growth is considered exponential, meaning that as \( x \) continues to increase, \( y \) will grow without bounds. There is no upper limit to how large \( y \) can grow as \( x \) becomes very large.
- **Domain and Range**: The domain of \( y = 2^x \) is all real numbers \( (-\infty, \infty) \). The range is \( (0, \infty) \) since \( y \) is always positive.
### Summary:
The graph of \( y = 2^x \) reflects exponential growth starting very close to the x-axis (but always above it) for negative \( x \) and rising steeply for positive \( x \). The curve approaches the x-axis as a horizontal asymptote and increases rapidly without bound as \( x \) increases.
### Steps to Graph \( y = 2^x \):
1. **Create a Table of Values**: Calculate \( y \) for several values of \( x \):
- For \( x = -3 \): \( y = 2^{-3} = \frac{1}{8} = 0.125 \)
- For \( x = -2 \): \( y = 2^{-2} = \frac{1}{4} = 0.25 \)
- For \( x = -1 \): \( y = 2^{-1} = \frac{1}{2} = 0.5 \)
- For \( x = 0 \): \( y = 2^0 = 1 \)
- For \( x = 1 \): \( y = 2^1 = 2 \)
- For \( x = 2 \): \( y = 2^2 = 4 \)
- For \( x = 3 \): \( y = 2^3 = 8 \)
This gives us the points:
\[
(-3, 0.125), (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), (2, 4), (3, 8)
\]
2. **Plot the Points**: On a Cartesian plane, plot these points.
3. **Draw the Curve**: Connect the points with a smooth, continuous curve that reflects the exponential growth.
### Description of the Behavior as \( x \) Increases:
- **Initial Behavior**: For negative values of \( x \), \( y = 2^x \) approaches zero but never actually reaches zero. This indicates that the function has a horizontal asymptote at \( y = 0 \).
- **Growth**: As \( x \) increases, \( y \) increases rapidly. The function doubles for each increase of 1 in \( x \) (e.g., \( y(1) = 2 \) and \( y(2) = 4 \) shows that \( y \) doubles).
- **Exponential Growth**: The growth is considered exponential, meaning that as \( x \) continues to increase, \( y \) will grow without bounds. There is no upper limit to how large \( y \) can grow as \( x \) becomes very large.
- **Domain and Range**: The domain of \( y = 2^x \) is all real numbers \( (-\infty, \infty) \). The range is \( (0, \infty) \) since \( y \) is always positive.
### Summary:
The graph of \( y = 2^x \) reflects exponential growth starting very close to the x-axis (but always above it) for negative \( x \) and rising steeply for positive \( x \). The curve approaches the x-axis as a horizontal asymptote and increases rapidly without bound as \( x \) increases.