Question
Age of x equals 4 to the X power graph
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GPT-4o mini
The graph of the function \( y = 4^x \) is an exponential function and has several key characteristics:
1. **Y-Intercept**: At \( x = 0 \), the value of \( y \) is \( 4^0 = 1 \). So the graph passes through the point (0, 1).
2. **Asymptote**: The graph approaches, but never actually touches, the horizontal line \( y = 0 \) as \( x \) approaches negative infinity. This means there is a horizontal asymptote at \( y = 0 \).
3. **Growth**: The function is always positive for all real values of \( x \) and increases rapidly as \( x \) becomes larger. This is typical of exponential growth.
4. **Behavior**:
- For negative values of \( x \), the graph decreases towards the asymptote but remains above the x-axis.
- For positive values of \( x \), the graph rises steeply to the right.
5. **Domain and Range**:
- **Domain**: All real numbers \( (-\infty, \infty) \)
- **Range**: All positive real numbers \( (0, \infty) \)
To visualize the graph of \( y = 4^x \):
- The graph starts at (0, 1), approaches the x-axis as \( x \) decreases, and rises steeply as \( x \) increases.
If you were to sketch it, the curve would start at (0, 1), fall towards the x-axis as \( x \) goes negative, and rise sharply as \( x \) goes positive.
If you need specific values or further analysis of the function, feel free to ask!
1. **Y-Intercept**: At \( x = 0 \), the value of \( y \) is \( 4^0 = 1 \). So the graph passes through the point (0, 1).
2. **Asymptote**: The graph approaches, but never actually touches, the horizontal line \( y = 0 \) as \( x \) approaches negative infinity. This means there is a horizontal asymptote at \( y = 0 \).
3. **Growth**: The function is always positive for all real values of \( x \) and increases rapidly as \( x \) becomes larger. This is typical of exponential growth.
4. **Behavior**:
- For negative values of \( x \), the graph decreases towards the asymptote but remains above the x-axis.
- For positive values of \( x \), the graph rises steeply to the right.
5. **Domain and Range**:
- **Domain**: All real numbers \( (-\infty, \infty) \)
- **Range**: All positive real numbers \( (0, \infty) \)
To visualize the graph of \( y = 4^x \):
- The graph starts at (0, 1), approaches the x-axis as \( x \) decreases, and rises steeply as \( x \) increases.
If you were to sketch it, the curve would start at (0, 1), fall towards the x-axis as \( x \) goes negative, and rise sharply as \( x \) goes positive.
If you need specific values or further analysis of the function, feel free to ask!
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