Question

Graph y equals (4)^x

Answer 1

To graph the function \( y = 4^x \), we can follow these steps:

1. **Understanding the Function**: The function \( y = 4^x \) is an exponential function with a base of 4. Exponential functions of this form will always be positive, and as \( x \) increases, \( y \) increases very rapidly. When \( x \) is negative, \( y \) approaches zero.

2. **Key Points**: Calculate some key points to help plot the graph:
- When \( x = -2 \):
\[
y = 4^{-2} = \frac{1}{4^2} = \frac{1}{16} \approx 0.0625
\]
- When \( x = -1 \):
\[
y = 4^{-1} = \frac{1}{4} = 0.25
\]
- When \( x = 0 \):
\[
y = 4^0 = 1
\]
- When \( x = 1 \):
\[
y = 4^1 = 4
\]
- When \( x = 2 \):
\[
y = 4^2 = 16
\]

3. **Plotting the Points**: You can plot the points \((-2, 0.0625)\), \((-1, 0.25)\), \((0, 1)\), \((1, 4)\), and \((2, 16)\) on a graph.

4. **Draw the Curve**: Connect these points smoothly to form the exponential curve. The curve should start close to zero for large negative \( x \) values, and rise sharply as \( x \) becomes positive.

5. **Behavior of the Graph**:
- The graph approaches the x-axis (but never touches it) as \( x \) approaches negative infinity, reflecting that \( y \) approaches 0.
- The graph increases rapidly as \( x \) increases, reflecting the nature of exponential growth.

In summary, the graph of \( y = 4^x \) is an upward-sloping curve that approaches the x-axis but never touches it, demonstrating the characteristic behavior of exponential functions.