Graph the equation shown below by transforming the given graph of the parent function.

y, equals, 4, dot, 2, to the power x

*y*=4⋅2*x*

The equation given is:

\[
y = 4 \cdot 2^x
\]

This is an exponential function of the form \( y = a \cdot b^x \), where \( a = 4 \) and \( b = 2 \). The parent function of this is \( y = 2^x \), and we are applying a vertical transformation to it by multiplying the output by 4.

### Steps to graph this:

1. **Parent Function**: Start with the graph of the parent function \( y = 2^x \). This is an increasing exponential curve that passes through the point (0, 1), since \( 2^0 = 1 \).

2. **Vertical Stretch**: The factor 4 in the equation causes a vertical stretch of the parent function by a factor of 4. This means that every point on the graph of \( y = 2^x \) is multiplied by 4. The new graph passes through the point (0, 4) instead of (0, 1), because \( y = 4 \cdot 2^0 = 4 \).

3. **Asymptote**: The graph will still have a horizontal asymptote at \( y = 0 \), as it never crosses or touches the x-axis.

4. **Other Points**:
   - At \( x = 1 \), the parent function gives \( y = 2 \). So for the transformed function, \( y = 4 \cdot 2 = 8 \).
   - At \( x = -1 \), the parent function gives \( y = \frac{1}{2} \), so the transformed function gives \( y = 4 \cdot \frac{1}{2} = 2 \).

Now that you understand the transformation, would you like to see the graph? I can generate it for you!

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Let me know if you'd like further details or have any questions. Here are 5 related questions:

1. How do you graph transformations of the parent function \( y = a \cdot b^x \)?
2. What happens if we change the base \( b \) in the equation \( y = a \cdot b^x \)?
3. How does a horizontal shift affect the graph of an exponential function?
4. Can exponential functions have a horizontal asymptote other than \( y = 0 \)?
5. How do you solve equations involving exponential functions?

**Tip**: Always check how multiplying the base function by constants impacts the graph—it often involves vertical or horizontal transformations.

Graph the equation shown below by transforming the given graph of the parent function. y = 4⋅2^x

Learn how to graph the exponential function y = 4⋅2^x by transforming the parent function y = 2^x. This problem covers exponential growth, graph transformations, and vertical stretches. Suitable for students in Grades 9-11.

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