https://file.aiquickdraw.com/mathbot/file/1731088417380qij92q08.jpg

\begin{array}{l}\text{Match the graph of the function with the function rule.}\\

y=2 \cdot 4^{x}\\

y=\frac{1}{2} \bullet 4^{x}\\

y=\frac{1}{4} \bullet 4^{x}\\

y=1 \bullet 4^{x}\end{array}

To determine which function rule matches the graph provided, let’s analyze the graph's characteristics:

The graph shows an exponential growth function that:

1. Starts near zero as \( x \) approaches negative values.
2. Increases rapidly as \( x \) becomes positive.

Each function provided is of the form \( y = c \cdot 4^x \), where \( 4^x \) is the base exponential function. The value of \( c \) affects the vertical stretch or compression of the graph.

### Analysis of Options
1. **\( y = 2 \cdot 4^x \)**: This would result in a function that grows relatively quickly, doubling the base \( 4^x \).
2. **\( y = \frac{1}{2} \cdot 4^x \)**: This function would grow, but at a slower rate compared to \( y = 4^x \) due to the \( \frac{1}{2} \) factor.
3. **\( y = \frac{1}{4} \cdot 4^x \)**: This function would also show exponential growth but at an even slower rate.
4. **\( y = 4^x \)**: This represents pure exponential growth without any scaling factor, making it the standard exponential curve.

### Matching with the Graph
The graph matches **\( y = 4^x \)** best. The function is an unmodified exponential growth without additional scaling, consistent with the rapid increase in the graph at positive \( x \) values.

### Final Answer
The function rule that matches the graph is:
\[
y = 4^x
\]

Would you like further details or have any questions?

Here are some related questions:
1. What is the effect of different values of \( c \) in the function \( y = c \cdot 4^x \)?
2. How does the base of the exponential function (4 in this case) impact the graph's growth rate?
3. What would happen if the base was a fraction instead, like \( y = 4^{-\frac{x}{2}} \)?
4. How can you determine the horizontal asymptote for exponential functions?
5. What would the graph look like if the function was \( y = -4^x \)?

**Tip:** Exponential growth functions with a base greater than 1 increase as \( x \) increases, while functions with a base between 0 and 1 decrease as \( x \) increases.

Match the graph of the function with the function rule:

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

\[ y = \frac{1}{2} \cdot 4^{x} \]

\[ y = \frac{1}{4} \cdot 4^{x} \]

\[ y = 1 \cdot 4^{x} \]

Match the function rule with an exponential growth graph. This problem explores exponential functions with a base of 4 and different scaling factors, highlighting how variations affect the graph's growth rate. Ideal for high school students in Grades 9-12.

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