Give a rule of the form ​f(x)equals=a Superscript xax to define the exponential function whose graph contains the given point.
​(a)​ (33​,2727​)
​(b)​ (negative 2−2​,3636​)
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Part 1
​(a) The graph of the exponential function ​f(x)equals=3 Superscript x3x passes through the point ​(33​,2727​).
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)
Part 2
​(b) The graph of the exponential function ​f(x)equals=enter your response here passes through the point ​(negative 2−2​,3636​).
​(Simplify your answer. Use integers or fractions for any numbers in the​ expression.)

To define the exponential function \( f(x) = a^x \) that passes through the given points, we can use the point \( (x_0, y_0) \) on the graph to find the value of \( a \). Let's break it down for both parts:

### Part 1: 
We are given the point \( (3, 27) \).

The general form of the exponential function is \( f(x) = a^x \). Since \( f(3) = 27 \), we can substitute these values into the equation:

\[
f(3) = a^3 = 27
\]

Now, solve for \( a \):

\[
a^3 = 27 \quad \Rightarrow \quad a = \sqrt[3]{27} = 3
\]

Thus, the exponential function is:

\[
f(x) = 3^x
\]

### Part 2:
We are given the point \( (-2, 36) \).

Again, using the general form \( f(x) = a^x \) and substituting \( f(-2) = 36 \), we get:

\[
f(-2) = a^{-2} = 36
\]

Now solve for \( a \):

\[
a^{-2} = 36 \quad \Rightarrow \quad \frac{1}{a^2} = 36 \quad \Rightarrow \quad a^2 = \frac{1}{36} \quad \Rightarrow \quad a = \frac{1}{6}
\]

Thus, the exponential function is:

\[
f(x) = \left( \frac{1}{6} \right)^x
\]

### Final Answers:
1. \( f(x) = 3^x \) for the point \( (3, 27) \).
2. \( f(x) = \left( \frac{1}{6} \right)^x \) for the point \( (-2, 36) \).

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Do you want details on how I solved for \( a \) in part (b)? Here are 5 related questions:

1. What are the characteristics of exponential functions?
2. How do we determine if a function is exponential from a set of points?
3. What happens when the base of an exponential function is a fraction?
4. How can we generalize exponential growth and decay functions?
5. How do transformations affect the graph of exponential functions?

**Tip:** For exponential functions, the base \( a \) determines the rate of growth or decay: \( a > 1 \) implies growth, while \( 0 < a < 1 \) implies decay.

Give a rule of the form f(x) = a^x to define the exponential function whose graph contains the given points.
(a) (3, 27)
(b) (-2, 36)

Part 1: The graph of the exponential function f(x) = 3^x passes through the point (3, 27).
(Simplify your answer. Use integers or fractions for any numbers in the expression.)

Part 2: The graph of the exponential function f(x) = ? passes through the point (-2, 36).
(Simplify your answer. Use integers or fractions for any numbers in the expression.)

Learn how to define exponential functions of the form f(x) = a^x that pass through given points. This problem involves finding the base of the exponential function using points (3, 27) and (-2, 36). Suitable for students in Grades 9-12.

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