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The problem is asking to find the equation of the exponential function of the form:

\[
y = C \cdot a^x
\]

given two points: \( (3, 45) \) and \( (6, 180) \).

### Step 1: Set up equations using the given points.

For the point \( (3, 45) \):
\[
45 = C \cdot a^3
\]

For the point \( (6, 180) \):
\[
180 = C \cdot a^6
\]

### Step 2: Divide the second equation by the first equation to eliminate \( C \).

\[
\frac{180}{45} = \frac{C \cdot a^6}{C \cdot a^3}
\]
\[
4 = a^3
\]

### Step 3: Solve for \( a \).

\[
a = \sqrt[3]{4}
\]
\[
a \approx 1.5874
\]

### Step 4: Substitute \( a \) into one of the original equations to solve for \( C \).

Using \( 45 = C \cdot (1.5874)^3 \):
\[
45 = C \cdot 4
\]
\[
C = \frac{45}{4} = 11.25
\]

### Step 5: Write the final equation.

The equation of the exponential function is:

\[
y = 11.25 \cdot (1.5874)^x
\]

Would you like further clarification on any steps?

---

Here are 5 related questions you might find interesting:
1. How does changing the base \( a \) affect the shape of the exponential curve?
2. What is the significance of the constant \( C \) in the exponential function?
3. How would the function change if one of the points was negative?
4. Can you express the exponential function using logarithms instead?
5. How do you find the inverse of an exponential function?

**Tip:** Always verify your solutions by plugging the points back into the final equation!

Write the equation of the exponential function, y = C * a^x, through the points (3, 45) and (6, 180).

Learn how to derive the equation of an exponential function through two points, using algebraic techniques. This step-by-step guide covers key concepts such as solving for C and a in the equation y = C * a^x. Suitable for high school students.

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