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Part 1
Find an approximate equation yequals=ab Superscript xbx of the exponential curve that contains the given pair of points. Verify your result with a graphing calculator.
​(99,323) and ​(29,28)


To find the equation of the exponential curve \( y = ab^x \) that passes through the points \( (99, 323) \) and \( (29, 28) \), we will follow these steps:

### Step 1: Set up the system of equations
For an exponential curve of the form \( y = ab^x \), we substitute the given points into the equation.

From the point \( (99, 323) \):
\[
323 = ab^{99}
\]

From the point \( (29, 28) \):
\[
28 = ab^{29}
\]

### Step 2: Solve the system of equations
We now have the following system of equations:
\[
\begin{aligned}
323 &= ab^{99} \\
28 &= ab^{29}
\end{aligned}
\]
To eliminate \( a \), divide the first equation by the second equation:
\[
\frac{323}{28} = \frac{ab^{99}}{ab^{29}}
\]

Simplifying the right-hand side:
\[
\frac{323}{28} = b^{70}
\]
So,
\[
b^{70} = \frac{323}{28}
\]
Now calculate the value of \( b \) by taking the 70th root:
\[
b = \left( \frac{323}{28} \right)^{\frac{1}{70}} \approx 1.0452
\]

### Step 3: Find the value of \( a \)
Substitute \( b \approx 1.0452 \) into one of the original equations, say \( 28 = ab^{29} \):
\[
28 = a(1.0452)^{29}
\]
First calculate \( 1.0452^{29} \approx 3.824 \):
\[
28 = a(3.824)
\]
Solve for \( a \):
\[
a = \frac{28}{3.824} \approx 7.32
\]

### Step 4: Write the equation
The approximate equation of the exponential curve is:
\[
y = 7.32(1.0452)^x
\]

### Step 5: Verify with a graphing calculator
You can verify this equation by plotting the points \( (99, 323) \) and \( (29, 28) \) along with the curve \( y = 7.32(1.0452)^x \) on a graphing calculator to check that the curve passes through both points.

Would you like more details or have any questions?

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Here are 5 related questions to extend your understanding:

1. How do you find the exponential model if you have more than two points?
2. What are the properties of exponential growth vs. decay?
3. How does changing the base \( b \) affect the shape of an exponential curve?
4. How do logarithmic functions relate to exponential functions?
5. Can you solve for \( b \) if the points were for a decreasing function?

**Tip:** When solving exponential equations, always look for ways to simplify the system by dividing equations to eliminate variables.

Find an approximate equation y = ab^x of the exponential curve that contains the given pair of points (99, 323) and (29, 28). Verify your result with a graphing calculator.

Learn how to find the approximate equation y = ab^x of an exponential curve that passes through the points (99, 323) and (29, 28). This problem covers exponential functions, including step-by-step solving and graphing verification. Ideal for students in Grades 10-12.

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