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:

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$$ on a graphing calculator to check that the curve passes through both points. Would you like more details or have any questions? --- Here are 5 related questions to extend your understanding: 1. How do you find the exponential model if you have more than two points? 2. What are the properties of exponential growth vs. decay? 3. How does changing the base $$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.