Let's address the questions in the worksheet.

Problem 8: Exponential Function Analysis

The equation given is $F(x) = a \cdot b^x$. We are tasked to:

1. Use the data points to write two equations.
2. Solve for the constants $a$ and $b$.

Given Data:

• $F(1) = 3.4$
• $F(7) = 9.1$

Step (a): Writing the equations

From $F(x) = a \cdot b^x$:

1. Using $F(1) = 3.4$: $a \cdot b^1 = 3.4$, so $a \cdot b = 3.4$.
2. Using $F(7) = 9.1$: $a \cdot b^7 = 9.1$.

Thus, the two equations are:

1. $a \cdot b = 3.4$
2. $a \cdot b^7 = 9.1$

Step (b): Solving for $a$ and $b$

Divide the second equation by the first to eliminate $a$: $\frac{a \cdot b^7}{a \cdot b} = \frac{9.1}{3.4}$ $b^6 = \frac{9.1}{3.4} \approx 2.676$ $b = \sqrt[6]{2.676} \approx 1.18$

Substitute $b \approx 1.18$ into the first equation: $a \cdot 1.18 = 3.4$ $a = \frac{3.4}{1.18} \approx 2.88$

Final Values:

• $a \approx 2.88$
• $b \approx 1.18$

Would you like detailed assistance on the other parts of the worksheet or further breakdowns?

Related Questions:

1. How can we verify the values of $a$ and $b$ using additional data points?
2. What does the parameter $b$ represent in the context of exponential growth or decay?
3. How does $a$ affect the graph of the function?
4. Can you identify whether the function represents growth or decay?
5. How would you rewrite the equation for predictions beyond the given $x$-values?

Tip:

Always double-check by substituting your $a$ and $b$ values into the original equations to ensure accuracy.