Let’s solve the problem step-by-step.

Given:

The function $f(x) = ab^x$ is an exponential function.

• $f(-2) = 19$
• $f(6) = 98$
• We need to find $f(4)$ to the nearest hundredth.

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

Since $f(x) = ab^x$, the two points give us:

1. $19 = ab^{-2}$
2. $98 = ab^6$

Step 2: Express $a$ in terms of $b$.

From the first equation: $19 = \frac{a}{b^2} \quad \Rightarrow \quad a = 19 b^2$

Step 3: Substitute $a = 19 b^2$ into the second equation.

$98 = (19 b^2) b^6 = 19 b^8$ $b^8 = \frac{98}{19} = 5.1579$

Step 4: Solve for $b$.

$b = \sqrt[8]{5.1579} \approx 1.216$

Step 5: Find $a$.

Substitute $b \approx 1.216$ into $a = 19 b^2$: $a = 19 (1.216)^2 = 19 \times 1.478 \approx 28.082$

Step 6: Write the equation of the function.

Now that we know $a$ and $b$, the function is: $f(x) = 28.082 \cdot (1.216)^x$

Step 7: Find $f(4)$.

$f(4) = 28.082 \cdot (1.216)^4$ $(1.216)^4 \approx 2.193$ $f(4) = 28.082 \times 2.193 \approx 61.61$

Final Answer:

$f(4) \approx 61.61$

Would you like a breakdown of any step or further explanation?

Here are 5 related questions:

1. What is the general formula for exponential functions and their properties?
2. How can you solve systems of equations involving exponentials?
3. What techniques exist for approximating roots, such as the 8th root used here?
4. How does changing the base $b$ affect the shape of the exponential curve?
5. Can exponential functions model growth and decay scenarios?

Tip: Exponential functions grow (or decay) at a rate proportional to their current value, making them useful for modeling population growth or radioactive decay.