To find the exponential equation in the form $y = ab^x$ that passes through the points $(0, 8)$ and $(1, 7)$, we’ll use these points to solve for the values of $a$ and $b$.

Step 1: Substitute the first point, $(0, 8)$

Since $x = 0$ and $y = 8$: $y = ab^x \Rightarrow 8 = ab^0$ Since $b^0 = 1$, this simplifies to: $a = 8$

So, the equation so far is: $y = 8b^x$

Step 2: Substitute the second point, $(1, 7)$

Now, with $x = 1$ and $y = 7$: $7 = 8b^1$ or simply: $7 = 8b$ Solving for $b$, divide both sides by 8: $b = \frac{7}{8}$

Final Equation

Now that we have $a = 8$ and $b = \frac{7}{8}$, the exponential equation is: $y = 8 \left(\frac{7}{8}\right)^x$

Would you like further details on the steps?

Related Questions

1. How do you determine if a function is exponential or not?
2. What happens to the graph if $b$ is greater than 1 in an exponential function?
3. How can you find an exponential equation given any two points?
4. What are the key differences between exponential growth and decay?
5. How can exponential functions be used to model real-life scenarios?

Tip

When $b$ is a fraction between 0 and 1, the exponential function represents decay, meaning the function value decreases as $x$ increases.