To find the equation of the exponential curve of the form $y = ab^x$ that passes through the points $(0, 7)$ and $(5, 60)$, we will follow these steps:

Step 1: Use the point $(0, 7)$

Substitute $x = 0$ and $y = 7$ into the general equation $y = ab^x$:

$7 = ab^0$

Since $b^0 = 1$, this simplifies to:

$7 = a$

Thus, $a = 7$.

Step 2: Use the point $(5, 60)$

Now substitute $x = 5$ and $y = 60$ into the equation $y = ab^x$, with $a = 7$:

$60 = 7b^5$

Solve for $b$:

$b^5 = \frac{60}{7} = 8.5714$

$b = \sqrt[5]{8.5714}$

Using a calculator:

$b \approx 1.56$

Step 3: Write the equation

Now that we have $a = 7$ and $b \approx 1.56$, the approximate equation of the exponential curve is:

$y = 7 \cdot (1.56)^x$

Verification

You can verify this equation using a graphing calculator or plotting software by inputting the equation and checking that it passes through the points $(0, 7)$ and $(5, 60)$.

---

Would you like further explanation, or do you have any additional questions?

Here are 5 related questions you might find helpful:

1. How would the equation change if the points were different?
2. What is the relationship between exponential growth and the base $b$?
3. How can we apply logarithms to solve for $b$?
4. How do exponential functions differ from linear functions?
5. What are some real-world examples of exponential growth?

Tip: Always check the units of your values when working with real-world problems to ensure the equation makes sense in context!