This table contains values of $x$ and $y$, where $y$ appears to follow an exponential relationship with $x$. To solve for $y$ when $x = 5$, let's determine the exponential curve of best fit in the form:

$y = a \cdot b^x$

Here:

• $a$ is the initial value (the value of $y$ when $x = 0$).
• $b$ is the base, representing the rate of change.

Step 1: Identify $a$

From the table, when $x = 0$, $y = 3$. Thus: $a = 3$

Step 2: Solve for $b$

We now use another point from the table to find $b$. For instance, using the point $(-1, 0.3)$, substitute into $y = a \cdot b^x$:

$0.3 = 3 \cdot b^{-1}$

Simplify: $0.3 = \frac{3}{b}$

Multiply through by $b$: $b = \frac{3}{0.3} = 10$

Thus, the equation becomes: $y = 3 \cdot 10^x$

Step 3: Find $y$ when $x = 5$

Substitute $x = 5$ into $y = 3 \cdot 10^x$:

$y = 3 \cdot 10^5$

Simplify: $y = 3 \cdot 100000 = 300000$

Final Answer:

When $x = 5$, $y = 300,000$.

Would you like me to elaborate on any step or verify the fit with more points?

Related Questions:

1. How do we derive the general formula for exponential growth or decay?
2. What happens to the value of $y$ as $x$ becomes very large or very small?
3. How do we calculate the equation of best fit using logarithms for exponential data?
4. What is the difference between exponential growth and linear growth?
5. What practical scenarios can be modeled with exponential functions?

Tip:

In exponential functions, the base $b > 1$ indicates growth, while $0 < b < 1$ indicates decay.