To find the equation of an exponential function from a given table of values, we can start with the general form of an exponential function:

$y = ab^x$

Where:

• $a$ is the initial value (when $x = 0$),
• $b$ is the base or growth factor,
• $x$ is the input variable,
• $y$ is the output.

Step 1: Extract Information from the Table

Here’s what the table suggests:

$x$	$y$
0	5
1	2.5
2	1.25
3	0.625

Step 2: Use the First Point

From the table, when $x = 0$, $y = 5$. This gives us the value of $a$:

$a = 5$

So the equation so far is:

$y = 5b^x$

Step 3: Solve for $b$ Using Other Points

Now, we can use the point where $x = 1$ and $y = 2.5$ to find $b$.

Substitute $x = 1$ and $y = 2.5$ into the equation:

$2.5 = 5b^1$

Solve for $b$:

$b = \frac{2.5}{5} = 0.5$

Step 4: Write the Final Equation

Now that we have both $a = 5$ and $b = 0.5$, the equation of the exponential function is:

$y = 5(0.5)^x$

Final Answer:

The equation of the exponential function is:

$y = 5(0.5)^x$

Would you like more details on how this was done or have any questions about the steps? Here's some related content you might find helpful:

1. How to find the general form of an exponential equation.
2. What is the significance of the base $b$ in exponential growth/decay?
3. How to identify exponential growth or decay from a table.
4. How to solve for $b$ with other points if $x = 0$ is not available.
5. The difference between exponential and linear functions.

Tip: If you have an exponential function and you're unsure about the base, always use two points and substitute them into the equation to find $b$.