The table represents a set of points on the graph of an exponential function. To find the equation of the exponential function, we use the general form of the equation:

$y = ab^x$

where:

• $a$ is the initial value (when $x = 0$),
• $b$ is the base of the exponential function, which determines the rate of growth or decay.

Step 1: Find $a$

From the table, when $x = 0$, $y = 1$. This gives us the value of $a$:

$y = ab^x \quad \text{when} \quad x = 0, \quad y = 1$ $1 = ab^0$ Since $b^0 = 1$, this simplifies to: $1 = a$

So, $a = 1$.

Step 2: Find $b$

Now, we need to find the value of $b$. We can use another point from the table, say when $x = 1$ and $y = 2$. Substituting into the equation:

$2 = 1 \cdot b^1$ $2 = b$

So, $b = 2$.

Step 3: Write the equation

Now that we have $a = 1$ and $b = 2$, the equation of the exponential function is:

$y = 2^x$

Final Answer:

The equation of the exponential function represented by the table is:

$y = 2^x$

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Do you want more details on how to solve this? Or have any questions?

Here are some follow-up questions:

1. How would the equation change if the table had a different set of values?
2. Can exponential growth functions also represent decay? How?
3. What happens if $b = 1$ in an exponential function?
4. How can we find the domain and range of the function $y = 2^x$?
5. How can logarithms help in solving for $x$ in exponential equations?

Tip: If you encounter an exponential function in real life, such as population growth or radioactive decay, you can use the base $b$ to determine how quickly the quantity increases or decreases.