Question

The graph shows an exponential function whose domain is the set of integers.
Write an equation for the exponential function in the form f(x)=a(b)x.
f(x)=
Graph: (1,-3)
(2,-9)

Answer 1

To find the equation of the exponential function \( f(x) = a(b)^x \), we can use the two points provided in the graph: \( (1, -3) \) and \( (2, -9) \).

1. **Substituting the first point into the equation:**
\[
f(1) = a(b)^1 = -3
\]
This simplifies to:
\[
ab = -3 \quad \text{(1)}
\]

2. **Substituting the second point into the equation:**
\[
f(2) = a(b)^2 = -9
\]
This simplifies to:
\[
ab^2 = -9 \quad \text{(2)}
\]

Now, we have a system of equations with:
- (1) \( ab = -3 \)
- (2) \( ab^2 = -9 \)

3. **Dividing equation (2) by equation (1)** to solve for \( b \):
\[
\frac{ab^2}{ab} = \frac{-9}{-3}
\]
Simplifying this, we get:
\[
b = 3
\]

4. **Now substitute \( b \) back into equation (1)** to solve for \( a \):
\[
a(3) = -3
\]
This gives us:
\[
a = -1
\]

5. **Now we can write the exponential function**:
\[
f(x) = -1(3)^x
\]

So, the equation for the exponential function is:
\[
f(x) = -1(3)^x
\]