Question

This graph shows an exponential function.
the two main points are (0,-2)(1,-3)

Write an equation for the function in the form f(x)=a(b)x.
f(x)=

Answer 1

To write the equation of an exponential function in the form \( f(x) = a(b)^x \) given the points (0, -2) and (1, -3), we can use the following steps:

1. **Determine the value of \( a \)**:
The point (0, -2) tells us about the y-intercept of the function. Since \( f(0) = a(b)^0 = a \), we have:
\[
a = -2.
\]

2. **Now, substitute \( a \) into the form**:
So now we have \( f(x) = -2(b)^x \).

3. **Use the second point (1, -3) to find \( b \)**:
Using the point (1, -3):
\[
f(1) = -2(b)^1 = -3.
\]
This simplifies to:
\[
-2b = -3.
\]
Dividing both sides by -2:
\[
b = \frac{3}{2}.
\]

4. **Write the final function**:
Now substitute \( a \) and \( b \) back into the equation:
\[
f(x) = -2\left(\frac{3}{2}\right)^x.
\]

Thus, the equation of the function is:
\[
f(x) = -2\left(\frac{3}{2}\right)^x.
\]