Question

The graph of an exponential function passes through the points (1, 1/16) and (3, 1/64) Write an equation for the function in the form y = a(b)x.
y =

Answer 1

To find the equation of an exponential function in the form \( y = a(b)^x \) that passes through the points \( (1, \frac{1}{16}) \) and \( (3, \frac{1}{64}) \), we will follow these steps:

1. **Substitute the points into the equation** to set up a system of equations.

For the point \( (1, \frac{1}{16}) \):
\[
\frac{1}{16} = a(b)^1 \implies a(b) = \frac{1}{16} \quad \text{(1)}
\]

For the point \( (3, \frac{1}{64}) \):
\[
\frac{1}{64} = a(b)^3 \implies a(b^3) = \frac{1}{64} \quad \text{(2)}
\]

2. **Express \( a \) from equation (1)**:
\[
a = \frac{1}{16b} \quad \text{(3)}
\]

3. **Substitute (3) into equation (2)**:
\[
\frac{1}{64} = \left(\frac{1}{16b}\right)(b^3)
\]
Simplifying this gives:
\[
\frac{1}{64} = \frac{b^3}{16b}
\]
\[
\frac{1}{64} = \frac{b^2}{16}
\]
Multiplying both sides by 16:
\[
\frac{16}{64} = b^2
\]
\[
\frac{1}{4} = b^2
\]
Taking the square root:
\[
b = \frac{1}{2} \quad \text{(we take the positive root since we are dealing with functions)}
\]

4. **Substitute \( b \) back into equation (3)** to find \( a \):
\[
a = \frac{1}{16 \cdot \frac{1}{2}} = \frac{1}{8}
\]

5. **Write the final equation** using the values of \( a \) and \( b \):
\[
y = \frac{1}{8} \left(\frac{1}{2}\right)^x
\]

Thus, the equation for the exponential function is:
\[
y = \frac{1}{8} \left(\frac{1}{2}\right)^x
\]