Question

The graph of an exponential function passes through the points (
–
1,10) and 1,
2
5
. Write an equation for the function in the form y=a(b)x.
y=

Answer 1

To find the equation of the exponential function in the form \( y = a(b)^x \) that passes through the points \((-1, 10)\) and \( (1, 25) \), we can follow the steps outlined below:

1. **Set up the system of equations**:
Using the points given, we can set up two equations based on the general form \( y = a(b)^x \).

For the point \((-1, 10)\):
\[
10 = a(b)^{-1} \quad \Rightarrow \quad 10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b
\]

For the point \((1, 25)\):
\[
25 = a(b)^{1} \quad \Rightarrow \quad 25 = ab
\]

2. **Substitute for \(a\)**:
Now substitute \( a = 10b \) into the equation \( 25 = ab \):
\[
25 = (10b)(b) = 10b^2
\]

This simplifies to:
\[
b^2 = \frac{25}{10} = 2.5 \quad \Rightarrow \quad b = \sqrt{2.5} = \frac{5}{\sqrt{10}} = \frac{\sqrt{10}}{4}
\]

3. **Calculate \( a \)**:
Now that we have the value of \(b\), we can find \(a\):
\[
a = 10b = 10 \cdot \sqrt{2.5}
\]

4. **Put it all together**:
The function can now be written as:
\[
y = a(b)^x = (10 \cdot \sqrt{2.5}) \left( \sqrt{2.5} \right)^x
\]

Simplifying gives us:
\[
y = 10 \cdot (2.5)^{\frac{x + 1}{2}}
\]

Thus, the exponential function that passes through the points \((-1, 10)\) and \((1, 25)\) is given by:

\[
\boxed{y = 10 \cdot (2.5)^{x+1}}
\]