Question

Write an exponential function in the form y, equals, a, b, to the power xy=ab
x
that goes through the points left parenthesis, 0, comma, 15, right parenthesis(0,15) and left parenthesis, 2, comma, 1215, right parenthesis(2,1215).

Answer 1

To find an exponential function in the form \( y = ab^x \) that goes through the points \((0, 15)\) and \((2, 1215)\), we can follow these steps:

### Step 1: Use the first point \((0, 15)\)
When \( x = 0 \):
\[
y = ab^0 = a \cdot 1 = a
\]
Given that \( y = 15 \), we have:
\[
a = 15
\]

### Step 2: Use the second point \((2, 1215)\)
Now we substitute \( a = 15 \) into the general form \( y = ab^x \):
\[
y = 15b^x
\]
Now substitute the point \((2, 1215)\):
\[
1215 = 15b^2
\]

### Step 3: Solve for \( b^2 \)
Divide both sides by 15:
\[
b^2 = \frac{1215}{15}
\]
Calculating the right side:
\[
b^2 = 81
\]
Taking the square root:
\[
b = 9
\]

### Step 4: Write the final function
Now we have \( a = 15 \) and \( b = 9 \). Thus, the exponential function is:
\[
y = 15 \cdot 9^x
\]

### Final answer
The exponential function is:
\[
\boxed{y = 15 \cdot 9^x}
\]