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, 10, right parenthesis(0,10) and left parenthesis, 3, comma, 3430, right parenthesis(3,3430).
x
that goes through the points left parenthesis, 0, comma, 10, right parenthesis(0,10) and left parenthesis, 3, comma, 3430, right parenthesis(3,3430).
Answers
GPT-4o mini
To find the exponential function in the form \( y = a b^x \) that passes through the points \( (0, 10) \) and \( (3, 3430) \), we can use these points to create a system of equations.
1. **Using the point \( (0, 10) \)**:
Plug in \( x = 0 \) and \( y = 10 \):
\[
10 = a b^0
\]
Since \( b^0 = 1 \):
\[
10 = a \implies a = 10
\]
2. **Using the point \( (3, 3430) \)**:
Now substitute \( x = 3 \) and \( y = 3430 \):
\[
3430 = 10 b^3
\]
Solving for \( b^3 \):
\[
b^3 = \frac{3430}{10} = 343
\]
3. **Finding \( b \)**:
Now take the cube root of 343:
\[
b = \sqrt[3]{343} = 7
\]
4. **Now we have both constants**:
\( a = 10 \) and \( b = 7 \).
Thus, the exponential function is:
\[
y = 10 \cdot 7^x
\]
This function can be used to find values of \( y \) at various \( x \), and it is verified since it passes through both given points.
1. **Using the point \( (0, 10) \)**:
Plug in \( x = 0 \) and \( y = 10 \):
\[
10 = a b^0
\]
Since \( b^0 = 1 \):
\[
10 = a \implies a = 10
\]
2. **Using the point \( (3, 3430) \)**:
Now substitute \( x = 3 \) and \( y = 3430 \):
\[
3430 = 10 b^3
\]
Solving for \( b^3 \):
\[
b^3 = \frac{3430}{10} = 343
\]
3. **Finding \( b \)**:
Now take the cube root of 343:
\[
b = \sqrt[3]{343} = 7
\]
4. **Now we have both constants**:
\( a = 10 \) and \( b = 7 \).
Thus, the exponential function is:
\[
y = 10 \cdot 7^x
\]
This function can be used to find values of \( y \) at various \( x \), and it is verified since it passes through both given points.