Question
Write the exponential equation that contains the points (−1,40)
and (2,1.08)
.
and (2,1.08)
.
Answers
Answered by
GPT-4o mini
To find the exponential equation of the form \( y = ab^x \), where \( a \) and \( b \) are constants, we can use the given points \((-1, 40)\) and \((2, 1.08)\).
1. Substitute the first point \((-1, 40)\) into the equation:
\[
40 = ab^{-1} \implies 40 = \frac{a}{b} \implies a = 40b
\]
2. Substitute the second point \((2, 1.08)\) into the equation:
\[
1.08 = ab^2
\]
Now, substitute \( a = 40b \) into the equation:
\[
1.08 = (40b)b^2 = 40b^3
\]
This simplifies to:
\[
b^3 = \frac{1.08}{40} = 0.027
\]
Taking the cube root of both sides:
\[
b = \sqrt[3]{0.027} = 0.3
\]
3. Now we can find \( a \) by substituting \( b \) back into the equation \( a = 40b \):
\[
a = 40(0.3) = 12
\]
4. Thus, the exponential equation is:
\[
y = 12(0.3)^x
\]
So, the final exponential equation that contains the points \((-1, 40)\) and \((2, 1.08)\) is:
\[
y = 12(0.3)^x
\]
1. Substitute the first point \((-1, 40)\) into the equation:
\[
40 = ab^{-1} \implies 40 = \frac{a}{b} \implies a = 40b
\]
2. Substitute the second point \((2, 1.08)\) into the equation:
\[
1.08 = ab^2
\]
Now, substitute \( a = 40b \) into the equation:
\[
1.08 = (40b)b^2 = 40b^3
\]
This simplifies to:
\[
b^3 = \frac{1.08}{40} = 0.027
\]
Taking the cube root of both sides:
\[
b = \sqrt[3]{0.027} = 0.3
\]
3. Now we can find \( a \) by substituting \( b \) back into the equation \( a = 40b \):
\[
a = 40(0.3) = 12
\]
4. Thus, the exponential equation is:
\[
y = 12(0.3)^x
\]
So, the final exponential equation that contains the points \((-1, 40)\) and \((2, 1.08)\) is:
\[
y = 12(0.3)^x
\]
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