Question
Write the exponential equation that contains the points (โ1,40)
(
โ
1
,
40
)
and (2,1.08)
(
2
,
1.08
)
.(1 point)
The equation is $$y=ยท.
(
โ
1
,
40
)
and (2,1.08)
(
2
,
1.08
)
.(1 point)
The equation is $$y=ยท.
Answers
Answered by
GPT-4o mini
To find the exponential equation that fits the points \((-1, 40)\) and \((2, 1.08)\), we start with the general form of an exponential equation:
\[
y = ab^x
\]
where \(a\) is the initial value (the value of \(y\) when \(x = 0\)), and \(b\) is the base of the exponential function.
### Step 1: Set Up the Equations
First, we will use the points to set up two equations.
For \((-1, 40)\):
\[
40 = ab^{-1} \implies 40 = \frac{a}{b} \implies a = 40b
\]
For \((2, 1.08)\):
\[
1.08 = ab^2
\]
### Step 2: Substitute \(a\) into the Second Equation
Now, we can substitute \(a = 40b\) into the second equation:
\[
1.08 = (40b)b^2
\]
\[
1.08 = 40b^3
\]
### Step 3: Solve for \(b\)
Now we solve for \(b\):
\[
b^3 = \frac{1.08}{40} \approx 0.027
\]
\[
b \approx 0.3
\]
### Step 4: Solve for \(a\)
Now, substitute \(b \approx 0.3\) back into \(a = 40b\):
\[
a = 40 \times 0.3 \approx 12
\]
### Step 5: Write the Equation
Now we can write the exponential equation:
\[
y = 12(0.3)^x
\]
### Final Answer
The exponential equation that contains the points \((-1, 40)\) and \((2, 1.08)\) is:
\[
y = 12(0.3)^x
\]
\[
y = ab^x
\]
where \(a\) is the initial value (the value of \(y\) when \(x = 0\)), and \(b\) is the base of the exponential function.
### Step 1: Set Up the Equations
First, we will use the points to set up two equations.
For \((-1, 40)\):
\[
40 = ab^{-1} \implies 40 = \frac{a}{b} \implies a = 40b
\]
For \((2, 1.08)\):
\[
1.08 = ab^2
\]
### Step 2: Substitute \(a\) into the Second Equation
Now, we can substitute \(a = 40b\) into the second equation:
\[
1.08 = (40b)b^2
\]
\[
1.08 = 40b^3
\]
### Step 3: Solve for \(b\)
Now we solve for \(b\):
\[
b^3 = \frac{1.08}{40} \approx 0.027
\]
\[
b \approx 0.3
\]
### Step 4: Solve for \(a\)
Now, substitute \(b \approx 0.3\) back into \(a = 40b\):
\[
a = 40 \times 0.3 \approx 12
\]
### Step 5: Write the Equation
Now we can write the exponential equation:
\[
y = 12(0.3)^x
\]
### Final Answer
The exponential equation that contains the points \((-1, 40)\) and \((2, 1.08)\) is:
\[
y = 12(0.3)^x
\]
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