(-1, 11) & (1, 3). y=2 asymptote. Find an exponential decay function

To find an exponential decay function that passes through the points \((-1, 11)\) and \((1, 3)\) with a horizontal asymptote at \(y = 2\), we can use the general form of an exponential decay function:

\[
y = a \cdot b^x + c
\]

Where:
- \(a\) is the vertical stretch factor,
- \(b\) is the decay factor (with \(0 < b < 1\) for decay),
- \(c\) is the horizontal asymptote (in this case, \(c = 2\)).

### Step 1: Modify the equation for the given asymptote
Since the horizontal asymptote is at \(y = 2\), the function becomes:

\[
y = a \cdot b^x + 2
\]

### Step 2: Substitute the given points into the equation
We can use the points \((-1, 11)\) and \((1, 3)\) to form a system of equations.

For \((-1, 11)\):
\[
11 = a \cdot b^{-1} + 2
\]
\[
11 - 2 = \frac{a}{b}
\]
\[
9 = \frac{a}{b} \tag{1}
\]

For \((1, 3)\):
\[
3 = a \cdot b^1 + 2
\]
\[
3 - 2 = a \cdot b
\]
\[
1 = a \cdot b \tag{2}
\]

### Step 3: Solve the system of equations
From equation \((2)\), we know:
\[
a \cdot b = 1
\]
\[
a = \frac{1}{b}
\]

Substitute this into equation \((1)\):
\[
9 = \frac{\frac{1}{b}}{b} = \frac{1}{b^2}
\]
\[
b^2 = \frac{1}{9}
\]
\[
b = \frac{1}{3}
\]

### Step 4: Find \(a\)
Using \(b = \frac{1}{3}\) in equation \((2)\):
\[
a \cdot \frac{1}{3} = 1
\]
\[
a = 3
\]

### Step 5: Write the final equation
Now that we have \(a = 3\) and \(b = \frac{1}{3}\), the exponential decay function is:

\[
y = 3 \cdot \left( \frac{1}{3} \right)^x + 2
\]

### Conclusion:
The exponential decay function is:

\[
y = 3 \cdot \left( \frac{1}{3} \right)^x + 2
\]

Do you have any questions or need further details?

Here are 5 questions that extend this topic:
1. How would the function change if the asymptote were \(y = 0\)?
2. What happens if you change the decay factor \(b\)?
3. How can you determine if a function is growth or decay from its formula?
4. How would you find the inverse of this function?
5. How can you interpret the meaning of the parameters \(a\), \(b\), and \(c\) in real-world contexts?

**Tip:** Exponential functions are powerful in modeling real-world decay or growth processes, such as radioactive decay, population growth, and depreciation.

Learn how to find an exponential decay function with a horizontal asymptote at y=2 that passes through the points (-1, 11) and (1, 3). This step-by-step solution explains key concepts in exponential functions, decay, and asymptotes. Suitable for high school students in Grades 10-12.

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