Question

(0, -2)Use the graph of the exponential decay function that models an exponential decay sequence to discuss the properties and determine the equation for the horizontal asymptote of the graph.

Answer 1

the equation for the graph is y=5(.3)^x -7

Answer 2

To discuss the properties and determine the equation for the horizontal asymptote of an exponential decay function, let's go through the characteristics of such a function.

1. **General Form**: The general form of an exponential decay function can be expressed as:
\[
y = a \cdot b^{x}
\]
where:
- \(a\) is the initial value (when \(x=0\)),
- \(b\) (the base) is a positive number between 0 and 1 (for decay).

2. **Characteristics**:
- The function decreases rapidly at first and then levels off.
- As \(x\) increases, \(y\) approaches 0 but never actually reaches it.
- The graph is continuous and smooth.

3. **Horizontal Asymptote**: The horizontal asymptote of an exponential decay function represents a value that the function approaches as \(x\) approaches positive infinity. For all exponential decay functions, the horizontal asymptote is at:
\[
y = 0
\]
This means that the function will never actually touch or cross the line \(y = 0\).

4. **Example with Specific Point (0, -2)**:
If you have the point (0, -2), this suggests the function has been translated or altered. The value at \(x = 0\) is -2 instead of the expected initial value from the standard form. This implies that the function can be adjusted to:
\[
y = a \cdot b^{x} - 2
\]
where \(a\) is the initial value that gives -2 when \(x = 0\) (i.e., setting \(a \cdot b^0 - 2 = -2\), which means \(a - 2 = -2\) therefore \(a = 0\)). So the function could look something like:
\[
y = 0 \cdot b^{x} - 2
\]

5. **Conclusion**: Regardless of the transformations applied, the horizontal asymptote remains at:
\[
y = -2
\]
This means that as \(x\) increases, the function will approach the line \(y = -2\).

In summary, the properties of the exponential decay function suggest it decreases but approaches a horizontal asymptote of \(y = 0\) unless shifted; in this case, based on the point (0, -2), we can deduce that the horizontal asymptote is at \(y = -2\).

Answer 3

Given the equation of the exponential decay function:

\[
y = 5(0.3)^x - 7
\]

we can analyze its properties, including the horizontal asymptote.

### Properties of the Function

1. **General Form**: This function is of the form:
\[
y = a \cdot b^x + k
\]
where:
- \(a = 5\) (the initial value),
- \(b = 0.3\) (the base, indicating a decay since \(0 < b < 1\)),
- \(k = -7\) (the vertical shift).

2. **Behavior**:
- As \(x\) increases (i.e., as \(x \to \infty\)), the term \(5(0.3)^x\) approaches 0. This is because \(0.3^x\) decreases towards 0 as \(x\) gets larger.
- Thus, the function approaches:
\[
y = 0 - 7 = -7
\]
- As \(x\) decreases (i.e., as \(x \to -\infty\)), \(5(0.3)^x\) becomes very large since \(0.3^x\) grows without bound.

3. **Horizontal Asymptote**:
The horizontal asymptote is the value that the function approaches as \(x\) becomes very large. Therefore, the horizontal asymptote for this function is:
\[
y = -7
\]

### Summary

- **Initial Value**: The function starts at \(y = 5(0.3)^0 - 7 = 5 - 7 = -2\) when \(x = 0\).
- **Decay**: The function decreases as \(x\) increases due to the decay factor of 0.3.
- **Horizontal Asymptote**: The horizontal asymptote of the graph is:
\[
y = -7
\]

This means that as \(x\) approaches infinity, the graph will get closer and closer to the line \(y = -7\).