(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

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 2

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

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 \(y = -7\).