Select All Statements That Are True Of The Function F ( X ) = − 5 ( 2 ) X F(x) = -5(2)^x F ( X ) = − 5 ( 2 ) X .- The Graph Is Increasing By A Constant Percent Rate Of Change Of 2.- The Graph Is Decreasing By A Constant Percent Rate Of Change Of 2.- As X X X Increases,

The given function is $f(x) = -5(2)^x$. This function represents an exponential curve, where the base is 2 and the coefficient is -5. The graph of this function will exhibit certain characteristics that are typical of exponential functions.

Increasing or Decreasing Rate of Change

To determine whether the graph is increasing or decreasing, we need to examine the coefficient of the exponential term. In this case, the coefficient is -5, which is negative. This means that the graph will be decreasing as $x$ increases.

Constant Percent Rate of Change

The rate of change of an exponential function is not constant, unlike linear functions. However, we can calculate the percent rate of change using the formula:

$$\text{Percent Rate of Change} = \left( \frac{f(x + h) - f(x)}{f(x)} \right) \times 100\%$$

where $h$ is a small positive value.

Let's calculate the percent rate of change for the given function:

$$\text{Percent Rate of Change} = \left( \frac{-5(2)^{x+h} + 5(2)^x}{5(2)^x} \right) \times 100\%$$

Simplifying the expression, we get:

$$\text{Percent Rate of Change} = \left( \frac{-5(2)^x \cdot 2^h + 5(2)^x}{5(2)^x} \right) \times 100\%$$

$$= \left( \frac{-5 \cdot 2^h + 1}{1} \right) \times 100\%$$

$$= \left( -5 \cdot 2^h + 1 \right) \times 100\%$$

As $h$ approaches 0, the percent rate of change approaches a constant value. However, this value is not 2, as stated in the problem. Instead, the percent rate of change is a function of $x$ and $h$.

Conclusion

Based on the analysis above, we can conclude that:

• The graph of the function $f(x) = -5(2)^x$ is decreasing as $x$ increases.
• The percent rate of change of the function is not constant and depends on the value of $x$ and $h$.

Therefore, the correct statement is:

• The graph is decreasing by a constant percent rate of change of 2 is false.
• The graph is decreasing by a constant percent rate of change is true.

Additional Discussion

The function $f(x) = -5(2)^x$ is an example of an exponential function with a negative coefficient. This type of function is commonly used to model population growth, chemical reactions, and other phenomena where the rate of change is not constant.

In this case, the negative coefficient indicates that the population or quantity is decreasing over time. The base of the exponential term, 2, represents the growth factor, which is the ratio of the population at time $t + 1$ to the population at time $t$.

The percent rate of change of the function is not constant, as it depends on the value of $x$ and $h$. This means that the rate of change is not the same at all points on the graph.

Real-World Applications

Exponential functions with negative coefficients have many real-world applications, including:

• Modeling population decline: The function $f(x) = -5(2)^x$ can be used to model the decline of a population over time.
• Chemical reactions: Exponential functions with negative coefficients can be used to model chemical reactions where the reactants are consumed at a constant rate.
• Financial modeling: Exponential functions with negative coefficients can be used to model the decline of an investment over time.

Conclusion

In conclusion, the function $f(x) = -5(2)^x$ is an example of an exponential function with a negative coefficient. The graph of this function is decreasing as $x$ increases, and the percent rate of change is not constant. The correct statement is that the graph is decreasing by a constant percent rate of change is true.

References

• [1] "Exponential Functions" by Math Open Reference
• [2] "Exponential Growth and Decay" by Khan Academy
• [3] "Exponential Functions with Negative Coefficients" by Wolfram MathWorld
  Q&A: Exponential Functions with Negative Coefficients =====================================================

Q: What is an exponential function with a negative coefficient?

A: An exponential function with a negative coefficient is a function of the form $f(x) = a \cdot b^x$, where $a$ is a negative constant and $b$ is a positive constant greater than 1. This type of function is commonly used to model population decline, chemical reactions, and other phenomena where the rate of change is not constant.

Q: What is the graph of an exponential function with a negative coefficient like?

A: The graph of an exponential function with a negative coefficient is a decreasing exponential curve. As $x$ increases, the value of the function decreases exponentially.

Q: How do you determine the rate of change of an exponential function with a negative coefficient?

A: The rate of change of an exponential function with a negative coefficient is not constant, unlike linear functions. However, you can calculate the percent rate of change using the formula:

$$\text{Percent Rate of Change} = \left( \frac{f(x + h) - f(x)}{f(x)} \right) \times 100\%$$

where $h$ is a small positive value.

Q: Can you give an example of an exponential function with a negative coefficient?

A: Yes, an example of an exponential function with a negative coefficient is $f(x) = -5(2)^x$. This function represents a decreasing exponential curve.

Q: What is the percent rate of change of the function $f(x) = -5(2)^x$?

A: The percent rate of change of the function $f(x) = -5(2)^x$ is not constant and depends on the value of $x$ and $h$. However, as $h$ approaches 0, the percent rate of change approaches a constant value.

Q: Can you give some real-world applications of exponential functions with negative coefficients?

A: Yes, exponential functions with negative coefficients have many real-world applications, including:

• Modeling population decline: The function $f(x) = -5(2)^x$ can be used to model the decline of a population over time.
• Chemical reactions: Exponential functions with negative coefficients can be used to model chemical reactions where the reactants are consumed at a constant rate.
• Financial modeling: Exponential functions with negative coefficients can be used to model the decline of an investment over time.

Q: How do you graph an exponential function with a negative coefficient?

A: To graph an exponential function with a negative coefficient, you can use a graphing calculator or a computer algebra system. You can also use a table of values to plot the function.

Q: Can you give some tips for working with exponential functions with negative coefficients?

A: Yes, here are some tips for working with exponential functions with negative coefficients:

• Make sure to identify the negative coefficient and the base of the exponential term.
• Use the formula for the percent rate of change to calculate the rate of change.
• Use a graphing calculator or a computer algebra system to graph the function.
• Use a table of values to plot the function.

Q: What are some common mistakes to avoid when working with exponential functions with negative coefficients?

A: Some common mistakes to avoid when working with exponential functions with negative coefficients include:

• Failing to identify the negative coefficient and the base of the exponential term.
• Using the wrong formula for the percent rate of change.
• Graphing the function incorrectly.
• Failing to use a table of values to plot the function.

Conclusion

In conclusion, exponential functions with negative coefficients are an important topic in mathematics and have many real-world applications. By understanding the properties and behavior of these functions, you can model and analyze complex phenomena in fields such as population decline, chemical reactions, and financial modeling.