Understanding the Negative Sign

Oct 13, 2025 by windailys.com 32 views

Understanding the Function n(x) = -e^(x) -5

The given function is n(x) = -e^(x) -5. To graph this function, we need to understand the transformations applied to the graph of y=e^x. The function n(x) can be broken down into three parts: a negative sign, a negative exponent, and a constant term.

Understanding the Negative Sign

The negative sign in front of the function indicates a reflection across the x-axis. This means that the graph of n(x) will be a reflection of the graph of y=e^x across the x-axis.

Understanding the Negative Exponent

The negative exponent in the function indicates a reflection across the y-axis. This means that the graph of n(x) will be a reflection of the graph of y=e^x across the y-axis.

Understanding the Constant Term

The constant term -5 in the function indicates a vertical shift of the graph of n(x) down by 5 units.

Graphing the Function

To graph the function n(x) = -e^(x) -5, we need to apply the transformations discussed above to the graph of y=e^x.

First, reflect the graph of y=e^x across the x-axis to get the graph of y=-e^x.
Next, reflect the graph of y=-e^x across the y-axis to get the graph of y=e^(-x).
Finally, shift the graph of y=e^(-x) down by 5 units to get the graph of n(x) = -e^(x) -5.

Domain and Range of the Function

The domain of a function is the set of all possible input values for which the function is defined. The range of a function is the set of all possible output values for which the function is defined.

For the function n(x) = -e^(x) -5, the domain is all real numbers, and the range is all real numbers less than or equal to -5.

In interval notation, the domain is (-∞, ∞) and the range is (-∞, -5].

Equation of the Asymptote

The equation of the asymptote of a function is the horizontal line that the function approaches as x approaches infinity or negative infinity.

For the function n(x) = -e^(x) -5, the equation of the asymptote is y = -5.

Conclusion

In conclusion, the function n(x) = -e^(x) -5 can be graphed by applying transformations to the graph of y=e^x. The domain and range of the function are all real numbers and all real numbers less than or equal to -5, respectively. The equation of the asymptote is y = -5.

References

[1] Calculus: Early Transcendentals, 8th Edition, James Stewart
[2] Calculus, 9th Edition, Michael Spivak

Discussion

Q: What is the main difference between the graph of y=e^x and the graph of n(x) = -e^(x) -5?

A: The main difference between the graph of y=e^x and the graph of n(x) = -e^(x) -5 is the reflection across the x-axis and the y-axis, and the vertical shift down by 5 units.

Q: How do you reflect the graph of y=e^x across the x-axis to get the graph of y=-e^x?

A: To reflect the graph of y=e^x across the x-axis, you need to multiply the function by -1. This will flip the graph of y=e^x across the x-axis, resulting in the graph of y=-e^x.

Q: How do you reflect the graph of y=-e^x across the y-axis to get the graph of y=e^(-x)?

A: To reflect the graph of y=-e^x across the y-axis, you need to replace x with -x in the function. This will flip the graph of y=-e^x across the y-axis, resulting in the graph of y=e^(-x).

Q: How do you shift the graph of y=e^(-x) down by 5 units to get the graph of n(x) = -e^(x) -5?

A: To shift the graph of y=e^(-x) down by 5 units, you need to subtract 5 from the function. This will move the graph of y=e^(-x) down by 5 units, resulting in the graph of n(x) = -e^(x) -5.

Q: What is the domain of the function n(x) = -e^(x) -5?

A: The domain of the function n(x) = -e^(x) -5 is all real numbers, which can be represented in interval notation as (-∞, ∞).

Q: What is the range of the function n(x) = -e^(x) -5?

A: The range of the function n(x) = -e^(x) -5 is all real numbers less than or equal to -5, which can be represented in interval notation as (-∞, -5].

Q: What is the equation of the asymptote of the function n(x) = -e^(x) -5?

A: The equation of the asymptote of the function n(x) = -e^(x) -5 is y = -5.

Q: How do you graph the function n(x) = -e^(x) -5?

A: To graph the function n(x) = -e^(x) -5, you need to apply the transformations discussed above to the graph of y=e^x. First, reflect the graph of y=e^x across the x-axis to get the graph of y=-e^x. Next, reflect the graph of y=-e^x across the y-axis to get the graph of y=e^(-x). Finally, shift the graph of y=e^(-x) down by 5 units to get the graph of n(x) = -e^(x) -5.

Q: What are some real-world applications of the function n(x) = -e^(x) -5?

A: The function n(x) = -e^(x) -5 has several real-world applications, including modeling population growth and decay, and modeling the spread of diseases. It can also be used to model the behavior of electrical circuits and the flow of fluids.

Q: How do you use the function n(x) = -e^(x) -5 to model population growth and decay?

A: To use the function n(x) = -e^(x) -5 to model population growth and decay, you need to replace x with the time variable t. This will give you a function that models the population growth or decay over time.

Q: How do you use the function n(x) = -e^(x) -5 to model the spread of diseases?

A: To use the function n(x) = -e^(x) -5 to model the spread of diseases, you need to replace x with the distance variable d. This will give you a function that models the spread of the disease over distance.

Q: What are some other functions that can be graphed by applying transformations to the graph of y=e^x?

A: Some other functions that can be graphed by applying transformations to the graph of y=e^x include:

y=e^(-x)
y=e^(2x)
y=e^(-2x)
y=e^(x)+5
y=e^(-x)+5

These functions can be graphed by applying the same transformations discussed above to the graph of y=e^x.