Horizontal and Vertical Analysis

Oct 13, 2025 by windailys.com 69 views

Introduction

In mathematics, a cubic function is a polynomial function of degree three, which means the highest power of the variable is three. The given function, $f(x) = x^3 - 15x^2 + 54x$ , is a cubic function that can be factored into its simplest form as $x(x-6)(x-9)$ . In this article, we will perform a horizontal and vertical analysis of this cubic function to understand its behavior and key characteristics.

Horizontal Analysis

Horizontal analysis involves examining the function's behavior as the input variable changes. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , we can analyze its behavior by looking at its factored form, $x(x-6)(x-9)$ .

Degree: The degree of a polynomial function is the highest power of the variable. In this case, the degree of the function is 3, which means the function will have a cubic behavior.
End Behavior: The end behavior of a function refers to its behavior as the input variable approaches positive or negative infinity. For the cubic function $f(x) = x^3 - 15x^2 + 54x$ , we can determine its end behavior by looking at the leading term, which is $x^3$ . As $x$ approaches positive infinity, $f(x)$ will also approach positive infinity. Similarly, as $x$ approaches negative infinity, $f(x)$ will also approach negative infinity.
Intercepts: The x-intercepts of a function are the points where the function crosses the x-axis. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , the x-intercepts are $(0,0)$ , $(6,0)$ , and $(9,0)$ . These points represent the values of $x$ where the function crosses the x-axis.

Vertical Analysis

Vertical analysis involves examining the function's behavior at specific points. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , we can analyze its behavior at the x-intercepts.

x-Intercepts: The x-intercepts of a function are the points where the function crosses the x-axis. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , the x-intercepts are $(0,0)$ , $(6,0)$ , and $(9,0)$ . These points represent the values of $x$ where the function crosses the x-axis.
Local Maxima and Minima: The local maxima and minima of a function are the points where the function changes from increasing to decreasing or vice versa. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , we can determine its local maxima and minima by looking at the factored form, $x(x-6)(x-9)$ . The local maxima and minima occur at the points where the function changes from increasing to decreasing or vice versa.

Key Information

Degree: The degree of a polynomial function is the highest power of the variable. In this case, the degree of the function is 3, which means the function will have a cubic behavior.
x-Intercepts: The x-intercepts of a function are the points where the function crosses the x-axis. In the case of the cubic function $f(x) = x^3 - 15x^2 + 54x$ , the x-intercepts are $(0,0)$ , $(6,0)$ , and $(9,0)$ . These points represent the values of $x$ where the function crosses the x-axis.

Conclusion

In conclusion, the horizontal and vertical analysis of the cubic function $f(x) = x^3 - 15x^2 + 54x$ provides valuable insights into its behavior and key characteristics. The function has a degree of 3, which means it will have a cubic behavior. The x-intercepts are $(0,0)$ , $(6,0)$ , and $(9,0)$ , which represent the values of $x$ where the function crosses the x-axis. The local maxima and minima occur at the points where the function changes from increasing to decreasing or vice versa.

References

[1] Cubic Functions. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/cubic-functions.html
[2] Horizontal and Vertical Analysis. (n.d.). Retrieved from https://www.mathopenref.com/horizontalverticalanalysis.html

Glossary

Degree: The degree of a polynomial function is the highest power of the variable.
End Behavior: The end behavior of a function refers to its behavior as the input variable approaches positive or negative infinity.
Intercepts: The x-intercepts of a function are the points where the function crosses the x-axis.
Local Maxima and Minima: The local maxima and minima of a function are the points where the function changes from increasing to decreasing or vice versa.
Q&A: Horizontal/Vertical Analysis of a Cubic Function =====================================================

Frequently Asked Questions

Q: What is the degree of a cubic function?

A: The degree of a cubic function is 3, which means the function will have a cubic behavior.

Q: What is the end behavior of a cubic function?

A: The end behavior of a cubic function refers to its behavior as the input variable approaches positive or negative infinity. For a cubic function, as x approaches positive infinity, f(x) will also approach positive infinity. Similarly, as x approaches negative infinity, f(x) will also approach negative infinity.

Q: What are the x-intercepts of a cubic function?

A: The x-intercepts of a cubic function are the points where the function crosses the x-axis. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, the x-intercepts are (0,0), (6,0), and (9,0).

Q: What are the local maxima and minima of a cubic function?

A: The local maxima and minima of a cubic function are the points where the function changes from increasing to decreasing or vice versa. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, we can determine its local maxima and minima by looking at the factored form, x(x-6)(x-9).

Q: How do I determine the degree of a polynomial function?

A: To determine the degree of a polynomial function, you need to look at the highest power of the variable. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, the highest power of the variable is 3, which means the degree of the function is 3.

Q: How do I determine the end behavior of a polynomial function?

A: To determine the end behavior of a polynomial function, you need to look at the leading term. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, the leading term is x^3. As x approaches positive infinity, f(x) will also approach positive infinity. Similarly, as x approaches negative infinity, f(x) will also approach negative infinity.

Q: How do I determine the x-intercepts of a polynomial function?

A: To determine the x-intercepts of a polynomial function, you need to set the function equal to zero and solve for x. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, we can set the function equal to zero and solve for x to find the x-intercepts.

Q: How do I determine the local maxima and minima of a polynomial function?

A: To determine the local maxima and minima of a polynomial function, you need to look at the factored form of the function. In the case of the cubic function f(x) = x^3 - 15x^2 + 54x, we can look at the factored form, x(x-6)(x-9), to determine the local maxima and minima.

Additional Resources

[1] Cubic Functions. (n.d.). Retrieved from https://www.mathsisfun.com/algebra/cubic-functions.html
[2] Horizontal and Vertical Analysis. (n.d.). Retrieved from https://www.mathopenref.com/horizontalverticalanalysis.html

Glossary

Degree: The degree of a polynomial function is the highest power of the variable.
End Behavior: The end behavior of a function refers to its behavior as the input variable approaches positive or negative infinity.
Intercepts: The x-intercepts of a function are the points where the function crosses the x-axis.
Local Maxima and Minima: The local maxima and minima of a function are the points where the function changes from increasing to decreasing or vice versa.