Select The Correct Graph.Consider The Following Polynomial Function: $f(x)=x^4-x^3-10x^2-8x$What Is The Graph Of This Function?

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Introduction

In mathematics, graphing polynomial functions is a crucial aspect of understanding their behavior and characteristics. A polynomial function is a function that can be written in the form of a sum of terms, where each term is a product of a constant and a variable raised to a non-negative integer power. In this article, we will analyze the given polynomial function f(x)=x4−x3−10x2−8xf(x) = x^4 - x^3 - 10x^2 - 8x and determine its graph.

Understanding the Polynomial Function

The given polynomial function is f(x)=x4−x3−10x2−8xf(x) = x^4 - x^3 - 10x^2 - 8x. To understand the graph of this function, we need to analyze its behavior and characteristics. Let's break down the function into its individual terms:

  • x4x^4 is a quartic term, which means it has a degree of 4.
  • −x3-x^3 is a cubic term, which means it has a degree of 3.
  • −10x2-10x^2 is a quadratic term, which means it has a degree of 2.
  • −8x-8x is a linear term, which means it has a degree of 1.

Determining the Graph

To determine the graph of the polynomial function, we need to consider the following factors:

  • Degree of the polynomial: The degree of the polynomial function is 4, which means it can have at most 4 real roots.
  • Leading coefficient: The leading coefficient of the polynomial function is 1, which means the graph will open upwards.
  • End behavior: As x approaches positive infinity, the graph will approach positive infinity. As x approaches negative infinity, the graph will approach negative infinity.
  • Intercepts: To find the x-intercepts, we need to set f(x) = 0 and solve for x. Similarly, to find the y-intercept, we need to set x = 0 and solve for f(x).

Finding the X-Intercepts

To find the x-intercepts, we need to set f(x) = 0 and solve for x. This means we need to solve the equation:

x4−x3−10x2−8x=0x^4 - x^3 - 10x^2 - 8x = 0

We can factor out an x from the equation:

x(x3−x2−10x−8)=0x(x^3 - x^2 - 10x - 8) = 0

This gives us two possible solutions:

  • x=0x = 0
  • x3−x2−10x−8=0x^3 - x^2 - 10x - 8 = 0

The first solution is the x-intercept at (0, 0). To find the other x-intercepts, we need to solve the cubic equation:

x3−x2−10x−8=0x^3 - x^2 - 10x - 8 = 0

Finding the Y-Intercept

To find the y-intercept, we need to set x = 0 and solve for f(x). This means we need to evaluate the function at x = 0:

f(0)=04−03−10(0)2−8(0)f(0) = 0^4 - 0^3 - 10(0)^2 - 8(0)

This simplifies to:

f(0)=0f(0) = 0

Therefore, the y-intercept is (0, 0).

Graphing the Function

Now that we have found the x-intercepts and y-intercept, we can graph the function. The graph will have the following characteristics:

  • X-intercepts: The graph will have x-intercepts at (0, 0) and two other points that satisfy the cubic equation.
  • Y-intercept: The graph will have a y-intercept at (0, 0).
  • End behavior: The graph will approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
  • Leading coefficient: The graph will open upwards since the leading coefficient is 1.

Conclusion

In conclusion, the graph of the polynomial function f(x)=x4−x3−10x2−8xf(x) = x^4 - x^3 - 10x^2 - 8x is a quartic function that has a degree of 4. The graph has x-intercepts at (0, 0) and two other points that satisfy the cubic equation, a y-intercept at (0, 0), and approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. The graph opens upwards since the leading coefficient is 1.

Graph Options

Based on the analysis, we can select the correct graph from the following options:

  • Graph 1: A quartic function with x-intercepts at (0, 0) and two other points that satisfy the cubic equation, a y-intercept at (0, 0), and approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
  • Graph 2: A cubic function with x-intercepts at (0, 0) and two other points that satisfy the cubic equation, a y-intercept at (0, 0), and approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
  • Graph 3: A quadratic function with x-intercepts at (0, 0) and two other points that satisfy the quadratic equation, a y-intercept at (0, 0), and approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.

Selecting the Correct Graph

Introduction

In our previous article, we analyzed the polynomial function f(x)=x4−x3−10x2−8xf(x) = x^4 - x^3 - 10x^2 - 8x and determined its graph. We also discussed the characteristics of the graph, including its x-intercepts, y-intercept, and end behavior. In this article, we will answer some frequently asked questions about selecting the correct graph.

Q: What is the degree of the polynomial function?

A: The degree of the polynomial function is 4, which means it can have at most 4 real roots.

Q: What is the leading coefficient of the polynomial function?

A: The leading coefficient of the polynomial function is 1, which means the graph will open upwards.

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

A: To find the x-intercepts, you need to set f(x) = 0 and solve for x. This means you need to solve the equation:

x4−x3−10x2−8x=0x^4 - x^3 - 10x^2 - 8x = 0

You can factor out an x from the equation:

x(x3−x2−10x−8)=0x(x^3 - x^2 - 10x - 8) = 0

This gives you two possible solutions:

  • x=0x = 0
  • x3−x2−10x−8=0x^3 - x^2 - 10x - 8 = 0

Q: How do I find the y-intercept of the polynomial function?

A: To find the y-intercept, you need to set x = 0 and solve for f(x). This means you need to evaluate the function at x = 0:

f(0)=04−03−10(0)2−8(0)f(0) = 0^4 - 0^3 - 10(0)^2 - 8(0)

This simplifies to:

f(0)=0f(0) = 0

Therefore, the y-intercept is (0, 0).

Q: What are the characteristics of the graph of the polynomial function?

A: The graph of the polynomial function has the following characteristics:

  • X-intercepts: The graph has x-intercepts at (0, 0) and two other points that satisfy the cubic equation.
  • Y-intercept: The graph has a y-intercept at (0, 0).
  • End behavior: The graph approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
  • Leading coefficient: The graph opens upwards since the leading coefficient is 1.

Q: How do I select the correct graph from the options?

A: To select the correct graph, you need to consider the characteristics of the graph, including its x-intercepts, y-intercept, and end behavior. Based on the analysis, the correct graph is Graph 1. This graph represents the quartic function f(x)=x4−x3−10x2−8xf(x) = x^4 - x^3 - 10x^2 - 8x with x-intercepts at (0, 0) and two other points that satisfy the cubic equation, a y-intercept at (0, 0), and approaches positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.

Q: What are the implications of selecting the correct graph?

A: Selecting the correct graph has important implications for understanding the behavior and characteristics of the polynomial function. By selecting the correct graph, you can gain a deeper understanding of the function's x-intercepts, y-intercept, and end behavior, which can be useful in a variety of applications, including mathematics, science, and engineering.

Conclusion

In conclusion, selecting the correct graph is an important aspect of understanding the behavior and characteristics of a polynomial function. By considering the characteristics of the graph, including its x-intercepts, y-intercept, and end behavior, you can select the correct graph and gain a deeper understanding of the function. We hope this article has been helpful in answering your questions about selecting the correct graph.