Which Function Has Two $x$-intercepts, One At $(0,0)$ And One At $(4,0)$?A. $f(x)=x(x-4)$ B. $f(x)=x(x+4)$ C. $f(x)=(x-4)(x-4)$ D. $f(x)=(x+4)(x+4)$

Mar 10, 2025 by ADMIN 166 views

**Understanding $x$-Intercepts in Polynomial Functions**

Introduction to $x$-Intercepts

In mathematics, an $x$-intercept is a point where a graph intersects the $x$-axis. This occurs when the value of $y$ is equal to zero. In other words, an $x$-intercept is a solution to the equation $f(x) = 0$, where $f(x)$ is a function. Understanding $x$-intercepts is crucial in various mathematical applications, including algebra, geometry, and calculus.

The Problem at Hand

We are given a problem that asks us to identify a function with two $x$-intercepts, one at $(0,0)$ and one at $(4,0)$. To solve this problem, we need to analyze each option and determine which function satisfies the given conditions.

Option A: $f(x)=x(x-4)$

Let's start by analyzing option A. The function $f(x)=x(x-4)$ can be expanded as $f(x)=x^2-4x$. To find the $x$-intercepts, we set $f(x)=0$ and solve for $x$. This gives us the equation $x^2-4x=0$. Factoring out $x$, we get $x(x-4)=0$. This equation has two solutions: $x=0$ and $x=4$. Therefore, option A satisfies the given conditions.

Option B: $f(x)=x(x+4)$

Now, let's analyze option B. The function $f(x)=x(x+4)$ can be expanded as $f(x)=x^2+4x$. To find the $x$-intercepts, we set $f(x)=0$ and solve for $x$. This gives us the equation $x^2+4x=0$. Factoring out $x$, we get $x(x+4)=0$. This equation has two solutions: $x=0$ and $x=-4$. Therefore, option B does not satisfy the given conditions.

Option C: $f(x)=(x-4)(x-4)$

Next, let's analyze option C. The function $f(x)=(x-4)(x-4)$ can be expanded as $f(x)=(x-4)^2=x2-8x+16$. To find the $x$-intercepts, we set $f(x)=0$ and solve for $x$. This gives us the equation $x^2-8x+16=0$. Unfortunately, this equation does not have real solutions, so option C does not satisfy the given conditions.

Option D: $f(x)=(x+4)(x+4)$

Finally, let's analyze option D. The function $f(x)=(x+4)(x+4)$ can be expanded as $f(x)=(x+4)^2=x2+8x+16$. To find the $x$-intercepts, we set $f(x)=0$ and solve for $x$. This gives us the equation $x^2+8x+16=0$. Unfortunately, this equation does not have real solutions, so option D does not satisfy the given conditions.

Conclusion

Based on our analysis, we can conclude that option A, $f(x)=x(x-4)$, is the correct answer. This function has two $x$-intercepts, one at $(0,0)$ and one at $(4,0)$. Understanding $x$-intercepts is crucial in various mathematical applications, and this problem demonstrates the importance of analyzing functions to identify their $x$-intercepts.

Understanding Polynomial Functions

Polynomial functions are a fundamental concept in mathematics, and understanding them is crucial in various mathematical applications. A polynomial function is a function that can be written in the form $f(x)=a_nx^n+a_{n-1}x{n-1}+\cdots+a_1x+a_0$, where $a_n\neq 0$ and $n$ is a non-negative integer.

Types of Polynomial Functions

There are several types of polynomial functions, including:

Linear functions: These are polynomial functions of degree 1, which can be written in the form $f(x)=ax+b$.
Quadratic functions: These are polynomial functions of degree 2, which can be written in the form $f(x)=ax^2+bx+c$.
Cubic functions: These are polynomial functions of degree 3, which can be written in the form $f(x)=ax^3+bx2+cx+d$.
Quartic functions: These are polynomial functions of degree 4, which can be written in the form $f(x)=ax^4+bx3+cx^2+dx+e$.

Properties of Polynomial Functions

Polynomial functions have several important properties, including:

Domain: The domain of a polynomial function is the set of all real numbers.
Range: The range of a polynomial function is the set of all real numbers.
Degree: The degree of a polynomial function is the highest power of the variable.
Leading coefficient: The leading coefficient of a polynomial function is the coefficient of the highest power of the variable.

Graphing Polynomial Functions

Graphing polynomial functions is an important concept in mathematics, and it has several applications in various fields, including science, engineering, and economics. To graph a polynomial function, we need to identify its $x$-intercepts, $y$-intercepts, and asymptotes.

$x$-Intercepts

The $x$-intercepts of a polynomial function are the points where the graph intersects the $x$-axis. To find the $x$-intercepts, we need to set the function equal to zero and solve for $x$.

$y$-Intercepts

The $y$-intercepts of a polynomial function are the points where the graph intersects the $y$-axis. To find the $y$-intercepts, we need to evaluate the function at $x=0$.

Asymptotes

The asymptotes of a polynomial function are the lines that the graph approaches as $x$ approaches infinity or negative infinity. To find the asymptotes, we need to analyze the degree of the polynomial function.

Conclusion

In conclusion, understanding polynomial functions is crucial in various mathematical applications. Polynomial functions have several important properties, including domain, range, degree, and leading coefficient. Graphing polynomial functions is an important concept in mathematics, and it has several applications in various fields. By analyzing the $x$-intercepts, $y$-intercepts, and asymptotes of a polynomial function, we can graph it accurately and understand its behavior.

Real-World Applications of Polynomial Functions

Polynomial functions have several real-world applications, including:

Science: Polynomial functions are used to model the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
Engineering: Polynomial functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Polynomial functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.
Computer Science: Polynomial functions are used to develop algorithms and data structures, such as sorting and searching algorithms.

Conclusion

In conclusion, polynomial functions are a fundamental concept in mathematics, and they have several real-world applications. By understanding polynomial functions, we can analyze and model complex systems, make predictions, and optimize performance. Polynomial functions are a powerful tool in various fields, and they continue to play a vital role in the development of new technologies and innovations.

Introduction

Polynomial functions are a fundamental concept in mathematics, and they have several real-world applications. In this article, we will answer some frequently asked questions about polynomial functions.

Q: What is a polynomial function?

A: A polynomial function is a function that can be written in the form $f(x)=a_nx^n+a_{n-1}x{n-1}+\cdots+a_1x+a_0$, where $a_n\neq 0$ and $n$ is a non-negative integer.

Q: What are the different types of polynomial functions?

A: There are several types of polynomial functions, including:

Linear functions: These are polynomial functions of degree 1, which can be written in the form $f(x)=ax+b$.
Quadratic functions: These are polynomial functions of degree 2, which can be written in the form $f(x)=ax^2+bx+c$.
Cubic functions: These are polynomial functions of degree 3, which can be written in the form $f(x)=ax^3+bx2+cx+d$.
Quartic functions: These are polynomial functions of degree 4, which can be written in the form $f(x)=ax^4+bx3+cx^2+dx+e$.

Q: What are the properties of polynomial functions?

A: Polynomial functions have several important properties, including:

Domain: The domain of a polynomial function is the set of all real numbers.
Range: The range of a polynomial function is the set of all real numbers.
Degree: The degree of a polynomial function is the highest power of the variable.
Leading coefficient: The leading coefficient of a polynomial function is the coefficient of the highest power of the variable.

Q: How do I graph a polynomial function?

A: To graph a polynomial function, you need to identify its $x$-intercepts, $y$-intercepts, and asymptotes. The $x$-intercepts are the points where the graph intersects the $x$-axis, the $y$-intercepts are the points where the graph intersects the $y$-axis, and the asymptotes are the lines that the graph approaches as $x$ approaches infinity or negative infinity.

Q: What are the real-world applications of polynomial functions?

A: Polynomial functions have several real-world applications, including:

Science: Polynomial functions are used to model the behavior of physical systems, such as the motion of objects under the influence of gravity or friction.
Engineering: Polynomial functions are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Polynomial functions are used to model the behavior of economic systems, such as the supply and demand of goods and services.
Computer Science: Polynomial functions are used to develop algorithms and data structures, such as sorting and searching algorithms.

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

A: To determine the degree of a polynomial function, you need to identify the highest power of the variable. For example, if the polynomial function is $f(x)=ax^3+bx2+cx+d$, then the degree of the function is 3.

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

A: To find the $x$-intercepts of a polynomial function, you need to set the function equal to zero and solve for $x$. For example, if the polynomial function is $f(x)=ax^2+bx+c$, then you need to solve the equation $ax^2+bx+c=0$ for $x$.

Q: How do I find the $y$-intercepts of a polynomial function?

A: To find the $y$-intercepts of a polynomial function, you need to evaluate the function at $x=0$. For example, if the polynomial function is $f(x)=ax^2+bx+c$, then you need to evaluate the function at $x=0$ to find the $y$-intercept.

Q: How do I find the asymptotes of a polynomial function?

A: To find the asymptotes of a polynomial function, you need to analyze the degree of the function. If the degree of the function is even, then the asymptote is the $x$-axis. If the degree of the function is odd, then the asymptote is the $y$-axis.

Conclusion

In conclusion, polynomial functions are a fundamental concept in mathematics, and they have several real-world applications. By understanding polynomial functions, you can analyze and model complex systems, make predictions, and optimize performance. Polynomial functions are a powerful tool in various fields, and they continue to play a vital role in the development of new technologies and innovations.