Which Represents A Quadratic Function?A. F ( X ) = − 8 X 3 − 16 X 2 − 4 X F(x) = -8x^3 - 16x^2 - 4x F ( X ) = − 8 X 3 − 16 X 2 − 4 X B. F ( X ) = 3 4 X 2 + 2 X − 5 F(x) = \frac{3}{4}x^2 + 2x - 5 F ( X ) = 4 3 ​ X 2 + 2 X − 5 C. F ( X ) = 4 X 2 − 2 X + 1 F(x) = \frac{4}{x^2} - \frac{2}{x} + 1 F ( X ) = X 2 4 ​ − X 2 ​ + 1 D. F ( X ) = 0 X 2 − 9 X + 7 F(x) = 0x^2 - 9x + 7 F ( X ) = 0 X 2 − 9 X + 7

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. It is often represented in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero. Quadratic functions are commonly used in various fields, including physics, engineering, and economics, to model real-world phenomena.

Understanding Quadratic Functions

A quadratic function can be represented in various forms, including the standard form, vertex form, and factored form. The standard form is the most common representation, where the function is written as $f(x) = ax^2 + bx + c$. The vertex form is represented as $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The factored form is represented as $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots of the equation.

Analyzing the Options

Now, let's analyze the given options to determine which one represents a quadratic function.

Option A: $f(x) = -8x^3 - 16x^2 - 4x$

This function is a cubic function, not a quadratic function, because the highest power of the variable is three. Therefore, option A does not represent a quadratic function.

Option B: $f(x) = \frac{3}{4}x^2 + 2x - 5$

This function is a quadratic function because the highest power of the variable is two. The coefficients of the terms are also correctly represented, with $a = \frac{3}{4}$, $b = 2$, and $c = -5$. Therefore, option B represents a quadratic function.

Option C: $f(x) = \frac{4}{x^2} - \frac{2}{x} + 1$

This function is not a quadratic function because the highest power of the variable is two, but the coefficients are not correctly represented. The function is actually a rational function, and the coefficients are not in the standard form of a quadratic function. Therefore, option C does not represent a quadratic function.

Option D: $f(x) = 0x^2 - 9x + 7$

This function is a linear function, not a quadratic function, because the coefficient of the $x^2$ term is zero. Therefore, option D does not represent a quadratic function.

Conclusion

In conclusion, the correct answer is option B: $f(x) = \frac{3}{4}x^2 + 2x - 5$. This function represents a quadratic function because the highest power of the variable is two, and the coefficients are correctly represented in the standard form of a quadratic function.

Key Takeaways

• A quadratic function is a polynomial function of degree two, where the highest power of the variable is two.
• A quadratic function can be represented in various forms, including the standard form, vertex form, and factored form.
• To determine if a function is a quadratic function, check if the highest power of the variable is two and if the coefficients are correctly represented in the standard form.

Real-World Applications

Quadratic functions have numerous real-world applications, including:

• Modeling the trajectory of a projectile
• Describing the motion of an object under the influence of gravity
• Representing the cost or revenue of a business
• Modeling the growth or decay of a population

Examples of Quadratic Functions

Some examples of quadratic functions include:

• $f(x) = x^2 + 2x - 3$
• $f(x) = 2x^2 - 5x + 1$
• $f(x) = x^2 - 4x + 4$

These functions can be used to model various real-world phenomena, such as the motion of an object, the growth of a population, or the cost of a business.

Solving Quadratic Equations

Quadratic equations are equations in which the highest power of the variable is two. They can be solved using various methods, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method for solving quadratic equations and is represented as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Conclusion

In the previous article, we discussed the concept of quadratic functions and identified the correct representation of a quadratic function among the given options. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic functions better.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, where the highest power of the variable is two. It is often represented in the form of $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero.

Q: What are the characteristics of a quadratic function?

A: The characteristics of a quadratic function include:

• The highest power of the variable is two.
• The function can be represented in various forms, including the standard form, vertex form, and factored form.
• The function has a parabolic shape, with a single turning point (vertex).
• The function can be graphed using a parabola.

Q: What are the different forms of a quadratic function?

A: A quadratic function can be represented in three different forms:

• Standard form: $f(x) = ax^2 + bx + c$
• Vertex form: $f(x) = a(x - h)^2 + k$
• Factored form: $f(x) = a(x - r_1)(x - r_2)$

Q: How do I determine if a function is a quadratic function?

A: To determine if a function is a quadratic function, check if the highest power of the variable is two and if the coefficients are correctly represented in the standard form.

Q: What are the applications of quadratic functions?

A: Quadratic functions have numerous real-world applications, including:

• Modeling the trajectory of a projectile
• Describing the motion of an object under the influence of gravity
• Representing the cost or revenue of a business
• Modeling the growth or decay of a population

Q: How do I solve a quadratic equation?

A: Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method for solving quadratic equations and is represented as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations and is represented as $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I graph a quadratic function?

A: A quadratic function can be graphed using a parabola. The graph of a quadratic function has a single turning point (vertex) and opens upward or downward.

Q: What are the key takeaways from this article?

A: The key takeaways from this article include:

• A quadratic function is a polynomial function of degree two.
• A quadratic function can be represented in various forms, including the standard form, vertex form, and factored form.
• A quadratic function has a parabolic shape, with a single turning point (vertex).
• A quadratic function can be graphed using a parabola.
• Quadratic functions have numerous real-world applications.

Conclusion

In conclusion, quadratic functions are an essential concept in mathematics, with numerous real-world applications. By understanding the properties and characteristics of quadratic functions, we can model and solve various problems in physics, engineering, economics, and other fields. This Q&A guide provides a comprehensive overview of quadratic functions and their applications.