Which Of The Following Equations Belongs To The Quadratic Function Family?A. $y=7(x+3.5)^{\frac{1}{2}}$B. $y=1.2(2)^x-8$C. $y=7.2 X-15$D. $y=2(x+1.5)^2-55$

Feb 27, 2025 by ADMIN 156 views

Which of the Following Equations Belongs to the Quadratic Function Family?

In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable (usually x) is two. Quadratic functions are commonly represented in the form of y = ax^2 + bx + c, where a, b, and c are constants. These functions have a parabolic shape and can be either upward-facing or downward-facing, depending on the value of a. In this article, we will explore which of the given equations belongs to the quadratic function family.

Quadratic functions are a fundamental concept in algebra and are used to model various real-world phenomena, such as the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth of a population. The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. The value of a determines the direction of the parabola, while the values of b and c determine the position of the vertex.

Now, let's analyze each of the given options to determine which one belongs to the quadratic function family.

Option A: $y=7(x+3.5)^{\frac{1}{2}}$

This equation represents a square root function, not a quadratic function. The exponent of the variable is 1/2, which means it is a square root function. Therefore, option A does not belong to the quadratic function family.

Option B: $y=1.2(2)^x-8$

This equation represents an exponential function, not a quadratic function. The variable is raised to the power of x, which means it is an exponential function. Therefore, option B does not belong to the quadratic function family.

Option C: $y=7.2 x-15$

This equation represents a linear function, not a quadratic function. The highest power of the variable is 1, which means it is a linear function. Therefore, option C does not belong to the quadratic function family.

Option D: $y=2(x+1.5)^2-55$

This equation represents a quadratic function. The variable is raised to the power of 2, which means it is a quadratic function. The equation can be rewritten in the standard form of a quadratic function as y = 2(x + 1.5)^2 - 55, which matches the general form of a quadratic function.

In conclusion, the equation that belongs to the quadratic function family is option D: $y=2(x+1.5)^2-55$ . This equation represents a quadratic function, which is a polynomial function of degree two. The other options do not belong to the quadratic function family, as they represent different types of functions, such as square root, exponential, and linear functions.

A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two.
Quadratic functions are commonly represented in the form of y = ax^2 + bx + c, where a, b, and c are constants.
The value of a determines the direction of the parabola, while the values of b and c determine the position of the vertex.
The equation $y=2(x+1.5)^2-55$ represents a quadratic function, while the other options do not belong to the quadratic function family.

For further reading on quadratic functions, we recommend the following resources:

Khan Academy: Quadratic Functions
Math Is Fun: Quadratic Functions
Wolfram MathWorld: Quadratic Function

In our previous article, we discussed the concept of quadratic functions and identified which of the given equations belongs to the quadratic function family. In this article, we will answer some frequently asked questions about quadratic functions to help you better understand and work with these functions.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants.

Q: What is the significance of the value of a in a quadratic function?

A: The value of a determines the direction of the parabola. If a is positive, the parabola opens upward, and if a is negative, the parabola opens downward.

Q: What is the significance of the values of b and c in a quadratic function?

A: The values of b and c determine the position of the vertex of the parabola. The vertex is the point where the parabola changes direction.

Q: How do I determine if a function is quadratic or not?

A: To determine if a function is quadratic or not, look for the highest power of the variable. If the highest power is 2, then the function is quadratic. If the highest power is 1, then the function is linear. If the highest power is 1/2, then the function is a square root function.

Q: What is the difference between a quadratic function and a polynomial function?

A: A polynomial function is a function that can be written in the form of y = a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where a_n, a_(n-1), ..., a_1, and a_0 are constants and n is a non-negative integer. A quadratic function is a special type of polynomial function where n = 2.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, first identify the vertex of the parabola. Then, use the vertex to determine the direction of the parabola. If the parabola opens upward, use a solid line to graph the function. If the parabola opens downward, use a dashed line to graph the function.

Q: What are some real-world applications of quadratic functions?

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

Modeling the trajectory of a projectile
Modeling the motion of an object under the influence of gravity
Modeling the growth of a population
Modeling the cost of producing a product

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

In conclusion, quadratic functions are an important concept in mathematics, and understanding them can help you better analyze and solve problems involving these functions. We hope this Q&A article has helped you better understand quadratic functions and their properties.

The general form of a quadratic function is y = ax^2 + bx + c.
The value of a determines the direction of the parabola.
The values of b and c determine the position of the vertex of the parabola.
Quadratic functions have many real-world applications.
To solve a quadratic equation, use the quadratic formula.