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

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 in the function 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. 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 situations, such as the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth or decay 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 and width of the parabola, while the value of c determines the vertical shift of the parabola.

Let's analyze each of the given equations to determine which one belongs to the quadratic function family.

Equation A: $y = 7.2x - 15$

This equation represents a linear function, not a quadratic function. A linear function is a polynomial function of degree one, which means the highest power of the variable in the function is one. The graph of a linear function is a straight line, not a parabola.

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

This equation represents an exponential function, not a quadratic function. An exponential function is a function of the form y = ab^x, where a and b are constants. The graph of an exponential function is an exponential curve, not a parabola.

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

This equation represents a square root function, not a quadratic function. A square root function is a function of the form y = a(x + b)^{\frac{1}{2}}, where a and b are constants. The graph of a square root function is a curve that opens upward or downward, but it is not a parabola.

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

This equation represents a quadratic function. The highest power of the variable in this function is two, which means it is a quadratic function. The graph of this function is a parabola that opens upward.

In conclusion, the equation that belongs to the quadratic function family is Equation D: $y = 2(x + 1.5)^2 - 55$. This equation represents a quadratic function, which is a polynomial function of degree two. The graph of this function is a parabola that opens upward.

• A quadratic function is a polynomial function of degree two, which means the highest power of the variable in the function is two.
• 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 and width of the parabola, while the value of c determines the vertical shift of the parabola.
• A quadratic function can be represented in various forms, including the standard form, vertex form, and factored form.

Quadratic functions have numerous real-world applications, including:

• Modeling the trajectory of a projectile
• Modeling the motion of an object under the influence of gravity
• Modeling the growth or decay of a population
• Modeling the stress on a beam or a bridge
• Modeling the motion of a pendulum

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In our previous article, we explored which of the given equations belongs to the quadratic function family. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable in the function is two. The general form of a quadratic function is y = ax^2 + bx + c, where a, b, and c are constants.

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 graph of a quadratic function is a parabola
• The parabola can open upward or downward
• The vertex of the parabola is the minimum or maximum point of the function

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

A: To determine if a function is quadratic, you can check the following:

• Check if the highest power of the variable is two
• Check if the function can be written in the form y = ax^2 + bx + c
• Check if the graph of the function is a parabola

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

A: A quadratic function can be represented in various forms, including:

• Standard form: y = ax^2 + bx + c
• Vertex form: y = a(x - h)^2 + k
• Factored form: y = a(x - r)(x - s)

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

• Determine the vertex of the parabola
• Determine the direction of the parabola (upward or downward)
• Plot the vertex and two points on either side of the vertex
• Draw a smooth curve through the points

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

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

• Modeling the trajectory of a projectile
• Modeling the motion of an object under the influence of gravity
• Modeling the growth or decay of a population
• Modeling the stress on a beam or a bridge
• Modeling the motion of a pendulum

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

• Write the equation in the form ax^2 + bx + c = 0
• Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
• Simplify the expression and solve for x

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: What is the significance of the quadratic formula?

A: The quadratic formula is significant because it allows us to solve quadratic equations, which are a fundamental concept in algebra. It is used to find the solutions to quadratic equations and is a crucial tool in mathematics and science.

In this article, we answered some frequently asked questions about quadratic functions. We discussed the characteristics of a quadratic function, how to determine if a function is quadratic, and the different forms of a quadratic function. We also discussed how to graph a quadratic function, the real-world applications of quadratic functions, and how to solve a quadratic equation using the quadratic formula. We hope this article has provided you with a better understanding of quadratic functions and their importance in mathematics and real-world applications.