Write a quadratic function whose zeros are and . – Welcome to the fascinating world of quadratic functions! In this comprehensive guide, we embark on a journey to unravel the intricacies of constructing a quadratic function with given zeros. As we delve into the concepts and applications of these functions, we invite you to join us on an enlightening adventure.
Our exploration begins with a thorough understanding of quadratic functions, their forms, and the significance of their coefficients. We then delve into the methods for finding zeros, empowering you with step-by-step strategies for factoring, using the quadratic formula, and completing the square.
1. Understanding Quadratic Functions
Quadratic functions are a type of polynomial function that has a degree of 2. They are characterized by their parabolic shape and can be expressed in the standard form:
$$f(x) = ax^2 + bx + c$$
where a, b, and c are real numbers and a ≠ 0.
1.1 Significance of the Coefficients
The coefficients in a quadratic function play a crucial role in determining its properties:
- Coefficient a:Governs the steepness and direction of the parabola. A positive value opens upward, while a negative value opens downward.
- Coefficient b:Determines the x-coordinate of the vertex, which is the point where the parabola changes direction.
- Coefficient c:Represents the y-intercept, which is the point where the parabola intersects the y-axis.
2. Finding Zeros of Quadratic Functions
Zeros, also known as roots, of a quadratic function are the x-values where the function equals zero. There are three methods to find zeros:
2.1 Factoring, Write a quadratic function whose zeros are and .
Factor the quadratic into two linear factors and set each factor equal to zero.
2.2 Quadratic Formula
Use the quadratic formula:
$$x = \frac-b ± \sqrtb^2
4ac2a$$
2.3 Completing the Square
Complete the square to rewrite the quadratic in the form:
$$f(x) = (x + h)^2 + k$$
where h and k are constants.
3. Creating Quadratic Functions with Given Zeros
To create a quadratic function with given zeros, p and q, use the standard form and substitute the zeros:
$$f(x) = a(x
- p)(x
- q)$$
Solve for a using any method, such as factoring or the quadratic formula.
4. Applications of Quadratic Functions
Quadratic functions have wide applications:
4.1 Modeling Parabolic Trajectories
Quadratic functions model the parabolic paths of projectiles, such as a thrown ball or a fired rocket.
4.2 Optimizing Profit
Businesses use quadratic functions to optimize profit, which is often represented as a parabola.
4.3 Solving Geometric Problems
Quadratic functions can be used to find the area or volume of certain geometric shapes, such as parabolas and circles.
5. HTML Table Representation
An HTML table can be used to organize the zeros and coefficients of a quadratic function:
| Coefficient | Value |
|---|---|
| a | |
| b | |
| c | |
| Zeros |
Expert Answers: Write A Quadratic Function Whose Zeros Are And .
What is the standard form of a quadratic function?
The standard form of a quadratic function is ax² + bx + c, where a, b, and c are real numbers and a ≠ 0.
How do I find the zeros of a quadratic function?
There are several methods to find the zeros of a quadratic function, including factoring, using the quadratic formula, and completing the square.
What are the applications of quadratic functions?
Quadratic functions have numerous applications in various fields, such as modeling parabolic trajectories, optimizing profit, and solving geometric problems.
