Deriving the Quadratic Function from Given Zeros: Examples and Applications

Understanding how to derive the equation of a quadratic function given its zeros is a fundamental concept in algebra, with wide-ranging applications in mathematics and science. This article will explore the process of finding a quadratic function with specific zeros, such as 7 and 3, and discuss the importance and significance of the quadratic function and quadratic equation.

Understanding Quadratic Functions and Zeros

A quadratic function is a polynomial of the second degree, typically expressed in the form:

f(x) ax2 bx c

Expressing a Quadratic Function in Terms of Its Zeros

When given the zeros of a quadratic function, denoted as R1 and R2, the function can be expressed in the form:

f(x) a(x - R1)(x - R2)

For example, if the zeros are 7 and 3, the quadratic function can be written as:

f(x) a(x - 7)(x - 3)

Determining the Specific Quadratic Function

To find a specific quadratic function, we can set the coefficient a to a specific value. Let's use a 1 for simplicity:

f(x) (x - 7)(x - 3)

Expanding this expression:

f(x) x2 - 3x - 7x 21

Which simplifies to:

f(x) x2 - 1 21

Alternative Representation and Simplification

Another method to represent the quadratic function is through its sum and product of roots. Given the roots α 7 and β 3, the sum of the roots (α β) and the product of the roots (αβ) are:

α β 7 3 10

αβ 7 * 3 21

Using these values, the quadratic function can be written in the form:

x2 - (α β)x αβ 0

Substituting the values:

x2 - 1 21 0

Infinite Number of Quadratic Functions

It is important to note that there are an infinite number of quadratic functions with the given zeros. This is because the coefficient a can be any non-zero real number. For example, the function:

y -3(x - 7)(x - 3)

is also a valid quadratic function with the same zeros:

y -3(x2 - 1 21)

Which simplifies to:

y -3x2 3 - 63

Conclusion

Deriving a quadratic function from its zeros is a valuable skill that is used in various mathematical applications. By understanding the relationship between the zeros and the coefficients of the quadratic function, we can express and manipulate quadratic functions with ease. Whether in algebra, calculus, or real-world applications, the knowledge of quadratic functions and their zeros is essential.