Deriving the Standard Equation of a Quadratic Function Given Its Vertex and a Point

Understanding and deriving the standard equation of a quadratic function when given its vertex and a point is a fundamental concept in algebra. This guide will walk you through the process with a detailed example, focusing on clarity, application, and relevance for SEO purposes.

Introduction to Quadratic Functions

A quadratic function is a polynomial function of degree two, typically expressed in the form (y ax^2 bx c). This form is known as the standard form. However, when given the vertex of the quadratic function, it is often more convenient to use the vertex form, which is (y a(x - h)^2 k), where ((h, k)) is the vertex of the parabola.

Deriving the Equation from Vertex and a Point

Given a quadratic function with vertex ( (1.5, 5) ) and that it passes through the point ( (2, 3) ), we can derive the standard form by following these steps:

Step 1: Write the Vertex Form

The vertex form of a quadratic function is:

[y a(x - h)^2 k]

Substitute the vertex ((1.5, 5)) into the equation:

[y a(x - 1.5)^2 5]

The function passes through the point ((2, 3)), so we can substitute (x 2) and (y 3) into the equation to find the value of (a):

[3 a(2 - 1.5)^2 5]

[3 a(0.5)^2 5]

[3 0.25a 5]

[3 - 5 0.25a]

[-2 0.25a]

[a -8]

Therefore, the vertex form of the quadratic function is:

[y -8(x - 1.5)^2 5]

Step 2: Convert to Standard Form

To convert the vertex form to standard form, expand the expression:

[y -8(x - 1.5)^2 5]

[y -8(x^2 - 3x 2.25) 5]

[y -8x^2 24x - 18 5]

[y -8x^2 24x - 13]

The standard form of the quadratic function is:

[y -8x^2 24x - 13]

Conclusion

This example demonstrates how to derive the standard form of a quadratic function given the vertex and a point on the parabola. The process involves using the vertex form, substituting the given values, solving for the coefficient (a), and then expanding the expression to obtain the standard form. This methodology is widely applicable and useful in various mathematical and real-world scenarios.

Further Reading and Resources

For more detailed information on quadratic functions, their forms, and applications, consider exploring the following resources:

- Math is Fun - Quadratic Functions - Khan Academy - Quadratic Equations in Standard Form - PurpleMath - Quadratic Formulas and Equations