Quadratic Functions

In this lesson, we will learn

• the different forms of quadratic functions
• general form
• factored form
• vertex form
• how to convert from general form to factored form.
• how to convert from the general form to the vertex form using the vertex formula.
• how to convert from the general form to the vertex form using completing the square.

Forms of Quadratic Functions

We can write quadratic functions in different ways or forms.

• General Form
• Factored Form
• Vertex Form

The general form of a quadratic equation is
y
= ax2 + bx + c where a, b and c are real numbers and a is not equal to zero.
For example, y = 2x2 + 5x − 30

The factored form of a quadratic equation is
y
= a(x + b)(x + c) where a, b and c are real numbers and a is not equal to zero.
For example, y = 2(x + 6)(x − 5).
The factored form is useful because we can see the x-intercepts (which are also the roots when the function is zero).
For example, the x-intercepts of y = a(x + b)(x + c) are (−b, 0) and (−c, 0)

The vertex form of a quadratic equation is
y
= a(x h)2 + k where a, h and k are real numbers and a is not equal to zero.
For example, y = 2(x + 6)2 − 5.
The vertex form is useful because we can see the turning point or vertex of the graph.
For example, the turning point or vertex of y = a(x h)2 + k is (h, k). If a is positive then it is a minimum vertex. It a is negative then it is a maximum vertex.

The following video looks at the various formats in which Quadratic Functions may be written as.

General Form to Factored Form

The following videos show how to change quadratic functions from general form to factored form.

General Form to Vertex Form by using the Vertex Formula

We can change a quadratic function from general form to vertex form by using the vertex formula.

Example of how to convert standard form to vertex form of a parabola equation.

General Form to Vertex Form by Completing the Square

We can change a quadratic function from general form to vertex form by completing the square.

The following video shows how to use the method of Completing the Square to convert a quadratic function from standard form to vertex form.

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