Quadratic Function Definition

 

In order to understand what a quadratic function is, let’s take a look at what a function is. A function takes a number as an input and, through a transformation on that number, results in an output. The most basic function is a linear function, structured like the following.

 

linear_function_explanation

 

The input of a function is usually called ‘x,’ while the output of a function is typically either ‘y’ or ‘f(x).’ Take a look at some examples of linear functions below.

x Function f(x)
1 f(x) = 10x + 30 40
-4 f(x) = -2x+1 9
3 f(x) = x + 50 53
10 f(x) = 2x 20

 

While a quadratic function is still a function, it is quite different from a linear function. The definitions of both are below.

Linear Function Quadratic Function
Standard Form f(x) = mx + b f(x) = ax^{2} + bx + c

OR

f(x) = a(x-h)^{2} + k

Relationship Linear Parabolic
Graph Straight line Parabola

Quadratic Function Properties

As you may have noticed, a quadratic function has two standard forms. The second equation is also known as the “vertex form.” You can take a look how to convert the standard form to the vertex form below.

Step Description Example
1 Start with the standard form f(x) = ax^{2} + bx + c f(x) = 3x^{2} + 6x + 1
2 For ease of notation, let f(x) be y y = 3x^{2} + 6x + 1
3 Move the numbers so all x’s are on one side y -1 = 3x^{2} + 6x
4 Simplify the right side so that the x^{2} term has a coefficient of 1 y -1 = 3(x^{2} + 2x)
5 We try to find the perfect square trinomial. Divide the ‘b’ term by 2, square it and that is the term we add to both sides b term transformation: (\frac{2}{2})^{2} = 1

y -1 +3(1) = 3(x^{2} + 2x + 1)

6 Just note, we add 3(1) on the left side because, in reality, we’re not adding 1 on the right side but 3 * 1.
7 Simplify the equation y - 2 = 3(x - 1)^{2}
8 Get the y term by itself again y = 3(x - 1)^{2} + 2

f(x) = 3(x - 1)^{2} + 2

As you can see, the resulting equation is in vertex form. The reason why this is called vertex form is because the h and k terms of the equation represent the coordinates for the vertex of the parabola.

Take our example from above. The (h,k) here is (1,2). Graphing the parabola, we get:

parabola_relationship

As you can see, the point where the parabola is mirrored can be found at point (1,2) on the graph. This is the definition of a vertex, where the line is defined as the axis of symmetry.

Factoring Quadratic Functions

Knowing how to factor a quadratic function can be one of the most important things that you will encounter in math relating to quadratic functions. When factoring a quadratic function, the most important rules of thumb to remember can be seen below.

1 Check for a common factor 4x^{2} + 2x = 2(x^{2} + x)
2 Find the two numbers where that equal to a*c and add up to b x^{2} + 3x - 4

- (4 * -1) = -4 = a*c

- (4 + -1) = 3 = b

These two rules can help us get to the desired output. It’s also important to keep in mind that there is sometimes what is called a perfect square. Take a look at the form below, which gives us the following:

quadratic_simplification

Take a look at an example of a perfect square below.

    \[ (x+3)^{2} = (x+3)(x+3) \]

    \[ (x+3)(x+3) = x^{2} + 3^{2} + 3x + 3x \]

    \[ a^{2} +2ab + b^{2} = x^{2} + 6x + 3^{2} \]

Vertex of Quadratic Function

When we talk about the vertex of a quadratic function, it is important to understand what we’re talking about. Imagine a parabola folded in half. The point where we fold it in half is called the axis of symmetry. This is because both sides of the fold contain a symmetrical part of the parabola.

The vertex of a quadratic function is the point on the graph of that function where, if we were to fold it, would correspond to the lowest or highest point. Take a look at the image below to get a better idea of the vertex.

In terms of what the point actually means, each coordinate corresponds to a different part of the function. Take a look at the table below for a description.

h Horizontal shift This point determines how many spaces to the right or left the graph has shifted from 0
k Vertical shift This point determines how many spaces up or down the graph has shifted from 0

Take a look at the image below to see the difference these coordinates make in the graph.

parabola_comparison

A B C
x^{2} (x-1)^{2} + 1 (x-1)^{2} + 5
(0,0) (1,1) (1,5)

Problem 1

You want to know the roots to a quadratic function. Given the following function, factor the quadratic function. After factoring the function, find the roots of the function.

    \[ 3x^{2} + 9x - 12 \]

Problem 2

Transform the following function from the standard form to the vertex form.

    \[ x^{2} + 3x - 4 \]

Problem 3

Describe the transformation that occurs from going to function A to function B. Give the vertex of both functions.

  • A: (x-1)^{2} + 1
  • B: (x-1)^{2} + 5

Solution Problem 1

Here is the solution to this problem:

    \[ 3x^{2} + 9x - 12 \]

    \[ 3(x^{2} + 3x - 4) \]

    \[ 3(x+4)(x-1) \]

    \[ 3(x+4)(x-1) = 0 \]

    \[ (x+4) = 0 \; and \; (x-1) = 0 \]

The roots are -4 and 1.

Solution Problem 2

Here is the solution:

    \[ y = x^{2} + 4x - 4 \]

    \[ y + 4 = x^{2} + 4x \]

    \[ y + 4 + (\frac{4}{2})^{2} = x^{2} + 4x + (\frac{4}{2})^{2} \]

    \[ y + 4 + 4 = x^{2} + 4x + 4 \]

    \[ y + 8 = (x+2)^{2} \]

    \[ y = (x+2)^{2} - 8 \]

Solution Problem 3

Going from (x-1)^{2} + 1 to (x-1)^{2} + 5 means going from vertex (1,1) to vertex (1,5). This means that while the shape of the parabola stays the same, we are shifting 5 units up for each point.

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Danica

Located in Prague and studying to become a Statistician, I enjoy reading, writing, and exploring new places.