March 31, 2021
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.
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.
|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) = a + bx + c |
f(x) = a + k
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.
|1||Start with the standard form f(x) = a + bx + c||f(x) = 3 + 6x + 1|
|2||For ease of notation, let f(x) be y||y = 3 + 6x + 1|
|3||Move the numbers so all x’s are on one side||y -1 = 3 + 6x|
|4||Simplify the right side so that the term has a coefficient of 1||y -1 = 3( + 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: = 1 |
y -1 +3(1) = 3( + 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|
|8||Get the y term by itself again||y = 3 + 2 |
f(x) = 3 + 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:
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||=|
|2||Find the two numbers where that equal to a*c and add up to b|| |
- (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:
Take a look at an example of a perfect square below.
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.
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.
Transform the following function from the standard form to the vertex form.
Describe the transformation that occurs from going to function A to function B. Give the vertex of both functions.
Solution Problem 1
Here is the solution to this problem:
The roots are -4 and 1.
Solution Problem 2
Here is the solution:
Solution Problem 3
Going from to 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.