March 31, 2021

Chapters

## 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.

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) = a + bx + c OR f(x) = a + 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) = 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.

A | B | C |

(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.

## Problem 2

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

## Problem 3

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

- A:
- B:

## 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.