# Quadratic Relationships

### Unit 3 - Standard Form

## Learning Goals

By the end of this unit you should be able to:

- determine max/min values of a quadratic relation using zeroes and symmetry
- convert a quadratic relation from standard form to vertex form (complete the square)
- solve quadratic equations using the quadratic formula
- solve real-world problems involving max/min

## Summary

An equation in Factored Form looks like this:

*y*=*ax*² +*bx*+*c***The Variables:**

- the value of
*a*gives you the shape and direction of opening - the value of
*c*is the y-intercept

To get the **x-intercepts**, solve using the **quadratic formula**.

The number inside the square root (*b*² - 4*ac*) of the quadratic formula is called the **Discriminant** (*D*). It helps us tell **how many solutions** the quadratic equation will have without having to use the whole formula.

When ** D < 0**, there will be

**no solutions**(x-intercepts). When

**, there will be**

*D*> 0**2**solutions. And when

**, there will be only**

*D*= 0**1**solution.

Use the **completing the square** method to convert standard form into **vertex form**.

## Quadratic Formula Example

## Completing the Square Example

## Example of a Word Problem

The path of a basketball after it is thrown in the air is given by the following equation:

** h = -0.25d² + 2d + 1.5**, where

**is the height and**

*h***is the horizontal distance in metres.**

*d*- What is the initial height of the basketball?
- What is the
**maximum height**reached by the basketball and at what**horizontal distance**does this occur at?

**Solutions:**