# Quadratic

### Grade10

## Introduction

**quadratic relationships**.

## Analyzing Quadratics

Parabola

In Quadratics the graphed equations is called a parabola.

Parabola:

- parabola is always symmetrical
*U*-shaped curve- Has Axis of symmetry
- Has Optimal value
- Has Vertex

Second differences

to determine if a table of values is quadratic you take the second differences. Quadratic relations have second differences that are constant.

## Vertex form

Vertex Form: y=a(x-h)^2+k

we are now going to be looking at equations which models a quadratic relation, vertex form. Vertex form of a quadratic relation determining things like the

- vertex of the parabola
- axis of symmetry
- parabola will open upwards or downwards
- optimal value
- step pattern

Axis of symmetry:

- The axis of symmetry divides a parabola into two congruent parts
- If one were to draw a line through the vertex of a parabola, one would have the axis of symmetry

Optimal Value:

- Optimal Value is a Minimum or a Maximum value on your graph
- If parabola opens up it a Minimum
- If parabola opens down it Maximum

Vertex:

- The vertex of the parabola is the point at which the curve changes direction
- it in the middle of the parabola
- axis of symmetry is your x value of vertex
- optimal value is y value of vertex

Step pattern:

- If your parabola has an (a) of 1, the three points on the parabola will be over: 1 up 1, over 2 up 4, over 3 up 9
- help you point other points on the graph

## Analyzing vertex form

A value - vertical stretch , compression , parabola opens up or down

H value- Horizontal shift (right or left) , x value of vertex , axis of symmetry

K value- Vertical shift y-intercept (up or down),y value of vertex , optimal value

## x-intercepts/zero

Steps

1. Set y=0: 0=-(x+5)^2 + 1

2.Move k value: -1=-(x+5)^2

3.Divide by the a value: -1/-1=-(x+5)^2/-1

4.Square root from both sides: 1=x+5

5. Move h value: -5+-1=x

6. 2 solution, solve for x: 1.-5+1= -4 2.-5-1=-6

points: (-4,0)(-6,0)

Vertex Form is one of the easiest ways to graph a quadratic equation. But it is possibly to change any other form into Vertex form like Factored Form to Vertex form

## Factored form

Factored form: y=a(x-r)(x-s)

quadratic relations can be expressed in factored form, where *a* is still the value that determine if the parabola opens up or down, and *r* and *s* are x-intercepts (zeroes).

X- intercepts/ zeroes :

- set (x-r)(x-s) to zeroes
- x-r=0 example x
**-**8=0 x=8 - x-s=0 example x+2=0 x=-2
- (8,0)(-2,0)

Axis of symmetry : Finding the x value of vertex

- x=r+s/2
- example 8+(-2)/2 = 6/2=3 X=3
- (3,?)

Optimal Value : Finding the y value of vertex

- Sub the x value of the vertex into the original equation
- Example y= 2(x-8)(x+2) - y=2(3-8)(3+2) - y =2(-5)(5) - y=
**-**50 - (3,-50)

when graphing factored form remember that you have 3 key point on your graph

- 2 x-intercepts
- vertex ( a.o.s and opt. value)

## Standard Form

Quadratic relations can also be expressed in Standard form, Standard form determines thing like

- Zeroes (quadratic formula)
- Discriminant(b^2-4ac)
- Axis of Symmetry (-b/2a)
- Optimal Value (sub in)
- Completing the square to turn to vertex form

## Quadratic formula (Zeroes)

x= -b+-square-root b^2-4ac/2a

here is an example of standard form turning into quadratic formula to solve for x

## Discriminant

b^2-4ac

determine how many solutions an equation has

- negative in the in the square root ( no solutions)
- positive in the square roots ( 2 solutions)
- 0 in the square root (1 solution)

## Axis of symmetry

finding the axis of symmetry is easier than going through the whole process of the quadratic formula

-4/2(1) - -4/2 = -2

X=-2

## Optimal value

- sub in the x value in to the equation
- if you can find your roots and vertex than you have all the points to graph

## Completing the square to turn to Vertex Form

Easy way to transform a Standard form equation into a Vertex Form equation.

## Factoring to turn to factored form

Common

Complex Trinomial

Factoring by grouping

Perfect Squares

Difference of Squares

## Factoring: Common

Common factor is very simple if your equation is 5c+10d you can factor out a 5 because 5 can go into 5 and 10. After diving your equation with 5 your left with 5(c+2d). If you equation had -8y^5 and -6y^3, you can factor out a -2y^3. You can factor out 2 because 2 can go into 8 and 6 but in order to get the correct answer you can factor out an -2 because both terms are negative. Also you have to factor out a y because both terms have a y but the question is how many y, always factor out the lowest exponent in this case the lowest exponent is ^3 so you factor out y^3. After diving your equation by -2y^3 your left with -2y^3(4y^2+3y)

## Factoring SimpleTrinomial

## Factoring :Complex Trinomial

## Perfect Squares

## Difference of Squares

ax-b

Two numbers in the expression, both numbers can be square rooted but the second bracket is negative symbol instead of a positive.example - 16x^2-81- (4x+9)(4x-9)

## Word Problems

- Motion
- Area

Motion Problems

When solving motion problems, a sketch is often helpful and a table can be used for organizing the information

## Area Problems

- slit the shape in two shapes
- create a separate area equation for each shape
- them add both