Introduction

In grade 9 you learned about linear relationships, and how the equation of a line in y=mx+b form can help us to make a sketch of the line. In grade 10 you are going to explore a different type of relationship called a quadratic relationships.

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

Analyzing vertex form

What affects does each thing have on the graph

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

Graphing vertex form part 1
Graphing vertex form part 2

x-intercepts/zero

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

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)
IMG 5724
You can change Factored Form to Standard Form but most of the time your changing Standard Form to Factored Form, when factoring.

Standard Form

standard Form=ax^2+bx+c

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

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

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

if you can't not factor then you use quadratic formula

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

-b/2a

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

b/2^2

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

Completing the Square and Vertex Form of Quadratic Equations

Factoring to turn to factored form

Common

SimpleTrinomial
Complex Trinomial

Factoring by grouping
Perfect Squares
Difference of Squares

Factoring: Common

Many different ways that you can factor an equation. To learning about other method of factoring, it is crucial that you know how to common factor.

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 Using the Great Common Factor, GCF - Example 1

Factoring SimpleTrinomial

Simple trinomial is more complex than common factoring. A simple trinomial is always going to have (x^2) they will never be a number in front of it for a simple trinomial. an example is x^2+9x+20. When finding your answers you want to think when multiplying what 2 numbers give me blank.
Factoring Simple Trinomials

Factoring :Complex Trinomial

Complex Trinomial is the same thing as Simple Trinomial its just that instead of have a 1 in front of x^2 the number is always going to be more than 1. an example is 3x^2-5x+6
Factoring Complex Trinomials

Perfect Squares

Perfect squares is an easier way to answer simple trinomials and complex trinomials. When using Perfect squares you square root ax and the c value of your equation. But if you get a decimal then you cannot use this method.
Factoring perfect square trinomials

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)

Algebra - Factoring Differences of Squares

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