# Grade 10 Quadratic Relations

### By: Prabhjot Gandham

## In The Real World

Quadratics are all around us, we see them almost everyday weather it be outside of your home. Some real world examples of quadratics are roller coaster drops, the Eiffel Tower in France as well as the Golden Gate Bridge located in San Francisco and some parabolas come naturally like a rainbow.

## Learning Goals

- Determine the finite differences (1st & 2nd) of a relation
- Describe the transformations and graph quadratic relations in vertex form
- Locate the characteristics of a parabola (i.e vertex, axis of symmetry, etc)

## First and Second Differences

## Vertex Form

*²*+k

*a*tells us the stretch on the parabola*h*tells us the horizontal translation or the x-value*k*tells us the translation of the y-value

Example: *y=-2(x+2)²-3*

- The
*a*or -2 tells us the vertical stretch by factor of 2 - The
*h*or +2 tells is the horizontal translation of 2 units left - The
*k*or -3 tells us the vertical translation down 3 units

__Axis of Symmetry (AOS): (Above as well)__

- The AOS is
*x=h*, so the*h*in the expression*y=a(x-h)²+k* - The AOS of the parabola above is -2

__Optimal Value: (Above as well)__

- It is written as
*y=k* - The optimal value of
*y=-2(x+2)²-3 would be -3*

__Transformation:__

They can occur in vertical or horizontal, vertical stretches and reflection

- Vertical Translation: The -3 in the expression shows the vertical translation. The negative indicates going downwards and the 3 indicates the units
- Horizontal Translation: The number inside the bracket, 2 shows the horizontal translation by 2 units. Since the number is positive it will move to the left but if it were negative then the parabola would move right.
- Vertical Stretch: The first 2 in the expression tell us the vertical stretch, if it was more than -1 and less than 1 it would be compressed.
- Reflection: The negative sign indicates the vertical reflection, if the number was a positive then the parabola would open upwards, since the number is a negative then the parabola would be flipped upside down.

__X-intercepts or Zeros:__

- You use this when needed to find x-intercept or zeros by setting y=0 and then solving

## Graphing Quadratic Functions In Vertex Form

## Transformations

## Factored Form

## Learning goals

- expand a binomial multiplied by a binomial
- create simplified expressions for perimeters, areas, and volumes
- common factor a polynomial
- factor both simple and complex trinomials
- factor perfect square trinomials and differences of squares

## Common Factoring

2x*2*+ 8x+4

take 2 common

2(x*2+4x+2)*

## Simple Trinomial

x*2+9x+20*

the sum has to be 9 but he product has to be 20 the numbers are 4 and 5

(x+4)(x+5)

## Complex Trinomial

*2+*13x+10

multiply a and c to get 40 and the sum has to be 13, the numbers are 8 and 5

4x*2*+8x+5x+10

4x(x+2)+5(x+2)

the factors are

(4x+5)(x+2)

## Difference of Square

(x*2-25)*

first square root the x2 and 25

(x-5)2

it would be one time plus and one time minus to get original equation

## Perfect Square

x2+6x+9

because the square root of the first term and second term multiply together then 2 should be the middle term

2(x*3)

2(3x)

6x

so it is a perfect square

## Word Problem

*²*+ 20t+ 60, where t is the time in seconds, and h is the height in meters.

a) Write the above formula in factored form

h=-5t*²* +20t +60

h= -5(t*²* -4 -12)

h= -5(t-6) (t+2)

b) When will the rock hit the water?

h= -5(t-6)(t+2)

- The rock will hit the ground at 6 seconds because since the 6 is negative you would put a positive instead because time cannot be negative

## Factoring a Complex Trinomial

## Graphing Using 3 Point Method

## Standard Form

## Learning Goals

- Convert a quadratic relation from standard form to vertex form (completing the square)
- Solve quadratic equations using the quadratic formula
- Use whats given in application questions to solve them

## Standard Form

- The form of this method is
*y=ax²+bx+c*

__Quadratic Formula:__

- This formula is the one you are suppose to use when doing standard form

## Using Quadratic Formula to Find X-Intercepts

**a=5**

**b=-7**

**c=2**

=-(-7)±√7² -4(5)(2)/2(5)

= 7±√49-40/10

=7±√9/10

=7±3/10

1. 7+3/10

**=1**

2. 7-3/10

**=0.40**

- First, you indicate what a,b and c is so in this case (a=5, b=-7 and c=2)
- Then, Sub into the equation
- Now, solve the numbers
- After than we get our two equations and x-intercepts

__Axis of Symmetry:__

- The formula for this is (-b/2a)

__Optimal Value:__

- For this we need to substitute the AOS with the original equation

__Completing the Square:__

- We use this to turn standard form into vertex form
- First factor them into the vertex form
- Place brackets around the numbers that are left
- Add and subtract the number from the brackets
- Write x and divide b by two then put squared outside of the bracket writing into vertex form