## 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

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

## Vertex Form

Vertex form equation is y=a(x-h)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

1. Vertical Translation: The -3 in the expression shows the vertical translation. The negative indicates going downwards and the 3 indicates the units
2. 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.
3. 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.
4. 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

How To Graph Using Vertex Form

## Transformations

Transformations - Graphing from Vertex 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

Example is...

2x2+ 8x+4

take 2 common

2(x2+4x+2)

## Simple Trinomial

example is...

x2+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

Example is 4x2+13x+10

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

4x2+8x+5x+10

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

the factors are

(4x+5)(x+2)

## Difference of Square

example is...

(x2-25)

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

example is...

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

-2x2+4x+5

## Word Problem

1. The height of a rock thrown from a walkway over a lagoon can be approximated by the formula h = -5t² + 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=-5t² +20t +60

h= -5(t² -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

Factored Form

## Graphing Using 3 Point Method

Graphing Factored Form of Quadratic Functions

## Learning Goals

1. Convert a quadratic relation from standard form to vertex form (completing the square)
3. Use whats given in application questions to solve them

## Standard Form

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

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

## Using Quadratic Formula to Find X-Intercepts

y=5x²-7x+2

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

## Completing the Square

Converting Standard to Vertex Form

## Maximizing Revenue Word Problems

Maximizing Revenue Word Problem (Completing the Square): Straightforward Worked Example!

## Reflection

Throughout the 3 quadratic units I have learned to answer, solve and graph using vertex form, factored form and standard form. For vertex form the h,k value give us the vertex. the axis of symmetry is represented as (x=h) and the optimal value is represented as (y=k). the "a" value tells me the direction of opening and if the parabola is going to be compressed or stretched. The "h" tell me the horizontal translation and the "k" tells me the vertical translation. The factored form equation is y=a(x-r)(x-s). the a value tells me the shape and opening of the parabola. The r and s values gives us the x-intercepts. Standard form provides the y - intercept and once you complete the square you get the vertex and once you use the quadratic formula you get the x-intercepts. I have learned that factoring connects to graphing because it gives the x - intercepts and then we can find the axis of symmetry and substitute into the original equation and solve for y. I also learned that we can convert from standard form to vertex form by completing the square as well and that if we expand vertex form then we get standard form. To end with, I think the quadratic units were a lot of fun and showed me how to do things I didn't even know.