# Quadratics

### Lets make this easier

## Quadratics

## Things you need to know

Vertex: Is the maximum or minimum point on a graph, it is the point where the graph changes direction.

Minimum/Maximum Value: The highest or lowest value on the parabola which is the highest or lowest point on the y-axis. This is labeled as y=__.

Axis of Symmetry: The vertical line which in goes through the centre of the parabola, this is labeled as x=__.

Y-Intercept: Is where the parabola intercepts the x-axis, labeled as (0, __).

X-Intercepts: Where the parabola intercepts the x-axis, it is labeled as (__, 0).

Zeros: The x-value which makes the equation equal to zero. Labeled as x=__.

## Parabola equations

## Type of equations

## 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=-4(x+6)²-8

- The a or -2 tells us the vertical stretch
- The h or +3 tells is the horizontal translation
- The k or -4 tells us the vertical translation

__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+3)²-4 would be -4

__Transformation:__

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

- Vertical Translation: The -4 in the equation above shows the vertical translation. The negative shows that its going to be downwards and the 4 shows the units
- Horizontal Translation: The number inside the bracket, 3 shows the horizontal translation which is by 3 units. If the number is positive it will move to the left but if the number was a negative then the parabola would move right.
- Vertical Stretch: The 2 in the equation tell us the vertical stretch
- Reflection: The negative sign shows the vertical reflection, if the number was a positive then the parabola would open upwards, If the number is a negative then the parabola would be flipped image which is upside down

__X-intercepts or Zeros:__

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

## y=2(x-4)-3

__The Step Pattern:__

To graph you need to find what the vertex is, in this case it is (-2,-3). Now that you know the vertex follow the step pattern which is over one up one, over two up four. Now using this we have to multiple the a value by -2. So instead of going over one up one and over two up four we are going to go over one down four and over two down 8 (1 multiplied by -2 is -2 and 4 multiplied by -2 is -8, since they are negative numbers we go down instead of up).

## Factored form

This is a form of quadratic relationships which is written as **y=a(x-s)(x-t)**

- To graph the standard form we need to find what the x-intercepts are, we also have to add up the two x-intercepts and divide them by two so that we can have the minimum/maximum value.
- Example: y=4(x-4)(x+4) so the x-intercepts are (4,0) and (-4,0), now you add and divide them 4+(-4)/2 =2. So now you would sub in the 3 as x
- This will figure out what the minimum/maximum values are and help to find the vertex

- y=2(x-4)(x+4)

- y=2(3-4)(3+4)

- y=2(-1)(7)

- y= 2(-7)
- y=-14

- So now knowing what the value is (-14), we can figure out the vertex which is (3, -14)
- Graphed as below

## Zeros or X-intercepts:

- Zeros are just used by setting the numbers to zero
- Example: y=0.5(x+3)(x-9), must set y to zero (y=0)

y=0.5(x+6)(x-18)

0=0.5(x+6)(x-18)

x+6= 0

x= -6

x-18= 0

x= 18

- The zeros or x-intercepts are -6 and 18

## Standard Form

The equation for this type of form is y=ax²+bx+c

## Zeroes

- When using zeroes the only thing you have to do is put the zero into the formula
- For example:

5x² -7x+4=0

a=5

b=-7

c=4

=-(-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 find what a,b and c is so in this equation (a=5, b=-7 and c=2)
- After, Sub a,b,c into the equation
- Then, you can solve the equation
- And finally we get our two equations and x-intercepts

## Completing the square

- We use this to turn standard form into vertex form
- First factor them so they are the vertex form
- Place brackets around the numbers that are remaining
- 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

## Axis of symmetry

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

## Complex Trinomials:

- The formula for this is a²+2ab+b² an example is 6x²+11x+4
- First you would find numbers that work with this for example

(3x+4)(2x+1)

(3x)(2x)= 6x

(3x)(1)= 3x

(4)(2x)= 8x

(4)(1)= 4

6x²+3x+8x+4

6x²+11x+4

- We got the like terms and put them together to find the final solution

## Simple Trinomial:

- Simple trinomials formula is x²+bx+c
- An example is g²+5g+6
- First I needed to place g in the brackets for the reason being that we need 2 g's so that we can get g². Now we look for the multiples of 6 (2x3,6x1), now we look at 5g and check what will give us 5, so (g+3)(g+2).

## factoring to turn to factored form

## Common Factoring:

- This is when you need to find out the GCF
- Example:

4x+8y

4(x+2y)

- What I did was divide everything by two, and putting it into brackets which was factored

## Difference of Squares:

- Squared terms
- Example: x²-4 would be (x-2)(x-2)

## Perfect Squares:

- Makes equations simpler when you are expanding

An example would be y=(x+3)²

## Key Things

__Vertex to Standard:__ Expand and gather the like terms

__Standard to Factored:__ Factor out as much as you can

__Factor to Vertex:__ Expand on equation and gather all the like terms possible

__Vertex to Factored:__ Need to make y to 0 (y=0) and then figure out the x-intercepts

__Factored to Standard: __Need to do gather terms and expand as well

__Standard to Vertex:__ Use the completing the square method

1. Write and Simplify an expression to represent the area of the given composite figure

=(x)3+(x+2)(x-2)

=3x+x²-2x+2x-4

=3x+x²-4

b. If the area of the shape is 36cm², determine the value of x.

A= x²+3x-4

36= x²+3x-4

0= x²+3x-40

0= (x+8)(x-5)

x= -8 OR x= 5

X=5 is the correct answer because it is not negative.

## 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 formula stated above in factored form

h=-5t² +20t +60

h= -5(t² -4 -12)

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

- The rock will hit the ground at 6 seconds because the 6 is negative number you would make the number a positive.

## Reflection

- When we started this unit I understood it, but it got more complicated for me, and I understood some but had but had difficulty with others. This was my first time I learnt about parabolas.
- I did well on the first quadratic quiz but as the the test came I just wasn't doing so well