# Quadratics 101

### By: Jagraj Kang

## Table of Contents

**Parabolas:**

-Vertex Form

-Axis of symmetry

-Optimal Value

-X-intercepts/Zeros/Roots

-Tranformations

-Step Pattern

-First difference

-Second differences

**Factored Form:**

-Axis of Symmetry

-Optimal Value

-Zeros/ X-intercepts

**Standard Form:**

-Quadratic Forumla

-Axis of Symmetry

-Optimal Value

-X-intercepts/Zeros

**Factoring:**

-Common Factoring

-Simple Trinomials

-Complex trinomials

-Completing the squares

**Word Problems**

## What is a Parabola?

## Vertex Form

- (h,k) is the vertex
- if "a" is positive, the parabola will be opening up and if it's negative, then its opening down
- if "a" is greater than 1 then it will be stretched and if its less than 1, then it will be compressed

## Axis of Symmetry

- "h" is the axis of symmetry. if "h" is positive in the equation, then it is a positive number and vice versa

## Optiaml value

- "k" is the optimal value
- if its positive, then it's above the x-axis and if its negative, then it's below the x-axis
- if its 0, then its on the x-axis

## X-intercepts/Zeros/Roots

y=4(x+4)^2-16

0=4(x+4)^2-16 Sub Y for 0

16=4(x+4)^2 Bring 8 over to the other side

16/2=(x+4)^2 Divide both sides by 4

4=(x+2)^2 Square both sides by - and +

+-2=x+2 calculate for the X's

-2=x+2

-2-2=x

x=-4

2=x+2

2-2=x

x=0

## Transformations

y=a(x-h)^2+k is the general equation for the quadratic.

- Where a controls if the parabola opens up or down.
- Where a controls if the parabola is stretched or compressed.
- Where h controls the horizontal shift.
- Where k controls the vertical shift.

All these are used when using transformations.

- If A is a positive number then the parabola will be opening up but if it is a negative it will be opening down.
- If the A value is greater than 1 then it is being stretched, if it is >1 but <0 then it is being compressed, if it is 0 then nothing changes.
- If the H value is increased by a number than the graph will shift up by that many points, but if it is decreased than it will shift down that many points.
- If the K value is increased than the graph shifts right by that many points, but if it is decreased than it shifts left by that many points.

## Step pattern

Another key thing to know is the step pattern. The step pattern is a basic rule to help you create your parabola. The step pattern is 1:1, 2:4. This means that from the vertex to create the parabola you have to use the step pattern. You have to move left or right 1 and then up or down 1, followed by left or right 2 and then up or down 4. The relation is to square the left right number. Thus if you move left or right 3 of the vertex you know that you have to go up or down 9. This pattern continues on and on.

## First Differences

## Second differences

## How to Graph form Vertex form

To graph from vertex form it is simple all you need to do is first find the vertex. Tom find the vertex all you have to do is look at the equation and see what the k-value is because it is going to be you y on the graph, also you look at your (h-value) which is inside the brackets which is you x value as shown above. Now you use those points to find you vertex.

Next look at the a-value which will tell you how much to move the next point. Now you would just use the step pattern and sub in the a-value. Plot the points on your graph and connect them.

## Factored Form y=(x-s)(x-r)

## Axis Of Symmetry

To find the axis of symmetry (AOS) you must first take the numbers inside the brackets out. y=(x-2)(x-5) to take out the coefficients you must first change the signs. -2 will become 2 and -5 will become 5. Next we must add the two number and divide it by two 2+5

=7/2

x=3.5

your axis of symmetry would therefore be 3.5.

## Optimal value

To find the optimal value from factored form you would sub in the axis of symmetry into the x's that are in the brackets and solve for y:

y=(x-2)(x-5)

y=(3.5-2)(3.5-5)

y=(1.5)(-1.5)

y=-2.25

In this case the Optimal value would be -2.25.

## Graphing in Factored Form

- Find A.O.S
- Sub it into the equation to get your y-value
- You have the vertex and x-intercepts and just connect the lines

## Standard Form y=ax^2+bx+c

## Quadratic formula

## Solving the Quadratic Formula

## Finding A.O.S

- Add both x-intercepts from the Quadratic formula
- Then divide it by 2

## Optimal Value

- Sub A.O.S in the original equation

## Discriminant

- D>0, 2 x-intercepts
- D<0, 0 x-intercepts
- D=0, 1 x-intercept

(discriminant formula is in red below)

## Factoring

## Common Factoring

## Simple Trinomials (S.F to F.F)

To solve for a simple trinomial:

- See if it is a simple trinomial
- If it is, find factors of 6 that can add to 5 (do this for all simple trinomials)
- Then put those numbers in (x+_)(x+_)

Example:

0=x^2 + 5x + 6

=2 x 3 = 6

=2 + 3 = 5

y=(x+2)(x+3)

## Complex Trinomials

- Step 1: Multiply the lead coefficient (2 in this case) by the constant term (-35 in this case) to get -70.
- Step 2: Find two numbers that multiply to make the product from step 1, but add to make the middle term coefficient (-3, in this case). Therefore, for our example, we need find the two numbers that multiply to make -70 but add to make -3. Of course, the numbers are -10 and 7.
- Step 3: Rewrite the original trinomial, replacing the middle term with two terms whose coefficients are the numbers from step 2.
- In other words,

2x2 -10x +7x -35.

- Step 4: Common factor the first two terms from step 3. Then, common factor the last two. Do the pairs separately; it won’t be the same common factor for the first two as for the last two.

- Step 5: Notice from Step 4 that, although the common factors you took out front don’t match, the brackets do match. Put the common factors in their own bracket, then rewrite:

- Step 6 (optional): Foil out your answer from Step 5 to check it.

Outer: 2x(-5)=-10x

Inner: 7(x)=7x

Last: 7(-5)=-35

- Add the four terms:

## Perfect Squares & Difference of Squares

- Perfect Squares are when you only need one bracket because that bracket squared gives you
- Difference of squares is similar to perfect square only instead of the same bracket the signs change but the number stays the same.

## Perfect Square

## Difference Of Squares

## Linking

All three forms: Vertex, factoring and standard can be changed to one another as shown below.

Also:

Vertex Form:

Vertex form can be converted to factored form by subbing Y=0 and then find find the factored form.Vertex Form can also be converted into standard form by expanding and simplifying the equation.

Graphing Vertex Form:

Vertex form us the vertex which we is helpful to graph. This is already given to us. Next we use the step pattern to graph the parabola.

Factored Form:

Factored form can be turned into vertex form by finding the vertex using the axis of symmetry and optimal value and then subbing those into the vertex form equation. It can also be turned into standard form by expanding and simplifying the equation.

Graphing Factored Form:

Factored form is the easiest to graph because the x intercepts are given. all you do is find the AOS and then the optimal value to graph the parabola by connecting the points.

Standard Form:

Standard form can be converted into vertex form by using the completing the square method. It can be converted into factored form by using the factoring methods seen above. Common, Simple, Complex, Different of squares, Perfect squares.

Graphing Standard Form:

Graphing Standard Form is hard so there are two ways to graph. Use the quadratic formula to find X intercepts or Complete the Square to get vertex form to graph or factor to turn into factored form then graph

## Word Problems

## A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway?

*x* + 12 + *x*

=2x+12

*x* + 16 + *x*

*=2x+16*

New area

(12 + 2*x*)(16 + 2*x*) = 285

192 + 56*x* + 4*x*2 = 285

4*x*2 + 56*x* – 93 = 0