# Quadratics

### Standard Form, Vertex Form, Factored Form.

## Analyzing quadratics

## Quadratic Terms

## Example

Vertex - (1,9)

Min/max value- Y=9

Axis of symm - x=1

X-intercept - (-2,0) (4,0)

Y-intercept - (-8,0)

## Vertex

## Vertex Form

In vertex form

The H in the equation is the axis of symmetry (x=h)

The K is the optimal value (y=k)

The A determines whether it will be stretched or compressed, and if its reflecting up or down

y=-1(x+3)²-2

As you can see The axis of symmetry is -3 because the h is +3. The Optimal value is -2.

And the parabola opens down cause of -1

## Step pattern

## Graphing Vertex Form

## Completing the square (Changing from standard form to vertex form)

When solving this equation the first thing you want to do is remove a common factor

Y=2(x²+12x)-3

Find the constant that must be added and subtracted to create a perfect square(12/2)²

Y=2(x²+6x+9-9)-3

Group the 3 terms that form a perf square to do this multiple 2 by the -9 and move the bracket

Y=2(x²+6x+9)-18-3

Factor perfect square and collect like terms

Y=2(x+3)²-21

## Graphing Vertex Form Part B

With a problem like this first you want to find your y-intercept so you set your X to 0

Y=-(0+5)²+1

Then you just solve it like a normal equation

Y=-(5)²+1

Y=-25+1

Y=-24

Therefore your answer is (0,-24)

## Graphing Vertex Form Part B

Now that you have your y-intercept you must find your zeros, to do this set your y to zero

0=-(X+5)²+1

Firstly move k to the other side (1)

-1=-(x+5)²

Then you divide each side by your A (-1)

1=(X+5)²

After that you move your H (5)

-5+/-1=X

Lastly you solve for X

X=-5+1,X=-5-2

X=-4,X=-6

(-4,0)(-6,0)

## Factored Form

## Graphing Factored Form

X-4=0, X=4. Then you must state the axis of symmetry by adding your x's and dividing by 2.

-2+4/2= 1.Once you have your axis of symmetry you sub it into your equation as you can see from the picture you have all the three points needed to complete your graph the vertex and the two x intercepts

## Word problems

## A) How many shoppers does the mall need to have, so it will have a revenue of zero

## B) At how many shoppers does the mall make the most money?

## A shop is selling Pens for 20 each, 300 customers are willing to buy. For every $5 increase 30 fewer customers are willing to buy.

## A) What is the maximum revenue the store can get?

Let X represent the amount of $5 increase

Then you write out your equation which would be

R=(300-30X)(20+5X)

This is the equation because the first part is the number of customers which is 300 then 30 fewer which translates to 300-30

The second part is 20+5x because it represents price which is currently 30 and increases by 5

Now you substitute your y value and find X which is shown below.

Now that you have your x=ints you add them and divide by two

10-4/2= 3

Finally to find the max revenue your plug your 3 into the first equation

R=(300-30(3))(20+5(3))

R=(210)(35)

R=7450

Therefore the maximum revenue the store can make is 7450