Standard Form

y = ax² + bx + c

Learning Goals

- How to obtain the Min/Max values of a quadratic relation in standard form by completing the square and changing it into vertex form.


- How to find the X-Intercepts of a quadratic relation using the Quadratic Formula

Summary of the Unit

- Over the Unit, we learned about two crucial components of quadratics, standard form, and the quadratic formula.


First, we'll start with standard form...

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This is standard form. Standard form is important because from standard form, we can obtain Vertex form, which gives us the min/max values of the parabola, plus one of its factors. How do we convert standard form to vertex form? We do something called...

"COMPLETING THE SQUARE"


This changes a quadratic relation in standard form


y = 2x² + 12x + 11


to vertex form


y = 2 (x+3)² - 7


Next, we have the Quadratic Formula

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The Quadratic Formula, when solved completely, will give us the variables of a parabola. The variables a, b, and c are the same variables from Standard Form. When used in conjunction with standard form and completing the square, we can find the


1. X - Intercepts


2. Vertex point


With these 3 points, we can graph our Parabola.


Finally, the last thing to cover is the discriminant. The discriminant is the last part of the quadratic formula, D = ² - 4ac


If D = 0, there is 1 x-intercept.

If D > 0, there are 2 x-intercepts.

If D < 0, there are 0 x-intercepts.

Quadratic Formula Example

Using the example above for completing the square, we have 2x² + 12x + 11

A = 2

B = 12

C = 11


Now in quadratic form, we get x =[ -12 +-12²-4(2)(11)]/2 (2)

x = [ -12 +-12²-4(22)]/4

x = [ -12 +-12²-88]/4

x = [ -12 +-144-88]/4

x = [ -12 +-56]/4

x = [ -12 +- 7.5]/4


= -12 + 7.5 = -4.5

-4.5/4 = -1.125


= -12 - 7.5 = -19.5

-19.5/4 = -4.875


X intercepts are (-1.125,0) and (-4.875,0)

Completing the Square example


1. Bracket ax² + bx, and factor NUMBERS ONLY.

y = 2 (x²+6x) + 11


2. Divide bx by 2 and then square it.

6x/2 = 3

3² = 9


3. Plug in number twice, once negative and once positive into brackets

y = 2(x²+6x+9-9) + 11


4. Bring out the negative term out of the bracket, and factor it by whatever factor is before the bracket.

y = 2(x²+6x+9) + 11 - 18


5. Square Root the first and third values of the bracket and then you're DONE.

y = 2(x²+6x+9) + 11 - 18


y = 2 (x+3)² - 7

Word Problem ( Completing the Square)

The flight of a toy rocket is modeled by the equation h = -5x² + 50x + 1.3, where h represents height in meters and x represents time in seconds. What is the max height of the toy rocket?


We need to complete the square in order to get vertex form, which will give us the vertex of the rocket.


h = -5(-x²-10x) +1.3

10/2 = 5

5² = 25

h = -5(x²-10x + 25 - 25) +1.3

h = -5(x²-10x + 25) +1.3 +125

h = -5(x²-10x + 25) +126.3

h = -5(10x + ) +126.3

h = -5(x + 5)²+126.3


Vertex is The max height of the rocket is 126.3 m.