Standard Form
y = ax² + bx + c
Learning Goals
- How to find the X-Intercepts of a quadratic relation using the Quadratic Formula
Summary of the Unit
First, we'll start with standard form...
"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
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
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)
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.