# Using Quadratics

### Author: Emma Malar

## Terms to Remember

Standard form of Quadratic formula: f(x)= ax^2 + bx + c

Axis of Symmetry: vertical line that goes through the middle of the parabola through the vertex, when x= the x-value of the vertex, use x= -b/2(a) to find AOS

X-intercepts: aka the zeros, where the graph crosses the x-axis, y=0, write in coordinate form (x,0)

Y-intercept: when x=0, vertical line crosses the y-axis, coordinate form (0,y)

Maximum/Minimum Point: Maximum point is when the graphs direction of opening is down, minimum point is when the graphs direction of opening is up.

Maximum/Minimum Value: just the y-coordinate of the vertex, ( just a # )

Quadratic Formula: **x = -b +/- **√**b^2 - 4ac**** / 2a**

Discriminant: what is under the square root, b^2 - 4ac, tells how many solutions there are, if it is greater than 0 there will be 2 solutions, less than 0 there will not be a solution, if it is equal to 0 there will be one solution.

## Example:

**h = -16t^2 + 15t +5,**where t is the time, h is final height, h0 is the initial height, and v0 is the initial velocity.

a. What is the maximum height the juggler can throw the ball?

**finding the y-value, time = x, maximum height is about 8.52 feet**

**x = -b / 2 (a)**

**x = - 15 / 2(-16)**

**x = -15 / -32**

**x = 15/ 32**

**x = .47**

**h = -16(.47)^2 + 15(.47) + 5**

**h = -16 * .2209 + 15(.47) + 5**

**h = -3.53 + 7.05 + 5**

**h = 8.5 feet**

b. How long does it take for the ball to reach its maximum height?

**finding the x-value, it takes about 0.47 seconds to reach the maximum height**

**h = -16t^2 + 15t + 5**

**x = -b / 2(a)**

**x = -15 / 2(-16)**

**x = -15 / -32**

**x = 15 / 32**

**x = .47 seconds**

c. How long does the juggler have to catch the ball in their other hand? Assume they catch the ball at the same height.

**finding positive zero, the catcher has about 1 second to catch the ball in their other hand**

**h = -16t^2 + 15t + 5**

**-15 + / - √ 15^2 - 4( -16)(5)**

**-****15 + / - ****√ 225 ****- 4( -16)(5)**

**-****15 + / - ****√ 225 + 69**

**-15 + **/ - √ 294

x = -15 + 17.15 / 2(-16)

x = -.07

x = -15 - 17.15 / 2(-16)

x = 1.00

d. How high will the ball be after 0.25 seconds?

**finding function value, about 7.75 feet after 0.25 seconds**

**h = -16(.25)^2 + 15(.25) + 5**

**h = -16 * .0625 + 15(.25) + 5 **

**h = -1 +3.75 + 5**

**h = 7.75 feet**

## 3 Examples of Quadratics in the Real-World

## 3 Examples solved by factoring or quadratic formula

12x^2 + 3x - 9 = 9 - 9

12x^2 + 3x - 9 = 0

(12x^2 + 3x -9) / 3

4x^2 + x - 3

4* - 3 = -12

-3 +4 = 1

(4x - 3) (4x + 4)

(4x+4) / 4

(4x - 3) ( x + 1)

4x - 3 = 0

4x -3 +3 = 0 +3

4x = 3

4x /4 = 3 /4

**x = 3/4 or**

x + 1 = 0

x + 1 - 1 = 0 - 1

**x = -1 **

2.) -6x = 7x^2 - x - 12

-6x +6x = 7x^2 - x + 6x -12

0 = 7x^2 + 5x - 12

(7x + 12) (7x -7)

(7x-7) / 7

x - 1 = 0

x - 1 + 1 = 0 +1

**x = 1 or**

7x + 12 = 0

7x +12 -12 = 0 -12

7x = -12

7x / 7 = -12 / 7

**x = -12 / 7**

3.) 5x^2 + 9x + 4 = 0

x = -b +/- √b^2 - 4ac / 2a

x = -9 + / - √9^2 - 4(5)(4)

x = -9 + / - √81 -20 *4

x = -9 + / - √81 - 80

x = -9 + / - √1

x = -9 + 1 / 2(5)

**x = -4/5 or**

x = -9 - 1 / 2(5)

**x = -1**

## Citations

*wikiHow*. N.p., 2016. Web. 29 Apr. 2016.

Gillani, S. "Beautiful Collection Of Bluebird Pictures". *Fun Peep*. N.p., 2015. Web. 29 Apr. 2016.

Simonetti, Giuseppe. "Microgravity Page". *Termserv.casaccia.enea.it*. N.p., 2016. Web. 30 Apr. 2016.

Meyer, Dan. "The Three Acts Of A Mathematical Story". *dy/dan*. N.p., 2011. Web. 30 Apr. 2016.

"Sscyear10mathsa - Quadratics". *Sscyear10mathsa.wikispaces.com*. N.p., 2016. Web. 30 Apr. 2016.