Using Quadratics

Author: Emma Malar

Terms to Remember

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


Vertex: the highest or lowest point of the parabola, Ex: (x,y)


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:

A juggler throws a ball unto the air. She catches the ball with her other hand 5 feet above the ground. Using the model 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

1.) 12x^2 + 3x = 9


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

Graph

Equation: y = x^2 - 4x -2


Vertex: (2, -6)


x = -b/ 2(a)

x = -(-4) / 2(1)

x = 4/2

x = 2


x = (2)^2 - 4(2) - 2

x = 4 - 4(2) - 2

x = 4 - 8 - 2

x = -6


Axis of Symmetry: 2


Y-intercept:


(0, -2)


Minimum or Maximum: Minimum

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Citations

"How To Find The Maximum Or Minimum Value Of A Quadratic Function Easily". 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.