Quadratics Review

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  • Topic: Quadratic equation, Quadratic function, Elementary algebra
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  • Published : December 11, 2012
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Unit 2: Quadratic Relations
January-04-12
1:07 AM
Standard Form: y=ax2+bx+c
 
Vertex Form: y = a(x – h)2 + k. 
 
Factored Form: y=ax-sx-t
 
Quadratic Formula:
x=-b±b2-4ac2a
 
Word Problems:
-Physics:
An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. Answer the following. a. Graph

b. Determine the maximum height the object reaches
c. Determine at what time the object reaches the ground
d. Determine the initial height of the object
 
Graph:

 
b.
y=-4.9t2+19.6t+58.8
x =-b2a
x =-19.62(-4.9)
x=-19.6-9.8
x=2
 
y=-4.9t2+19.6t+58.8
y=-4.922+19.62+58.8
y=-19.6+39.2+58.8
y=78.4
∴ the maximum height the object reaches is 78.4 meters after two seconds  
c.
y=-4.9t2+19.6t+58.8
y =-4.9t2-4t-12
y =-4.9t -6(t+2)
t1 =6 t2=-2
tip: always check if the equation is factorable. It saves time.  
∵ this problem is about time than a negative time doesn't make sense. ∴ the object lands after 6 seconds.  
d. the initial height is the y-intercept which is 58.8 meters.  
-Finance:
You run a canoe-rental business on a small river in Ohio. You currently charge $12 per canoe and average 36 rentals a day. An industry journal says that, for every fifty-cent increase in rental price, the average business can expect to lose two rentals a day. Use this information to attempt to maximize your income. What should you charge?  

y=12+0.5x36 -2x
y=432 -24x+18x-x2
y=-x2+6x+432
x =-b2a
 
x =-62(-1)
x =-6-2
x =3
y=-x2+6x+432
y=-32+6(3)+432
y=-9+18+432
y=441
∴ to maximize revenue I should charge $13.5.
 
-Geometry:
You have a 500-foot roll of fencing and a large field. You want to construct a rectangular playground area. What are the dimensions of the largest such yard? What is the largest area?  
Let "L" be the length
Let "W" be the width

2l+2w=500
22l=-2w+5002

l=-w+250 
A=l∙w
A=-w+250w
A=-w2+250w
w =-b2a 
w =-(250)2(-1)
w =125
A=-(125)2+250(125)
A=-15625+31250
A=15625
∴ the dimensions for the largest area are 125 x 125.
The maximum area is 15625 cm2 
 
Sample equation:
y = -3x2 + 21x + 24
 
Properties:
* a factor is -3
* b is 12
* c or y-intercept is 8
* quadratic equation is a maximum since the a factor is negative * since a >1  than the graph is vertically stretched * x-axis is at 3.5
* The equation is translated 3.5 units to the right and 60.75 units up  
Table of Values:
(you can find the table of values by plugging in the x-value in the equation or by using an online application)  
http://www.wolframalpha.com/widgets/gallery/view.jsp?id=117cdc17fa822afc77008dd9ba74b6af x-value | y-value| First difference | Second difference | | | ignore| ignore|
| | | ignore| | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | since this is a second degree equation, the second differences are the same.  

tip: the second difference is "a" of the equation multiplied by 2. For example the value of a...
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