|
Question 637209: I was never taught parabolas in school, so if someone could explain them too me and how to solve these problems, it would be much appreicated.
A parabola with equation y = ax^2 + bx passes through the two points (-1,3) and (4,-2).
(a) Use algebra to find its equation.
Hint: Set up a system of two simultaneous equations.
(b) Is the point (2,-3) on the parabola? Justify your answer with working.
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website! A parabola with equation y = ax^2 + bx passes through the two points (-1,3) and (4,-2).
(a) Use algebra to find its equation.
Hint: Set up a system of two simultaneous equations.
(b) Is the point (2,-3) on the parabola? Justify your answer with working.
**
Graph of a parabola: y=ax^2+bx+c
3 equations are needed to solve, not 2
..
(1) 3=a(-1)^2+b(-1)+c
(2) -2=a(4)^2+b(4)+c
(3) -3=a(2)^2+b(2)+c
..
(1) 3=a-b+c
(2) -2=16a+4b+c
(3) -3=4a+2b+c
..
(1) 3=a-b+c
(2) -2=16a+4b+c
subtract to eliminate c
(4) 5=-15a-5b
..
(2) -2=16a+4b+c
(3) -3=4a+2b+c
subtract to eliminate c
(5) 1=12a+2b
..
(4) 5=-15a-5b
(5) 1=12a+2b
..
(6) 10=-30a-10b
(7) 5= 60a+10b
add to eliminate b
(8) 15=30a
a=15/30=1/2
2b=1-12a=1-6=-5
b=-5/2
c=3-a+b=3-1/2-5/2=0
..
a=1/2
b=-5/2
c=0
..
equation:
y=(1/2) x^2-(5/2)x+0
y=x^2/2-5x/2
..
Since point (2,-3) was used in developing the equation, it is on the curve of the parabola.
note: In developing the equation of a parabola the constant c must also be considered and you would need 3 variables and 3 simultaneous equations. In this case c turned out to be =0, but this is not aways the case.
|
|
|
| |
