SOLUTION: Given the following 3 points, find the particular equation. Graph the parabola. Label the axis of symmetry, vertex, y-intercept and its symmetric partner, and the x-intercepts.

Algebra ->  Rational-functions -> SOLUTION: Given the following 3 points, find the particular equation. Graph the parabola. Label the axis of symmetry, vertex, y-intercept and its symmetric partner, and the x-intercepts.       Log On


   

Question 358803: Given the following 3 points, find the particular equation. Graph the parabola. Label the axis of symmetry, vertex, y-intercept and its symmetric partner, and the x-intercepts.
(3,138) (10, 292) (15, 282)
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
y=ax%5E2%2Bbx%2Bc
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138=a%283%29%5E2%2Bb%283%29%2Bc
1.9a%2B3b%2Bc=138
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292=a%2810%29%5E2%2Bb%2810%29%2Bc
2.100a%2B10b%2Bc=292
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282=a%2815%29%5E2%2Bb%2815%29%2Bc
3.225a%2B15b%2Bc=282
Subtract eq. 1 from eq. 2 to eliminate c,
100a%2B10b%2Bc-9a-3b-c=292-138
91a%2B7b=154
4.13a%2Bb=22
Subtract eq. 1 from eq. 3 to eliminate c,
225a%2B15b%2Bc-9a-3b-c=282-138
216a%2B12b=144
5.18a%2Bb=12
Subtract eq. 4 from eq. 5 to eliminate b
18a%2Bb-13a%2Bb=12-22
5a=-10
highlight%28+a=-2%29
Then,
-36%2Bb=12
highlight%28+b=48%29
Then,
-18%2B144%2Bc=138
highlight%28+c=12%29
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highlight_green%28y=-2x%5E2%2B48x%2B12%29
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Convert to vertex form, y=a%28x-h%29%5E2%2Bk, by completing the square.
y=-2%28x%5E2-24x%29%2B12
y=-2%28x%5E2-24x%2B144%29%2B12%2B2%28144%29
y=-2%28x-12%29%5E2%2B12%2B288
y=-2%28x-12%29%5E2%2B300
The vertex is (12,300).
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The vertex lies on the axis of symmetry x=12.
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The y-intercept is,
y=-2%280%29%5E2%2B48%280%29%2B12=12
Its symmetric partner would be
x=12%2B12=24
y=12
(24,12)
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The x-intercepts are,
-2%28x-12%29%5E2%2B300=0
2%28x-12%29%5E2=300
%28x-12%29%5E2=150
x-12=0+%2B-+sqrt%28150%29
x=12+%2B-+5%2Asqrt%286%29
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