Alyssa N.
asked • 01/15/20Finding equation with line passing through 3 points
Passing through 0.-16 and -2,0 and -3,2. Please show all steps.
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1 Expert Answer
Barry M. answered • 01/16/20
Tutor
4.9
(19)
Professor, CalTech Grad; Many Years Tutoring Math, SAT/ACT Prep, Chem
Let's try for a parabola. (At first glance, it doesn't look like a circle could be formed from the 3 points--maybe an ellipse or hyperbola could work)
In standard form, y = ax2 + bx + c.
The points are given as 0.-16 and -2,0 and -3,2.
Given that when x = 0, y = -16, it's clear that c = -16. Plugging in the other values into the standard equation:
0 = 4a - 2b - 16 and
2 = 9a - 3b - 16
Dividing these 2 equations by 2 and 3, respectively, and moving around the variables gives
2a - b = 8 and
3a - b = 6
Solving for a and b gives a = -2, b = -12. The equation will then be y = -2x2 - 12 x - 16.
Arnold V.
tutor
Tutor Barry M. gave an elegant answer that parabola y = -2x^2 - 12 x - 16 passes through the given points (0.-16) and (-2,0) and (-3,2). In a similar way we find the curves described by cubic functions y = a(x + b)^3 + c that pass through the same three points. The two solutions exit and they are:
y = -2(x + 2)^3
and
y=[-2(x+4)^3 +16]/7.
Report
01/20/20
Barry M.
tutor
Yes. Thank you for the compliments. I think that infinitely many cubic and higher degree polynomials will also work, as there will be more coefficients than constraints (equations). Maybe if you have time, see what results if you try an ellipse or hyperbola. On second thought, it's probably going to be very time consuming.
Report
01/20/20
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Arturo O.
The 3 given points are not in the same line. Compare the slopes of the line passing through the first 2 points and the line passing through the last 2 points. The slopes are different, and hence the lines are different.01/15/20