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Find the equation of the parabola with points (-3,15), (0,-6), & (2,10)

Find the equation of the parabola with points (-3,15), (0,-6), & (2,10)-example-1
User DirkyJerky
by
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2 Answers

20 votes
20 votes

Answer:


y = 3\, x^(2) + 2\, x - 6.

Explanation:

In general, the equation of a parabola is in the form
y = a\, x^(2) + b\, x + c for some constants
a,
b, and
c, where
a \\e 0.

Let
y = a\, x^(2) + b\, x + c\! denote the equation of this parabola for some constants
a,
b, and
c where
a \\e 0. A point
(x_(0),\, y_(0)) is on this parabola if and only if the equation of this parabola holds after substituting in
x = x_(0) and
y = y_(0):


y_(0) = a\, {x_(0)}^(2) + b\, x_(0) + c.

Thus, each of the three distinct points on this parabola would give a equation about
a,
b, and
c:

  • The equation for
    (-3,\, 15) would be
    15 = (-3)^(2)\, a + (-3)\, b + c.
  • The equation for
    (0,\, 6) would be
    6 = 0^(2)\, a + 0\, b + c.
  • The equation for
    (2,\, 10) would be
    10 = 2^(2)\, a + 2\, b + c.

Simplify the equations:


\left\lbrace\begin{aligned}& 9\, a - 3\, b + c = 15 \\ & c = 6 \\ & 4\, a + 2\, b + c = 10\end{aligned}\right..

Solve this linear system of three equations and three unknowns for
a,
b, and
c:


\left\lbrace \begin{aligned} & a = 3 \\ & b = 2 \\ & c = (-6) \end{aligned}\right..

Therefore, the equation of this parabola would be:


y = 3\, x^(2) + 2\, x - 6.

User Max Bolingbroke
by
2.5k points
13 votes
13 votes

Answer:

(d) y = 3x² +2x -6

Explanation:

The equation of a parabola through thee points can be found different ways. One is to use a quadratic regression tool. Another is to write and solve linear equations in the coefficients. (3 equations in 3 unknowns).

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Here, we can eliminate answer choices that don't work to arrive at the correct answer choice.

The point (0, -6) is the y-intercept of the function. That means the value of the constant term in the quadratic is -6. (Eliminates A and C.)

The y-values of the other two points are both greater than -6, indicating the parabola opens upward. That means the leading coefficient is positive. (Eliminates B.)

The only reasonable choice is D:

y = 3x² +2x -6

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Additional comment

You get the same answer if you use a regression tool.

Find the equation of the parabola with points (-3,15), (0,-6), & (2,10)-example-1
User Kristiyan Varbanov
by
3.0k points