Question

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

Answer 1

`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`.

Answer 2

(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.