Question

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

Answer 1

Answer:

  (d)  y = 3x² +2x -6

Step-by-step 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.

Answer 2

Answer:

$y = 3\, x^{2} + 2\, x - 6$.

Step-by-step 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 $a \ne 0$.

Let $y = a\, x^{2} + b\, x + c\!$ denote the equation of this parabola for some constants $a$, $b$, and $c$ where $a \ne 0$. 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:

$y = 3\, x^{2} + 2\, x - 6$.