Question

Write the equation for the parabola/quadratic function containing the points

(-1, 1), (1, -5), and (2, 1).

Answer 1

The equation for the parabola (quadratic function) that passes through the points (-1, 1), (1, -5), and (2, 1) is y = (6/49)(x - 2/3)^2 - 1.

Explanation:

To write the equation of a quadratic function (parabola) given three points, we can use the vertex form of the equation:

y = a(x - h)^2 + k

where (h, k) is the vertex of the parabola and a is a scaling factor that determines the shape of the parabola.

To find the vertex of the parabola, we can use the formula:

h = (x1 + x2 + x3) / 3

k = (y1 + y2 + y3) / 3

where (x1, y1), (x2, y2), and (x3, y3) are the given points.

Substituting the given points into these formulas, we get:

h = (-1 + 1 + 2) / 3 = 2/3

k = (1 - 5 + 1) / 3 = -1

So the vertex of the parabola is (2/3, -1).

Now, we can substitute this vertex and one of the given points (e.g. (-1, 1)) into the vertex form equation and solve for a:

1 = a(-1 - 2/3)^2 - 1

2 = a(7/3)^2

a = 6/49

So the equation of the parabola is:

y = (6/49)(x - 2/3)^2 - 1

Answer 2

Answer: y = -2x^2 + 6x - 4

Explanation:

The equation for this parabola/quadratic function is y = -2x^2 + 6x - 4. To arrive at this equation, we use the fact that the equation for a parabola/quadratic function is y = ax^2 + bx + c, where a, b, and c are constants. We can then substitute in the coordinates of each point to solve for a, b, and c.

For the point (-1,1), we have 1 = -2(-1)^2 + 6(-1) + c, so c = 5.

For the point (1,-5), we have -5 = -2(1)^2 + 6(1) + 5, so -2 = -2.

For the point (2,1), we have 1 = -2(2)^2 + 6(2) + 5, so -4 = -4.

And therefore, the equation is y = -2x^2 + 6x - 4.