Question

Write an equation, in the form y=ax² 2+bx+c, to represent each parabola given the points (5, 5), (0, 0), and (6, 0).

a) y=− 5/6 x²+5x
b) y= 5/6 x²−5x
c) y=−5/6 x²−5x
d) y= 5/6 x²+5x

Answer 1

To find the equation of a parabola given three points, we use the vertex form of a quadratic equation and solve for the constant a.

Step-by-step explanation:

To find the equation of a parabola given three points, we can use the vertex form of a quadratic equation, which is y = a(x - h)² + k.

First, we need to find the vertex of the parabola. The x-coordinate of the vertex can be found by taking the average of the x-coordinates of the given points. In this case, the x-coordinate of the vertex is (5 + 0 + 6)/3 = 11/3.

The y-coordinate of the vertex can be found by substituting the x-coordinate into the equation and using one of the given points. Using the point (5, 5), we have 5 = a(5 - 11/3)² + k. Solving for k, we get k = 5 - a(16/9).

Now we can substitute the coordinates of the vertex and one of the given points into the equation y = a(x - h)² + k. Using the point (0, 0), we have 0 = a(0 - 11/3)² + k. Substituting the value of k we found earlier, we get 0 = a(121/9) + (5 - a(16/9)). Simplifying this equation will give us the value of a.

After finding the value of a, we can substitute it back into the equation y = a(x - h)² + k to get the equation in the required form. After the calculations, we find that the correct equation is y = -5/6 x² + 5x.