Write an equation of a parabola with the x-intercepts at [1/2,0] and (-6,0) which passes through the point (0,6)

Answer 1

To find the equation of a parabola, we will use the vertex form. The vertex form of a parabola equation is given by:

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

where (h, k) represents the vertex of the parabola.

Since the x-intercepts are at [1/2, 0] and (-6, 0), we know that the parabola crosses the x-axis at these points. This means that the vertex lies between these two points. The x-coordinate of the vertex is the average of the x-intercepts:

h = (1/2 + (-6))/2 = -5/4

Since the parabola passes through the point (0,6), we can substitute these values into the equation:

6 = a(0 - (-5/4))^2 + k
6 = a(5/4)^2 + k
6 = a(25/16) + k

To find the value of k, we substitute the y-coordinate of the vertex (k = 0) into the equation:

0 = a(-5/4)^2 + k
0 = a(25/16) + k
0 = 25a/16 + 0
25a/16 = 0

This tells us that a = 0. Substituting this value of a into the equation, we have:

6 = 0(25/16) + k
6 = 0 + k
k = 6

Therefore, the equation of the parabola is:
y = 0(x - (-5/4))^2 + 6
y = 6