How do you find the quadratic equation of a parabola?

To find the quadratic equation of a parabola, we need to determine the values of the coefficients a, b, and c in the general form of a quadratic equation, which is y = ax^2 + bx + c. There are a few different methods you can use to find these coefficients, depending on the information you have about the parabola.

One common method is to use the vertex form of the quadratic equation, which is y = a(x – h)^2 + k, where (h, k) represents the coordinates of the vertex. To find the vertex form, we need to determine the values of h and k.

Step 1: Calculate the x-coordinate of the vertex, h, using the formula: h = -b/2a. This formula is derived from the fact that the x-coordinate of the vertex lies on the axis of symmetry, which is given by the equation x = -b/2a.

For example, let’s say we have the quadratic equation y = 2x^2 + 4x + 3. By comparing this equation to the general form y = ax^2 + bx + c, we can see that a = 2, b = 4, and c = 3.

Using the formula, h = -b/2a, we can substitute the values of a and b to find h: h = -4/(2*2) = -4/4 = -1.

So, the x-coordinate of the vertex is -1.

Step 2: Calculate the y-coordinate of the vertex, k, by substituting the value of h into the original quadratic equation and solving for y.

Using the example equation y = 2x^2 + 4x + 3 and the x-coordinate of the vertex, h = -1, we can substitute -1 for x:

K = 2(-1)^2 + 4(-1) + 3
= 2(1) – 4 + 3
= 2 – 4 + 3
= 1 – 4 + 3
= 0.

So, the y-coordinate of the vertex is 0.

Now that we have the coordinates of the vertex (h, k) = (-1, 0), we can substitute these values into the vertex form of the quadratic equation, y = a(x – h)^2 + k, to find the quadratic equation of the parabola.

Using the values of h = -1 and k = 0 in the vertex form, we have:

Y = a(x – (-1))^2 + 0
= a(x + 1)^2.

To determine the value of the coefficient a, we can substitute any other point on the parabola into this equation. For example, let’s say we have the point (2, 12) that lies on the parabola.