To find the equation of a parabola, we need to know the coordinates of its vertex and one other point on the parabola. In this case, we have the coordinates of the y-intercept and one of the x-intercepts. The vertex is the point where the parabola reaches its highest or lowest point, and it can be found by finding the x-coordinate of the point where the two lines intersect. In this case, we can solve for x in the first two equations to find that the x-coordinate of the vertex is 4. We can then substitute this value into any of the equations to find the y-coordinate of the vertex, which is 0.
Now that we have the coordinates of the vertex, we can use the general form of a parabola equation, y = ax^2 + bx + c, to solve for the values of a, b, and c. We can substitute the coordinates of the vertex (4,0) into the equation to get 0 = a(4)^2 + b(4) + c. Simplifying this equation, we get 16a + 4b + c = 0.
Next, we can use the third equation to find the x-coordinate of the other intercept. We can solve for x in the third equation to get x = 4. This means that the other x-intercept is also 4, and we can substitute this value into the equation to find the y-coordinate of the intercept, which is 12.
Now we have two points on the parabola, the vertex (4,0) and the other intercept (4,12). We can substitute these coordinates into the general form of the parabola equation to get two equations: 0 = 16a + 4b + c and 12 = 16a + 4b + c. We can then solve these equations simultaneously to find the values of a, b, and c.
Once we have the values of a, b, and c, we can substitute them back into the general form of the equation, y = ax^2 + bx + c, to get the equation of the parabola. In this case, the equation is y = -3x^2 + 8x + 32.
I hope this explanation helps you understand how to find the equation for a parabola. Remember to always find the coordinates of the vertex and one other point
