Find Equation of Parabola Given Zero and Point

[LoveandHate]

stupid question, but if i know a zero of a parabola and a point it goes through, how do i find it's equation?

 

[rock.freak667]

stupid question, but if i know a zero of a parabola and a point it goes through, how do i find it's equation?

y=ax²+bx+c

sub the points into the general equation and solve for the constants a,b and c.

 

[LoveandHate]

thanks, but what point do i use?

 

[Mark44]

You might not have enough information. You're going to need three points to solve for the three unknowns, a, b, and c.

If you know the vertex and one other point, you can find a third point. Because of the symmetry of the parabola (which I'm assuming opens up or down, not left or right), if you have a point at (x₁, y₁), there will be another point across the axis of symmetry of the parabola. It will have the same y value, but will have a different x value.

 

[LoveandHate]

well i assumed, because they only gave me one zero, that that was the vertex of the the parabola. but the discriminant is larger than zero, so it has two. i was going to use the vertex form of the equartion, but that obviously won't work now.
the answer in my book is y=-2x^2-3x+3. this equation works for the point given (1, -2), but it does not work for the zero given (3,0).

i am an honours math student, but i cannot seem to figure this one out!

 

[Mark44]

It would be helpful if you gave us the exact wording of the problem.

Using the two points you gave--(1, -2) and (3, 0), and assuming that the vertex was at (1, -2), I found an equation of a parabola that goes through both points, namely y = (1/2)(x - 1)² - 2. It's easy to check that both points satisfy this equation.

If you know the vertex (h, k) and one other point, you can find the equation of a parabola. The form to use is y - k = a(x - h)². You can find a by substituting the point that isn't the vertex into this equation, which is what I did.

 

[LoveandHate]

yeahh. i completely understand what you did. i did that too, but i don't want to assume that (1, -2) is the vertex if they don't tell me.
the question says: "Determine the equation of a quadratic function that satisfies each set of conditions. b) x-intercept 3, and passing through the point (1, -2).
The answer they give is y=-2x^2-3x+3.

 

[Mark44]

That equation works if it is the y-intercept that is 3 - the point (0, 3). It doesn't work if an x-intercept is 3 -- the point (3, 0).