Parabolas in Vertex Form
Question One: A parabola goes through the point (0-2) and intersects the x-axis at (-2,0) and (2,0). What is the equation for the parabola?
Question Two: A parabola goes through the point (0,-2) and intersects the x axis at (-3.,0) and (3,0). What is the equation for the parabola?
Question Three: A parabola has a vertex at (3,5). When x is 1 the value of y is 7. Since the parabola is symmetric this means that when x is 5 y is also 7. What is the equation of this parabola?
Answer to Question One: We know that the parabola that goes through point (0,-2) can be written in the form
y=ax2 - 2
How do we know this? First, the axis of symmetry is x=0. Subtracting 2 from ax2 means that when x=0 y=-2.
We know that a22-2=0
This means a=0.5
When b=0 the quadratic formula reduces to (-4ac)0.5/ 2a
Plug a=0. and c=-2 into this formula and observe.
(-4 * (0.5) * (-2))0.5/ 2 *0.5
= 2 or -2
So a=0.5 gives us the correct root of 2 or -2.
The equation of the parabola is
Answer to Question Two: This parabola can also be written in the form
y=ax2 - 2
But we know that the coefficient of x2 is less than 0.5 because the parabola is flatter than the one in question 1.
To solve for the coefficient a, set y=0 at x = 3 or at x=-3.
We get 9a-2=0 or a=2/9.
Again, when b=0 the quadratic formula reduces to (-4ac)/2a
To confirm this answer is correct we plug a= (2/9) and c=-2 into this formula to solve for the root, which should be -3 or 3.
quadratic formula to solve for the root, which should be -3 or 3.
(-4 * (2/9) * -2)0.5/( 2 * (2/9))
3 or -3.
So a=2/9 gives us the correct roots of 3 or -3.
Answer to Question Three: The parabola in question 3 has the same shape as the parabola in question one. The coefficient of X2 (a) of both parabolas is 0.5.
However, the vertex of the parabola in question 3 is (3,5). The formula for the parabola in vertex form can be written as
Y = a (X-h)2 + k
We have values for a, h, and k. Our equation is
Y = 0.5 ( X-3)2 + 5
Expanding to get the equation in standard form gives us
Y = 0.5x2 -3x + 9.5
If you plug x=1 or X=5 into the above equation you will get y=7.
Note: What if I asked you to find the equation of a parabola with a vertex (3,5) that also went through points (1,7) and (5,7)? I would first find the formula for the identical parabola with vertex (0,2) and with points on the x-axis (-2,0) and (2,0). Once I found the coefficient of x2 for this identical parabola I would shift it to (3,5) to get the desired equation.
Other resource on this topic:
Below are links to other free resource on graphing parabolas and finding vertices.