**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?

**Answers**:

**Answer to Question One**: We know that the parabola that goes through point (0,-2) can be written in the form

y=ax

^{2 }- 2
How do we know this?
First, the axis of symmetry is x=0.
Subtracting 2 from ax

^{2}means that when x=0 y=-2.
We know that a2

^{2}-2=0
Or 4a-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

Y=0.5x

^{2 }-2**Answer to Question Two**: This parabola can also be written in the form

y=ax

^{2 }- 2
But we know that the coefficient of x

^{2 }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.

Observe

(-4 * (2/9) * -2)

^{0.5}/( 2 * (2/9))
(4/3)/(4/9)

or

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 X

^{2}(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 x

^{2 }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.

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