Monday, December 7, 2015

Parabolas in Vertex Form

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=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


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=0.5x2 -2


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.


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 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.











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