**Assignment 2:**

** Investigating the Relationship between the Two Standard Forms of the Graph of a Parabola**

**y**** = ax^{2} + bx + c**

*y* = *a*(*x* - *h*)^{2} + *k*

by

Ángel M. Carreras Jusino

Goals:

- Show the derivation of the equation
y=fromax^{2}+bx+c-y=a(xand vice-versa.h)^{2}+k- Show the relationship between the parameters involved in both forms of the graph of a parabola.

From

y=ax^{2}+bx+ctoy=a(x-h)^{2}+k

To derive

y=a(x-h)^{2}+kfromy=ax^{2}+bx+cwe use the technique of completing the square.

Therefore comparing the two formulas we have that

From

y=a(x-h)^{2}+ktoy=ax^{2}+bx+cTo derive

y=ax^{2}+bx+cfromy=a(x-h)^{2}+kwe need to expand the equation.Therefore comparing the two formulas we have that

Now, how these information help us to understand better the standard forms of the graph of a parabola?

Lets see what type of direct information provides each form that the other does not provide and what advantages have one form over the other.

The form

y=ax^{2}+bx+cprovides they-intercept of the graph, the point (0,c), and the quadratic formula is based in the values ofa,b, andcto find the zeros of the graph.Example. The graph of

y= 2x^{2}- 4x- 6 hasy-intercept (0, -6) and using the quadratic formula its zeros are

The form

y=a(x-h)^{2}+kprovides the coordinate of the vertex of the parabola, the point (h,k), the symmetry axis, the linex=h, and the zeros of the graph can easily found by solving byx.Example. The graph of

y= 2(x- 1)^{2}- 8 has vertex on the point (1, -8), the symmetry axis is the linex= 1 and the zeros are

Note that both examples produce the same graph, now lets see how the information provided by one form can be gathered from the other.

- The form
y=a(x-h)^{2}+kprovides the coordinates of the vertex and the equation of the symmetry axis, information thaty=ax^{2}+bx+cdoes not provide. From what was discussed at the beginning, when we were deriving the equations, we got that.

Therefore from the values of the parameters

a,bandcfrom the equation of the formy=ax^{2}+bx+cwe can obtain the coordinates of the vertex (h,k) and the equation of the symmetry axisx=h.

- The form
y=ax^{2}+bx+cprovides the location of they-intercept, information thaty=a(x-h)^{2}+kdoes not provide. From the previous discussion of the derivation of the equations we got thatc=ah^{2}+k.Therefore from the values of the parametersa,h, andkfrom the equation of the formy=a(x-h)^{2}+kwe can obtain the location of they-intercept (0,c).In conclusion, from the two standard forms of the graph of a parabola can be gathered the same information after a little manipulation of the parameters.