Math Assignment: Investigating the Quadratic Function

Algebra II

Dr. Garciano

18 September 2008

The purpose of this math assignment is to investigate quadratic functions and to be able to understand how constant terms and coefficients in functions affect the final product of a graph by displaying the results using different families.

A good example to start this investigation off is by graphing functions in the family y=x²+k where k is a constant term. After doing so, I will state the coordinates of the vertex for each function using the data shown on the graph.

Looking at the graph above, you could see that there are three different functions displayed. From looking at this, you could determine the vertices for each of them. The coordinates of the vertex of each equation are:

y=x² : (0,0)

y=x²+3 : (0,3)

y=x²−2 : (0,-2)

Also, the significance in the constant term k can be viewed from the 3 equations on the graph. The position of the graph would vary depending on its value. This is because k in the equation y= x²+k is represented as the y-intercept. Therefore, k will affect the position of the graph vertical-wise. However at the same time, the value of k will not affect the shape of the graph because you could see that they’re all the same and because there is no variable that determines it.

A certain transformation is used for this graph from  y=x² to become y=x²+k. It is a vertical translation since all three functions are the same except for their positions on the graph which is spread out vertically.

Now, I will graph another set of functions of the family y=(x-h) ² where h is a constant term. From plotting this, I will obtain facts on how the value of  h will have an effect on the position and shape of the graph.

Again after graphing each function, the vertices become apparent which are:

y=x² : (0,0)

y=(x-2)²  : (2,0)

y=(x+3)² : (-3,0)

The position of the graph would vary depending on the value of h. This is because h determines where on the x-axis the graph is suppose to be. Therefore, h will affect the graph horizontal-wise. However again, h has nothing to do with the shape of the graph, but only its position on the graph.

All that is different between the three equations are their positions on the graph. As you can see, they are all on the same x-axis, however, they are spread out horizontally. Therefore the ...