Dana - 2007: Pre Calculus

Dana Edwards and Brett Diemer 10/10/07

Pre calc- Ms. Vega

Transformations Extra Assignment 

Given the function: f(x)=ax^2 + bx + c, with the ability to change the values of a, b, and c, Brett and I came to the following conclusions about the relationship between the three variables:

When a=0 and b=0, the graph should be a horizontal line at the value of the constant, or f(x)=c. However, because the program used does not permit one to enter a value of exactly 0, we entered a=-0.02 and b=-0.02, resulting in a very open, downward facing parabola whose vertex was slightly to the left of (a more negative x-value) and slightly above (a more positive y value) the value of c on the y-axis. 

We set a to .04, b to 1.04, and adjusted the value of c. When c=0, the graph looks almost exactly like the line f(x)=x, and when the value of c is changed, the graph becomes essentially f(x)=x+c. This is because, with the quadratic part taken away, the value of a being near zero, only the values of b and c play a part in the appearance of the graph. When a remained at 0.04, b was changed to -1.05, and the value of c was adjusted, the graph became the same as the previous, but reflected over the x and y axes, or f(x)=-x+c.

Things got really exciting when Brett and I set the value of a to 1.01 and adjusted the values of b and c. B and c both acted to change the position of the vertex of the graph, but did not affect the shape of the parabola. For a given value of b, c simply shifts the vertex by its value. For instance, if the vertex is at (-1,-2), and a value of 3 is entered for c, the y-value of the vertex will increase by three, so it will become (-1,1). B, on the other hand, affects the position of the vertex in a more complex fashion. It moves the vertex along the parabola’s x-axis reflection, so in our case, along the graph f(x)= -1.01x^2 +c.  The x-value of the vertex is shifted by –b/2, and the y-value is shifted by -(b/2)^2.  For instance, if the vertex is at (0,0), and 4 is put in for the value of b, the vertex will move to (-(4/2), -(4/2)^2), or (-2,-4).

A, b, and c are all unique and interesting variables that affect the graph of f(x)=ax^2 + bx + c. A affects the orientation and stretch (I hesitate to use the word slope, because the slope is always changing on a parabola) of the parabola, while b and c affect the position of the vertex of the graph, and therefore shift every value of the graph by certain amounts. A is a stretch/compression and possibly x-axis reflecting transformation, while b and c are shifting transformations.