Exploring Second Degree Equations

by

Mallory Thomas

Objective: Graph the parabola

Overlay a new graph by replacing x by (x-4). Below are the equations and the graphs.

Then, change the equation to move the vertex of the graph into the second quadrant.

Now, change the equation to produce a graph concave down that shares the same vertex.

Finally, generalize what you have observed. We could write the equation of a parabola in a generalized form like

In this case a, b, and c are integers, while x and y remain the variables. Through the othe rparts of this exploration we can see that the sign of a determines the concavity of the parabola. If a is positive then the parabola is concave up, while if a is negative then the parabola is concave down. The value of b moves the parabola to the left or to the right. If the value of b is negative then the parabola is moved to the right, or in the positive x direction. If the value of b is negative then the parabola moves to the left, or the negative x direction. The c-value moves the parabola up or down. When the c-value is positive the parabola moves up, or in the positive y direction. When the value of c is negative, then the parabola moves down, or in the negative y direction.