Lisa Brock

Assignment 3

The Affects of a, b, and c on

To best examine the affects of a, b, and c on the curve , we must examine each variable individually while holding the others constant.

Let's begin with a.  Let b = 1 and c.

Each graph goes through the point (0,1).  When a = 0, the graph is a line instead of a parabola.  When a = 0, the  term becomes zero, turning the equation into a linear equation.  When a is positive, the graph is concave up.  When a is negative, the graph is concave down.  As the absolute value of a increases the parabola becomes more steep and narrow.  Also, the position of the vertex changes as a is varied.

Therefore, a determines the direction of concavity, the steepness of the parabola, and the position of the vertex.

Now let's take a look at b.

Each parabola goes through (0, 1).  Each graph is concave up.  Each graph is of equal steepness.  The vertex shifts as b changes.  As b increases in the positive direction, the graph shifts farther to the left.  As b increases in the negative direction, the graph shifts farther to the right.

Therefore, b has an affect on the position on the vertex.  Since the value of a also has an affect on the position of the vertex, the values of both a and b will determine the position of the vertex.

Let's see what affect c has on the graph.  Let a = 1 and b = 1.  This results in the equation .  Graph this equation for different values of c.

Each graph is concave up.  Each graph is of equal steepness.  The vertex has shifted vertically but not horizontally.

Therefore, c affects the y-coordinate of the vertex.

In summary, a, b, and c all determine the position of the vertex.  c appears to affect only the y-coordinate of the vertex.  b appears to affect both coordinates of the vertex.  a appears to affect both coordinates of the vertex.  a also determines the direction of concavity and the steepness of the parabola.