Explorations With Parabolas

# By Donna Greenwood

A parabola is graphed in the xy plane as the equation y = ax2 + bx + c, for any rational numbers a, b and c. Looking at how the graphs change as two of the values are held constant while the third varies provides interesting insights.

The first set of graphs hold b (=1) and c (=1) constant as a is varied.

The yellow line is of a=0. The parabola degenerates into a line, as we would expect. The gray and black graphs are of a = -.01 and .01 respectively. The red, blue and aqua graphs are of a = .5, 1 and 10 respectively. The magenta and green graphs are of a = -1 and -5. The sign of the x2 term determines whether the resulting parabola is a “positive parabola” or a “negative parabola,” that is, whether the parabola opens up or down. The coefficient of the x2 term determines how “wide” the parabola is. Smaller coefficients yield more open, gently sloping parabolas, while larger values create narrow, sharply steep curves.

These graphs all intersect at (0, 1), the y-intercept of the graphs. If c were equal to 0 for all these equations, the intersection point would be the origin.

The second set of graphs hold a and c constant at 1 as b is varied.

The magenta graph is b = -3; the red is b = 1, the blue is b = -.5, the green is b = 5; the aqua is b = 3, the yellow is b = 0; the gray is b = .1 and the black is b = -.1. From this we can see that negative b terms (magenta, blue and black) have their vertices in positive x quadrants ? I and IV, while equations with positive b terms have vertices in negative x quadrants ? II and III. This makes sense when you remember that the x-coordinate of the vertex is given by the equation ?b/2a. The y-coordinate of the vertex is found by replacing x in the original equation and solving for y. So, in large part, b determines where the vertex of the graph lies.
As with the first set of graphs, these graphs all intersect at (0, 1) which leads to the conjecture that the c term determines where the graphs intersect.

Finally, looking at the special case where b = 0 (the yellow graph, which is in-between the gray and black graphs) doesn’t really show anything new. The equation of the graph is y = x2 + 1, with the vertex at (0, 1).

The third set of graphs hold a and b constant as c is varied.

These graphs don’t intersect as the other sets of equations did. This is because the c term governs the y-intercept, and that term varies in this set of graphs. The variation in the c term moves the parabolas up or down the y-axis. The y-intercept for a quadratic in the form y = ax2 + bx + c has its y-intercept at (0, c).
These parabolas all have the same shape and orientation, due to the constant a and b terms.