Examining a Parabola

Examine the graphs for the parabola y=ax^2+bx+c for different values of a, b, and c (they can be any rational number). To make it easier for the reader, all graphs will include y=x^2+x+1 in grey as a standard.

Examination when we vary “a”:

As the graph shows, when “a” is altered, graph is stretched or shrunk. We can see as “a” increases, the graph is shrunk. When “a” is negative, the graph is inverted.

The case where “a” is a fraction (less than 1 and greater than -1) was examined too, but the results support the prior conclusion: the graph is stretched or shrunk. Here we see that the smaller the absolute value of “a” is, the more stretched the graph is. Similar to before, when “a” is negative, the graph is inverted.

Therefore, by having “a” greater than or equal to one, the graph will shrink. The large “a” is, the more the graph will shrink. When “a” is less than 1 and greater than negative one, the graph will stretch.

Examination when we vary “b”:

When “b” is altered, the placement of the graph changes (the graph is shifted to a new x-value). The higher the value of “b”, the lower the graph becomes on the graph. When “b” is positive, the parabola is on the negative x-axis side of the graph. When “b” is negative, the graph is reflected over the y-axis (it is not on the positive x-axis side). Note that two graphs have been shown using the same equations to illustrate the parabolas are the same, just shifted.

After examining this, I wondered what would happen if b was a fraction less than 1 and greater than -1. Hence, the next graph below shows my further exploration. Upon construction, I could see that fractions shift the graph in the same manner as before, except the graph is shifted up not down. Again, when the fractional “b” is negative, the graph is reflected over the y-axis.

Examination when we vary “c”:

When “c” is altered, the graph shifts to a different y-value (the x-value stays the same). When “c” is positive (and greater than 1), the graph is moved to a higher y-value. The higher the value of “c”, the further the graph is moved up. When “c” is negative (and less than -1), the graph is moved downwards.

In this examination, having a fractional “c” that is less than 1 and greater than -1 does effect the results. When “c” is greater than 0 and less than 1, the graph is shifted to the down along the y-value. The larger the fraction, the further the graph is shifted. When “c” is less than 0 and greater than -1, the graph is shifted to the down along the y-value too but a greater distance than if the fraction was positive. Again, the larger the fraction, the further the graph is shifted.