Exploration of Parabolas
This is an exploration of parabolas as we change the quadratic coefficient, the linear coefficient, and the constant term. We will analyze how each of these terms affects the shape of the parabola.
First, we will keep the quadratic and linear terms a constant value of one and vary the constant term from -5 to -1.
a = 1, b = 1, c = -5... -1
As you can see, the vertex of the parabola translates vertically and approaches the x-axis as c increases.
Again, if we maintain the quadratic and linear terms as one, the parabola will translate vertically as c increases.
a = 1, b = 1, c = 0...5
Now we will investigate what happens to the parabola as we change the linear term. I will keep the quadratic term coefficient as one and for simplicity, make the constant term zero.
a = 1, b = -5...0, c = 0
As you can see, the linear term affects the position of the vertex (horizontally and vertically). However, as b decreases, contrary to what one would think, the vertex translates horizontally to the right and also translates vertically in an upward direction.
a = 1, b = 0...5, c = 0
As b increases, the vertex of the parabola translates to the left also translates in a downward motion. Furthermore, the translations of the parabola are reflections across the y-axis.
Now we will investigate what happens to the parabola as we change the quadratic coefficient and keep the linear and constant terms zero (again, for simplicity).
a = -5...0, b = 0, c = 0
First of all, we can clearly see that the parabola has been reflected across the x-axis. This is a result of the negative coefficient. Secondly, the parabolas “shrink” as a decreases. For instance, a = -5 is “wide” compared to a = -2. Obviously for a = 0 the equation would become y = 0. This coefficient determines whether the parabola is vertically compressed or stretched.
a = 1...5, b = 0, c = 0
For a values of 1 through 5, the parabola “shrinks” as a increases. Please note that the vertex of these parabolas has been at the origin since the coefficient of the quadratic term does not influence the location of the vertex, but only the width of the parabola.