Parabola

Behavior of parabolas and quadratic equations

The following are various parabolas illustrating the effect of the lead coefficient and the vertex on the shape of the parabola and the number of roots.

Note: this page is also a demonstration of the capabilities of SVG. Note that each parabola is a different hyperlink with its own tooltip title tag. This could not be done with an image map as the parabolas cross-over each other. Because SVG allows each parabola to be its own object, the links are individual to the curve. This page also requires that the MathML math fonts be installed for proper display of embedded MathML code. Parabolic path tags built from an OpenOffice Calc spreadsheet.

Axes

These are the x and y axes. The horizontal midline of the chart is the x-axis, the vertical midline of the chart is the y-axis. The major grid lines, in violet-red, are at an interval of 5. The minor gridlines are at unit intervals.

The effect of varying k, which also varies only c

y=x²+12x+41
The vertex form for this parabola is (y-5)=(x-(-6))². As k decreases, c also decreases at the same rate and the parabola is moved vertically. In some sense, c controls the vertical position of a parabola.

y-=x²+12x+36
The vertex form for this parabola is y=(x-(-6))²

y=x²+12x+27
The vertex form for this parabola is (y-(-9))=(x-(-6))²

The effect of varying the lead coeficient a

y=x²
The vertex form for this parabola is y=x². The lead coefficient "a" controls the "span" of the parabola without affecting the vertex location in any way. A larger lead coefficient results in a "narrower" parabola, a smaller lead coefficient results in a "wider" parabola. This can be seen in the formulation (y-k)=a(x-k)² and the related form $(y - k) = \frac{1}{4 p} {(x - h)}^{2}$

y=2x²
The vertex form for this parabola is y=2x²

y=0.5x²
The vertex form for this parabola is y=0.5x²

The imaginary roots are the roots of the reflected function

y=x²-14x+53
The vertex form for this parabola is (y-4)=(x-7)². The roots are 7 ± 2i. This parabola is a "reflection" of the one below. The imaginary roots (x-intercepts, zeros) occur at the "reflection" of the roots of the "reflected" parabola below. The word imaginary in this case can also be taken to refer to the "image" of the roots.

y=-x²+14x-45
The vertex form for this parabola is (y-4)=-(x-7)². The roots are 7 ± 2, or x=5 and x=9.

Notes on roots

Roots, also known as x-intercepts or zeros of the function, can be found from either the quadratic form of an equation y=ax²+bx+c or from the vertex form of the equation $(y - k) = a {(x - h)}^{2}$ .

The roots from the quadratic form can be found from $x = \frac{- b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

For the vertex form the roots are: $x = h \pm \sqrt{\frac{- k}{a}}$ .

Although this form is rarely presented in algebra texts, the radical is in many ways much more informative than the discriminant $\sqrt{b^{2} - 4 a c}$ . The form makes more immediately clear that real roots only occur if either k or a are negative, but not both. That $\frac{- b}{2 a}$ is indeed the axis of symmetry is also more clear in the vertex root equation where h is used in lieu of $\frac{- b}{2 a}$ .

Algebra fall 2006 • IS 201 index • Lee Ling • COMFSM