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Families of Curves

The concept of "curve families" is very important in mathematics, as there are many situations in which there are an entire range of answers to a particular equation. The most prominent examples occur in the realm of differential equations - equations involving derivatives of functions - where the graph of the solutions might include a very large number of different curves.

For example, on the right we see that there are multiple curves, all with different equations, that are solutions for the differential equation . These curves can be thought of as a moderately complex "family," all sharing one thing in common: they are solutions to the same equation. However, families can have more simple things in common as well. In this tutorial we will look at a few families and discuss how their "things in common" are manifested graphically.

(From Elementary Differential Equations and Boundary Value Problems,William E. Boyce and Richard C. DiPrima, 1992, p. 26).

Linear Families

 Perhaps the simplest of families are lines which have an attribute in common. Since all non-vertical lines can defined by their slope and y-intercept in the form y = mx+b, we can "link" a group of lines together by holding one of these characteristics constant and manipulating the other.

Same slope, different intercepts

On the left we see what happens when we take a linear "base equation" (in this case, the equation y = x), and increase its b-value, thus shifting its y-intercept. The result in the animation is that the line moves up the y-axis. Reversing the process then sends the line back to its original position.

The animation, then, represents a family of parallel lines all sharing one thing in common: they have a slope of 1. (The size of the family? Infinite: we could keep increasing (or decreasing) the b-value for as long as we would like!) By changing the y-intercept - or, more generally, by changing any parameter of a curve - we come up with a family that is joined by a common characteristic but differentiated from each other in some other way.

Same y-intercept, different slopes

Now we take the base equation y = x + 1 and change its m-value, thus changing the slope while keeping the y-intercept constant. The result can again be seen in the animation: each line hits the y-axis in the same place (at (0,1)), but we get the appearance of a rotating line.

This family, then, shares the "anchor" of the same y-intercept, but the lines then increase at different rates from there. By changing the curve's other parameter, we have created a family connected in a different way: they have a common point instead of a common slope. (The size of the family is still infinite; we could pick any m-value from to for our slope.)

What if we then combined these curve transformations? By taking the line y = x, rotating it, returning, and then shifting the y-intercept, we could create two related families - cousins? - whose common member was our beginning equation. The result is below, with the "rotate" family in red and the "change the b-value" family in blue.

Quadratic Families

 We now examine second-degree equations, beginning with parabolas of the form . If we set a = 1, and then change the value of k, we can see (in the first graph below) that we get a family of parabolas whose width and direction of opening are all the same. However, the vertex of each parabola in the family will be different - an analagous case to our parallel lines example above. In the same way, if we set k = 0 and allow the value of a to vary (but only looking at positive values), we have the family of parabolas on the right - same vertex at (0,0), but with varying widths.
                                

Motion around a circle

We now introduce the parameter h, which allows us to shift the parabola's vertex along the x-axis. The full equation is now ; in our example we will again set a = 1 but let both h and k vary.
However, if we just allowed h and k to be anything, we would get a family of parabolas, all opening upward, with uniform width and vertices . . . anywhere. The resulting graph would be rather difficut to decipher, so let's restrict the h and k values by looking only at the parabola family whose vertices lie on the circle . The result is a family whose members seem to be rotating around the circle itself. (For an alternative interpretation of a parabola rotating around a circle, scroll down to the end of this page.)

Is a circle an ellipse?

We conclude our look at quadratics by comparing the graphs of the circle and the family of ellipses . By allowing the value of k to vary, we see from the graph below that you could argue that circles are members of the ellipse family, for setting the value of k = 1 in this case gives us the unit circle.

Approximations to a function

 A curve family does not have to consist only of equations of the same degree, for the members may be related in more complex ways. We can consider the Taylor approximations of a function to be a type of family, one where each successive approximation is a more complex equation than the previous one. We will look at the first 6 Taylor approximations to y= cos x at x = 0.
As can be seen in the animation, the result is a family whose members increase significantly in their complexity: from the first approximation, y = 1, and in the second, to   in the last. The visual result is that the blue curves seem to "melt" into the red graph of the cosine function, coming closer and closer to convergence with it.

Extra-Terrestrial Families from the 3rd Dimension

 Just kidding. However, there are families of curves even when graphing in 3-D. Moreover, many 3-D surfaces themselves are really families of 2-dimensional curves, such as the graph of below.
This paraboloid can be generated by simply spinning the parabola around the z-axis. As shown on the right, when viewed from any angle, the paraboloid's 2-D cross-sections are parabolas.
There are, of course, families of surfaces as well. From a "base" equation of , we can create a family by rotating the surface around the z-axis, as below. This family shares the same general shape, and every member has a vertex on the circle in the xz-plane.


And, finally, we can create families that are related in more subtle ways than just using vertical shifts or rotations. Below we see several members of the family , noting that each change in the parameter k seems to "tilt" the paraboloid ever more.

This just in . . .

After glancing at the Math 696 News page . . . we have re-interpreted the phrase "motion around a circle," and the result is on the right. Now we take a parabola with a = 1, h = 0, and k = 3, and rotate the whole thing around the circle . Now we have a family of parabolas whose equations are not even all of the form , thanks to the varying orientations of their axes of symmetry.