Function Families, Parameters and Locus Curves
A function family fa(x) contains a parameter a, and calculus answers how the shape of the graph changes with a: for which a there are extrema, how many roots the equation has, and on which curve (locus) the special points such as vertices and extrema lie. In brief: for fa(x) = x2 − 2ax + 3 the vertex is (a, 3 − a2). As a varies, the vertices lie on the parabola y = 3 − x2, the locus of the vertices. For fk(x) = x3 − 3kx, there are two extrema when k > 0 and none when k ≤ 0.
Parameter problems combine algebra and calculus: you treat the parameter as a constant when differentiating and then study how the results depend on it. The elimination of the parameter between the coordinates of a point gives the curve on which the point moves. This guide explains how to analyse families, count extrema and roots depending on a parameter, find loci of special points by eliminating the parameter and gives a table of families, with worked examples.
What is a function family?
A family (or pencil) of functions is a set {fa} obtained by letting a parameter a take different values. Examples: the lines y = mx + 2 (a family through the point (0, 2)), the parabolas y = x2 + c (all translates), the exponentials y = ax (all pass through (0, 1)), and the parabolas y = x2 − 2ax + 3.
When differentiating, the parameter is treated like any other constant: for fa(x) = x3 − ax, fa′(x) = 3x2 − a. Then the analysis has a case distinction depending on a.
How do extrema depend on the parameter?
Example 1: a family of cubics
fk(x) = x3 − 3kx. f′ = 3x2 − 3k = 3(x2 − k).
- k < 0: f′ > 0 always, the function is increasing, no extrema.
- k = 0: f′ = 3x2 ≥ 0, no extrema (the graph of x3 has a horizontal inflection at 0).
- k > 0: critical points x = ±√k. f″ = 6x: at x = −√k, f″ < 0, a local maximum with f = 2k√k. At x = √k, f″ > 0, a local minimum with f = −2k√k.
For k = 1: max (−1, 2) and min (1, −2). For k = 4: max (−2, 16) and min (2, −16).
Example 2: x + a/x
fa(x) = x + a/x for x > 0. f′ = 1 − a/x2. For a > 0, f′ = 0 at x = √a: f′ < 0 before and > 0 after, so a minimum with f(√a) = 2√a. For a ≤ 0, f′ > 0 for x > 0, so no extremum. This is the AM–GM inequality x + a/x ≥ 2√a.
Example 3: an exponential family
fa(x) = ex − ax. f′ = ex − a. For a > 0, f′ = 0 at x = ln a, a minimum since f″ = ex > 0, with value a − a ln a = a(1 − ln a). For a ≤ 0 there are no critical points. For a = e: the minimum is e(1 − 1) = 0, so ex ≥ ex for all x, with equality at x = 1, the tangent line at (1, e).
How do you find the locus of special points?
The special point (for example, the vertex or a local extremum) has coordinates that depend on the parameter, x = X(a) and y = Y(a). Eliminate a to obtain a relation between x and y, the equation of the locus.
Example 4: vertices of parabolas
fa(x) = x2 − 2ax + 3. The vertex: x = a, y = 3 − a2. Eliminate a = x: y = 3 − x2. The vertices of all parabolas lie on the parabola y = 3 − x2.
Example 5: extrema of the cubics
For fk(x) = x3 − 3kx (k > 0): the minimum is (√k, −2k√k) and the maximum is (−√k, 2k√k). With x = ±√k we have k = x2 and y = −2k√k = −2x2√k. For the minimum (x = √k > 0): y = −2x3. For the maximum (x = −√k < 0): y = 2x2|x| = −2x3. So all extrema lie on the curve y = −2x3.
Example 6: the family x e−ax
fa(x) = x e−ax for a > 0. f′ = e−ax(1 − ax) = 0 at x = 1/a, with f = 1/(ae). A maximum (f′ changes from + to −). The coordinates: x = 1/a, y = 1/(ae) = x/e. The locus of the maxima is the line y = x/e (x > 0).
Example 7: the inflection points
For fa(x) = x3 − 3ax2 + 4: f″ = 6x − 6a = 0 at x = a, with y = a3 − 3a3 + 4 = 4 − 2a3. Eliminating: y = 4 − 2x3. The inflection points lie on that cubic.
How many roots does an equation have depending on the parameter?
To count the solutions of g(x) = k, treat the graph of g and the horizontal line y = k: the number of intersections depends on k. The local extrema of g give the critical values.
Example 8: the equation x3 − 3x = k
The function g(x) = x3 − 3x has a local maximum 2 at x = −1 and a local minimum −2 at x = 1. So: for |k| > 2, one real solution; for k = ±2, two solutions (one double); for −2 < k < 2, three solutions. For k = 0: x = 0, ±√3.
Example 9: the equation ex = ax
Equivalent to a = ex/x (for x ≠ 0). The function h(x) = ex/x has h′ = ex(x − 1)/x2, a minimum at x = 1 on x > 0 with h(1) = e. So for 0 < a < e there is no solution on x > 0, for a = e one solution (x = 1) and for a > e two solutions. (For a < 0, there is one negative solution.)
Example 10: the number of solutions of x + 4/x = k for x > 0
The minimum of x + 4/x is 4 at x = 2. For k < 4: no solutions. k = 4: one (x = 2). k > 4: two solutions. For k = 5: x2 − 5x + 4 = 0, x = 1 or 4 ✓.
For which parameters is a function monotonic?
The function is non-decreasing if f′(x) ≥ 0 for all x, which for a quadratic f′ requires a nonpositive discriminant.
Example 11
fa(x) = x3 + ax2 + 3x. f′ = 3x2 + 2ax + 3. It is ≥ 0 for all x if the discriminant 4a2 − 36 ≤ 0, i.e., |a| ≤ 3. For |a| > 3, f has two critical points and is not monotonic. For a = 4: f′ = 3x2 + 8x + 3 has roots (−8 ± √28)/6 ≈ −0.45 and −2.22.
Example 12: a family with a fixed point
All members of fa(x) = ax2 + (1 − a)x pass through (0, 0) and (1, 1) for any a (since fa(1) = a + 1 − a = 1). The tangent slope at x = 0 is f′(0) = 1 − a, which depends on a.
Table of families and loci
| Family | Special point | Coordinates | Locus |
|---|---|---|---|
| x2 − 2ax + 3 | vertex | (a, 3 − a2) | y = 3 − x2 |
| x3 − 3kx, k > 0 | extrema | (±√k, ∓2k√k) | y = −2x3 |
| x e−ax, a > 0 | maximum | (1/a, 1/(ae)) | y = x/e |
| x + a/x, a > 0 | minimum | (√a, 2√a) | y = 2x |
| x3 − 3ax2 + 4 | inflection | (a, 4 − 2a3) | y = 4 − 2x3 |
| ex − ax, a > 0 | minimum | (ln a, a − a ln a) | y = ex(1 − x) |
Where do these ideas come from?
Families of curves were studied from the start of analytic geometry: Descartes and Fermat considered equations with parameters, and Leibniz and the Bernoullis in the 1690s studied envelopes of families of lines and curves. Locus problems were central to Greek geometry, and Descartes used the method of eliminating a parameter to find the curves traced by moving points.
Where are they used?
In physics and engineering, parameters represent physical constants (a mass, a resistance), and the family shows how the behaviour changes. Bifurcation theory studies how the number of equilibria changes with a parameter, like the transition from no extrema to two extrema at k = 0 in Example 1. In economics, the family of cost curves with a parameter describes how the optimal production depends on the price of a resource.
Which mistakes are common?
Forgetting the case distinction
Extrema of x3 − 3kx exist only for k > 0. Always examine the sign of the parameter.
Differentiating with respect to the parameter
When finding extrema in x, treat a as a constant: fa′(x), not ∂f/∂a.
Not restricting the locus
The locus of the extrema in Example 6 is only the part x > 0 of the line, since a > 0. Mention the restriction.
Losing solutions when eliminating the parameter
Check the domain of the parameter: for fk, the maxima are at x < 0 and the minima at x > 0, both on y = −2x3.