Parameter Changes in Families of Curves

Introduction

A formula containing a parameter describes a whole family of curves, not just one graph. Instead of sketching each member separately, we can ask which features change as the parameter varies and which remain fixed.

For a rational function, useful features include its intercepts, critical points, vertical and slant asymptotes, and concavity. A parameter can move these features, make them appear or disappear, or cause a special cancellation. The goal is to track these changes systematically.

We will use the family

where \(a\) is a real parameter. The domain excludes \(x=1\) for every value of \(a\), including when the formula can be simplified there. This family has two important transition values, \(a=-1\) and \(a=0\).

A Useful Form of the Family

Polynomial division separates the part that describes the slant behavior from the parameter-dependent remainder:

so, for \(x\ne1\),

Write \(c=a+1\) when describing the features that depend on this remainder. The form \(x+1+c/(x-1)\) makes the asymptotes and derivatives particularly clear.

Intercepts and Asymptotes

The \(x\)-intercepts must satisfy \(x^2+a=0\), but a solution is an intercept only if it belongs to the domain. Thus, for \(a<0\), the algebraic candidates are \(x=\pm\sqrt{-a}\); for \(a=0\), the candidate is \(x=0\); and for \(a>0\), there are no real candidates. At \(a=-1\), the candidate \(x=1\) is excluded, so it is not an intercept. The \(y\)-intercept is always defined, because \(x=0\) is in the domain, and its value is \(f_a(0)=-a\).

For \(a\ne-1\), the term \((a+1)/(x-1)\) is unbounded as \(x\) approaches \(1\), so \(x=1\) is a vertical asymptote. At \(a=-1\), that term vanishes and the formula simplifies to \(x+1\) wherever the original function is defined. The graph then has a hole at \((1,2)\), not a vertical asymptote. For every \(a\), the remainder \(f_a(x)-(x+1)=(a+1)/(x-1)\) tends to zero as \(x\) tends to either infinity, so \(y=x+1\) is the slant asymptote when \(a\ne-1\). At \(a=-1\), the graph lies on that line except for the hole.

Critical Points and Concavity

Differentiate the rewritten form on its domain:

If \(a+1>0\), solving \(f_a'(x)=0\) gives \((x-1)^2=a+1\), hence two critical numbers \(x=1-\sqrt{a+1}\) and \(x=1+\sqrt{a+1}\). The derivative is positive when \(|x-1|>\sqrt{a+1}\) and negative when \(0<|x-1|<\sqrt{a+1}\). Therefore the left critical point is a local maximum, and the right critical point is a local minimum.

Their values can also be found exactly. Set \(s=\sqrt{a+1}>0\), so \(a+1=s^2\). At \(x=1-s\),

At \(x=1+s\),

These substitutions verify the critical-point heights. If \(a+1<0\), then \(f_a'(x)=1-(a+1)/(x-1)^2>0\) everywhere in the domain, so there are no critical numbers. If \(a=-1\), the derivative is \(1\) everywhere in the domain, also with no critical numbers.

The second derivative determines concavity. If \(a+1>0\), then \(f_a''(x)<0\) for \(x<1\) and \(f_a''(x)>0\) for \(x>1\): the graph is concave down on the left and concave up on the right. If \(a+1<0\), these signs reverse. At \(a=-1\), the function is the line \(x+1\) on its domain and its second derivative is zero. A concavity change across \(x=1\) when \(a\ne-1\) does not give an inflection point, because \(x=1\) is not in the domain.

Why the Parameter Thresholds Occur

The feature changes follow directly from the equations defining them. The condition \(a=-1\) makes the remainder \(a+1\) vanish. That removes the vertical asymptote and leaves a hole in the original domain. The critical-point equation requires \(a+1\ge0\); two distinct critical numbers occur for \(a>-1\), and none occur for \(a\le-1\). The concavity direction on each side likewise depends on the sign of \(a+1\).

For the intercepts, the equation \(x^2+a=0\) changes from two real roots to one repeated root and then no real roots as \(a\) increases through \(0\). The exceptional value \(a=-1\) must also be checked against the domain. Together these observations identify the two values at which the qualitative picture changes: \(a=-1\) and \(a=0\).

A Full Example with Two Critical Points

The Special Values \(a=-1\) and \(a=0\)

At \(a=0\), the numerator is \(x^2\), so the only \(x\)-intercept is \(x=0\), with a repeated root. Since \(a+1=1\), the critical numbers are \(x=0\) and \(x=2\); in this case the intercept at \(x=0\) is also a critical point. For \(a<0\), there are two real numerator roots, except that at \(a=-1\) the root \(x=1\) is excluded. For \(a>0\), there are no real numerator roots.

A Common Pitfall

A common error is to cancel a factor and then silently restore the excluded point. For example, at \(a=-1\), writing \((x^2-1)/(x-1)=x+1\) and concluding that the graph is the complete line incorrectly includes \((1,2)\). The original denominator is zero at \(x=1\), so that point remains outside the domain after simplification.

Another tempting mistake is to call a change in concavity across \(x=1\) an inflection point. For \(a\ne-1\), \(x=1\) is a vertical asymptote and is not a point on the curve. An inflection point must be a point of the graph where concavity changes; a missing point cannot qualify.

Why Parameter Tracking Matters

A parameter study replaces many separate sketches with a small number of calculations and threshold checks. First identify domain restrictions. Then solve the equations for intercepts and critical numbers, and inspect the parameter values where their solutions change. Finally, use the asymptote and concavity formulas to describe the intervals between those features.

This approach is useful whenever a family of curves is controlled by one or more constants. It distinguishes ordinary movement, such as a changing intercept height, from a genuine qualitative change, such as the loss of an asymptote or the creation of critical points.