Locus curve (curve discussion)

A locus is a curve on which all points of a given family of functions lie that fulfill a certain property. In a curve discussion , the locus curves of extreme points or turning points of the graphs of a family of functions are often sought.

calculation

To calculate the locus, the coordinates of the relevant points (e.g. all low points of a family of functions) are first determined as a function of the respective parameter (e.g. ). The equation for the coordinate is then solved for the parameter and inserted into the functional equation, which leads to the elimination of the parameter: The equation for the locus remains. Alternatively, the equation for the coordinate can also be determined first and the equation adjusted according to the parameter inserted into it. ${\ displaystyle x}$ ${\ displaystyle a}$ ${\ displaystyle x}$ ${\ displaystyle y}$

Examples

Extreme points of a family of curves

The extreme points (high and low points) of the family of functions given by have the coordinates (with ). The curve with the equation is the locus of all extreme points, since all extreme points of the individual function graphs lie on this curve. ${\ displaystyle f_ {a} (x) = x ^ {3} -3a ^ {2} x}$ ${\ displaystyle x}$ ${\ displaystyle x = \ pm a}$ ${\ displaystyle a \ neq 0}$ ${\ displaystyle y = -2x ^ {3}}$

Turning points of a family of curves

If you z. B. the locus for all turning points of the family of functions

{\ displaystyle f_ {t} (x) = tx ^ {4} -4t ^ {2} x ^ {2}}

With

{\ displaystyle t> 0}

want to determine, proceed as follows:

Determine turning points:
${\ displaystyle w_ {1} = + {\ sqrt {\ tfrac {2t} {3}}}}$ and ${\ displaystyle w_ {2} = - {\ sqrt {\ tfrac {2t} {3}}}}$
Since the function graph is axially symmetrical to the -axis, you can continue working with a single turning point. ${\ displaystyle y}$
${\ displaystyle x}$ -Write coordinate in equation:
${\ displaystyle x = {\ sqrt {\ tfrac {2t} {3}}}}$
${\ displaystyle x}$ Solve equation for parameter : ${\ displaystyle t}$
${\ displaystyle t = {\ tfrac {3x ^ {2}} {2}}}$
Insert the equation of into the function equation : ${\ displaystyle t}$
${\ displaystyle y = {- 7 {,} 5x ^ {6}}}$

The locus for all inflection points of the functions thus has the equation . ${\ displaystyle f_ {t}}$ ${\ displaystyle o _ {\ text {w}} (x) = {- 7 {,} 5x ^ {6}} \, \ ,; x \ neq 0}$

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