Mathematics, 07.07.2019 10:30 MonkeyBurroughs

Place the steps to sketch the graph of a rational function in the appropriate order

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solution: the steps to sketch the graph of rational function are (1) find x-intercept, (2) find y-intercepts, (3) find vertical asymptotes, (4) find horizontal asymptotes, (5) find coordinates of extreme points and (6) plot all points on coordinate plane and join them by a free hand curve.

explanation:

let the rational function is defined as $r(x)=\frac{p(x)}{q(x)}$ .

the steps to sketch the graph of rational function are,

(1) find x-intercept: to find x-intercept set numerator equal to zero, $p(x)=0$ and find the real values of x.

(2) find y-intercept: to find y-intercept put x equal to zero, $x=0$ and find the values of y.

(3) find vertical asymptote: to find vertical asymptotes set denominator equal to zero, $q(x)=0$ and find the real values of x.

(4) find horizontal asymptotes: to find horizontal asymptotes notice the degree of numerator and denominator.

(i) if $deg(p(x))< deg(q(x))$ then the horizontal asymptote is $y=0$ .

(ii) if $deg(p(x))=deg(q(x))$ then the horizontal asymptote is a horizontal line $y=\frac{a}{m}$ , where a is the leading coefficient of p(x) and m is the leading coefficient of q(x).

(iii) if $deg(p(x))=deg(q(x))+1$ then the there is not horizontal asymptote exist but there is a slant asymptote with slope line $\frac{a}{m}$ , where a is the leading coefficient of p(x) and m is the leading coefficient of q(x).

(iv) if $deg(p(x))> deg(q(x))+1$ then the there is not horizontal asymptote exist.

(5) find the coordinates of extreme points: to find the coordinates of extreme points find the critical points by $r'(x)=0$ where r'(x) is the first derivative of r(x). then find second derivative if $r'(x)> 0$ then it will give point of minima and if $r'(x)< 0$ then it will give point of maxima. put these values of x in the original function to find the y-coordinate of extreme points.

(6) plot all points on coordinate plane and join them by a free hand curve.

therefore, the steps to sketch the graph of rational function are (1) find x-intercept, (2) find y-intercepts, (3) find vertical asymptotes, (4) find horizontal asymptotes, (5) find coordinates of extreme points and (6) plot all points on coordinate plane and join them by a free hand curve.

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