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MATH / The graph of the rational function (p(x))/(q(x)) is shown below, with a horizontal asymptote at y=0. If q(x) is quadratic, p(2)=2, and q(−1)=18, find p(x)+q(x). [asy] size(8cm); im…

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problem

The graph of the rational function p(x)q(x)\frac{p(x)}{q(x)} is shown below, with a horizontal asymptote at y=0y = 0. If q(x)q(x) is quadratic, p(2)=2p(2)=2, and q(−1)=18q(-1) = 18, find p(x)+q(x).p(x) + q(x). [asy] size(8cm); import graph; Label f; f.p=fontsize(6); real f(real x) {return 2*(x-1)/(3*(x-2)*(x-1));} int gridsize = 5; draw((-gridsize,0)--(gridsize,0), black+1bp, Arrows(8)); draw((0,-gridsize)--(0, gridsize), black+1bp, Arrows(8)); label("xx", (gridsize, 0), E); label("yy", (0, gridsize), N); label("00", (0,0),SE, p=fontsize(8pt)); for (int i=-gridsize+1; i<0; ++i){ label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt)); label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));} for (int i=1; i<=gridsize-1; ++i){ label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt)); label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));} draw(graph(f,-5,.85)); draw(graph(f,1.15,1.85)); draw(graph(f,2.15,5)); draw((2,-5)--(2,5), dashed); draw(circle((1,-2/3),.15)); [/asy]
Plain-text mathematical notation (without MathML)
The graph of the rational function (p(x))/(q(x)) is shown below, with a horizontal asymptote at y=0. If q(x) is quadratic, p(2)=2, and q(−1)=18, find p(x)+q(x).
[asy]
size(8cm);
import graph;

Label f; 
f.p=fontsize(6); 

real f(real x) {return 2*(x-1)/(3*(x-2)*(x-1));}

int gridsize = 5;
draw((-gridsize,0)--(gridsize,0), black+1bp, Arrows(8));
draw((0,-gridsize)--(0, gridsize), black+1bp, Arrows(8));
label("x", (gridsize, 0), E);
label("y", (0, gridsize), N);
label("0", (0,0),SE, p=fontsize(8pt));
for (int i=-gridsize+1; i<0; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}
for (int i=1; i<=gridsize-1; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}



draw(graph(f,-5,.85));
draw(graph(f,1.15,1.85));
draw(graph(f,2.15,5));
draw((2,-5)--(2,5), dashed);
draw(circle((1,-2/3),.15));

[/asy]
Original LaTeX notation
The graph of the rational function $\frac{p(x)}{q(x)}$ is shown below, with a horizontal asymptote at $y = 0$. If $q(x)$ is quadratic, $p(2)=2$, and $q(-1) = 18$, find $p(x) + q(x).$
[asy]
size(8cm);
import graph;

Label f; 
f.p=fontsize(6); 

real f(real x) {return 2*(x-1)/(3*(x-2)*(x-1));}

int gridsize = 5;
draw((-gridsize,0)--(gridsize,0), black+1bp, Arrows(8));
draw((0,-gridsize)--(0, gridsize), black+1bp, Arrows(8));
label("$x$", (gridsize, 0), E);
label("$y$", (0, gridsize), N);
label("$0$", (0,0),SE, p=fontsize(8pt));
for (int i=-gridsize+1; i<0; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}
for (int i=1; i<=gridsize-1; ++i){
    label("$"+string(i)+"$",(i,0),S, p=fontsize(8pt));
    label("$"+string(i)+"$",(0,i),E, p=fontsize(8pt));}



draw(graph(f,-5,.85));
draw(graph(f,1.15,1.85));
draw(graph(f,2.15,5));
draw((2,-5)--(2,5), dashed);
draw(circle((1,-2/3),.15));

[/asy]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
2b950598-96c7-59e1-a5d6-6a19b31ca350
task_key
intermediate~5falgebra--test--2b950598-96c7-59e1-a5d6-6a19b31ca350
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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