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MATH / The graph of y=(p(x))/(q(x)) is shown below, where p(x) and q(x) are quadratic. (Assume that the grid lines are at integers.) [asy] unitsize(0.6 cm); real func (real x) return (-(x…

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problem

The graph of y=p(x)q(x)y = \frac{p(x)}{q(x)} is shown below, where p(x)p(x) and q(x)q(x) are quadratic. (Assume that the grid lines are at integers.) [asy] unitsize(0.6 cm); real func (real x) { return (-(x + 5)*(x - 4)/(x - 2)^2); } int i; for (i = -8; i <= 8; ++i) { draw((i,-8)--(i,8),gray(0.7)); draw((-8,i)--(8,i),gray(0.7)); } draw((-8,0)--(8,0)); draw((0,-8)--(0,8)); draw((2,-8)--(2,8),dashed); draw((-8,-1)--(8,-1),dashed); draw(graph(func,-8,1.9),red); draw(graph(func,2.1,8),red); limits((-8,-8),(8,8),Crop); [/asy] The horizontal asymptote is y=−1,y = -1, and the only vertical asymptote is x=2.x = 2. Find p(−1)q(−1).\frac{p(-1)}{q(-1)}.
Plain-text mathematical notation (without MathML)
The graph of y=(p(x))/(q(x)) is shown below, where p(x) and q(x) are quadratic.  (Assume that the grid lines are at integers.)

[asy]
unitsize(0.6 cm);

real func (real x) {
  return (-(x + 5)*(x - 4)/(x - 2)^2);
}

int i;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0));
draw((0,-8)--(0,8));
draw((2,-8)--(2,8),dashed);
draw((-8,-1)--(8,-1),dashed);
draw(graph(func,-8,1.9),red);
draw(graph(func,2.1,8),red);

limits((-8,-8),(8,8),Crop);
[/asy]

The horizontal asymptote is y=−1, and the only vertical asymptote is x=2.  Find (p(−1))/(q(−1)).
Original LaTeX notation
The graph of $y = \frac{p(x)}{q(x)}$ is shown below, where $p(x)$ and $q(x)$ are quadratic.  (Assume that the grid lines are at integers.)

[asy]
unitsize(0.6 cm);

real func (real x) {
  return (-(x + 5)*(x - 4)/(x - 2)^2);
}

int i;

for (i = -8; i <= 8; ++i) {
  draw((i,-8)--(i,8),gray(0.7));
  draw((-8,i)--(8,i),gray(0.7));
}

draw((-8,0)--(8,0));
draw((0,-8)--(0,8));
draw((2,-8)--(2,8),dashed);
draw((-8,-1)--(8,-1),dashed);
draw(graph(func,-8,1.9),red);
draw(graph(func,2.1,8),red);

limits((-8,-8),(8,8),Crop);
[/asy]

The horizontal asymptote is $y = -1,$ and the only vertical asymptote is $x = 2.$  Find $\frac{p(-1)}{q(-1)}.$

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
fc97ac86-3036-5fb6-8002-5f80ecc5d188
task_key
intermediate~5falgebra--test--fc97ac86-3036-5fb6-8002-5f80ecc5d188
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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