Question

if f has a vertical asymptote given by x = 10, a horizontal asymptote y = 0, y-intercept at -2, and no x-intercept, then write the equation of a rational function f(x) = \frac{p(x)}{q(x)} having the indicated properties, in which the degrees of p and q are as small as possible. graph the function using a graphing utility to verify that it has the required properties.

write the equation of the rational function. choose the correct answer below.
a. f(x) = \frac{20}{x + 10}
b. f(x) = \frac{20}{x - 10}
c. f(x) = \frac{10}{x + 10}
d. f(x) = \frac{10}{x - 10}

use a graphing utility to graph the function obtained in the previous step. choose the correct graph below. each graph is shown in a viewing rectangle -20,20,4 by -20,20,4

Answer:

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<reasoning>

Determine the denominator from the vertical asymptote

Using the Vertical Asymptotes Determination knowledge point
\[

$$\begin{aligned} &\text{Vertical asymptote at } x = 10 \implies q(x) = x - 10 \end{aligned}$$

\]

Determine the numerator from the horizontal asymptote and intercepts

Using the Horizontal Asymptotes and X-Intercept of Rational Function knowledge points
\[

$$\begin{aligned} &\text{Horizontal asymptote } y = 0 \implies \text{degree}(p) < \text{degree}(q) \implies \text{degree}(p) = 0\\ &\text{No x-intercept} \implies p(x) = a \quad (a e 0)\\ &f(x) = \frac{a}{x - 10} \end{aligned}$$

\]

Solve for the constant using the y-intercept

Using the Y-Intercept of Rational Function knowledge point
\[

$$\begin{aligned} &f(0) = -2 \implies \frac{a}{0 - 10} = -2\\ &-\frac{a}{10} = -2 \implies a = 20\\ &f(x) = \frac{20}{x - 10} \end{aligned}$$

\]

Identify the correct graph

Using the Vertical Asymptotes Determination and Y-Intercept of Rational Function knowledge points
\[

$$\begin{aligned} &\text{Vertical asymptote: } x = 10 \quad (\text{dashed vertical line to the right of the y-axis})\\ &\text{Horizontal asymptote: } y = 0 \quad (\text{approaches x-axis})\\ &f(0) = -2 \quad (\text{crosses y-axis below the origin})\\ &x > 10 \implies f(x) > 0 \quad (\text{upper-right branch})\\ &x < 10 \implies f(x) < 0 \quad (\text{lower-left branch}) \end{aligned}$$

\]
Comparing the options, Graph B shows the vertical asymptote at \(x = 10\), the y-intercept at \((0, -2)\), and the branches in the correct quadrants.
</reasoning>

<answer>

Question 1

<mcq-option>(A) \(f(x) = \frac{20}{x + 10}\)</mcq-option>
<mcq-correct>(B) \(f(x) = \frac{20}{x - 10}\)</mcq-correct>
<mcq-option>(C) \(f(x) = \frac{10}{x + 10}\)</mcq-option>
<mcq-option>(D) \(f(x) = \frac{10}{x - 10}\)</mcq-option>

Question 2

<mcq-option>(A) branch in upper-left and lower-right, asymptote at x = -10</mcq-option>
<mcq-correct>(B) branch in upper-right and lower-left, asymptote at x = 10</mcq-correct>
<mcq-option>(C) branch in upper-left and lower-right, asymptote at x = 10</mcq-option>
<mcq-option>(D) branch in upper-right and lower-left, asymptote at x = -10</mcq-option>
</answer>

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"elements": [
{
"type": "line",
"params": [[10, -20], [10, 20]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 1.5,
"dash": 2,
"name": "x = 10",
"withLabel": true,
"labelPosition": [10.5, 15]
}
},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, -20, 9.9],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2.5
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},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, 10.1, 20],
"properties": {
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"strokeWidth": 2.5
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},
{
"type": "point",
"params": [[0, -2]],
"properties": {
"name": "(0, -2)",
"size": 4,
"color": "#5583F2",
"withLabel": true,
"labelPosition": [-4, -2]
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}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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