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\\)}

Explanation:

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<reasoning>

Analyze vertical asymptote and denominator

Using the Vertical Asymptotes Determination knowledge point
\[

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

\]

Analyze horizontal asymptote and degree

Using the Horizontal Asymptotes knowledge point
\[

$$\begin{aligned} &\text{Horizontal asymptote } y = 0 \implies \text{degree}(p) < \text{degree}(q)\\ &\text{Since } \text{degree}(q) = 1, \text{ we must have } \text{degree}(p) = 0 \implies p(x) = a \end{aligned}$$

\]

Analyze x-intercept and numerator

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

$$\begin{aligned} &f(x) = \frac{a}{x - 10}\\ &a e 0 \implies \text{No x-intercepts, which matches the given condition.} \end{aligned}$$

\]

Determine constant using y-intercept

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

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

\]
</reasoning>

<answer>
<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>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[10, -10], [10, 10]],
"properties": {
"name": "x = 10",
"strokeColor": "#F2557F",
"strokeWidth": 1.5,
"dash": 2,
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, -10, 9.5],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2.5
}
},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, 10.5, 20],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2.5
}
},
{
"type": "point",
"params": [[0, -2]],
"properties": {
"name": "(0, -2)",
"size": 4,
"strokeColor": "#5583F2",
"fillColor": "#5583F2",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
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"Rational Functions"
]
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</post_analysis>

Answer:

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<reasoning>

Analyze vertical asymptote and denominator

Using the Vertical Asymptotes Determination knowledge point
\[

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

\]

Analyze horizontal asymptote and degree

Using the Horizontal Asymptotes knowledge point
\[

$$\begin{aligned} &\text{Horizontal asymptote } y = 0 \implies \text{degree}(p) < \text{degree}(q)\\ &\text{Since } \text{degree}(q) = 1, \text{ we must have } \text{degree}(p) = 0 \implies p(x) = a \end{aligned}$$

\]

Analyze x-intercept and numerator

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

$$\begin{aligned} &f(x) = \frac{a}{x - 10}\\ &a e 0 \implies \text{No x-intercepts, which matches the given condition.} \end{aligned}$$

\]

Determine constant using y-intercept

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

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

\]
</reasoning>

<answer>
<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>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[10, -10], [10, 10]],
"properties": {
"name": "x = 10",
"strokeColor": "#F2557F",
"strokeWidth": 1.5,
"dash": 2,
"withLabel": true
}
},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, -10, 9.5],
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"strokeColor": "#8C55F2",
"strokeWidth": 2.5
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{
"type": "functiongraph",
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{
"type": "point",
"params": [[0, -2]],
"properties": {
"name": "(0, -2)",
"size": 4,
"strokeColor": "#5583F2",
"fillColor": "#5583F2",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0]
}
</plot>

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