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
\[

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

\]

Determine the numerator from x-intercept and horizontal asymptote

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

$$\begin{aligned} &\text{No x-intercept } \implies p(x) = a \quad (a e 0 \text{ is a constant})\\ &f(x) = \frac{a}{x - 10}\\ &\text{Degree of } p(x) = 0 < \text{Degree of } q(x) = 1 \implies \text{Horizontal asymptote is } y = 0 \end{aligned}$$

\]

Solve for the constant using the y-intercept

Using the Y-Intercept of Rational Function and Rational Function Evaluation knowledge points
\[

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

\]

Identify the correct graph

Using the Rational Function Evaluation knowledge point
\[

$$\begin{aligned} &\text{For } x > 10, \quad f(x) > 0 \quad (\text{curve is in the upper half-plane})\\ &\text{For } x < 10, \quad f(x) < 0 \quad (\text{curve is in the lower half-plane})\\ &\text{Vertical asymptote is at } x = 10 \quad (\text{to the right of the y-axis})\\ &\text{This matches the graph in option D} \end{aligned}$$

\]
</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) curve with vertical asymptote at x = -10</mcq-option>
<mcq-option>(B) curve with vertical asymptote at x = -10</mcq-option>
<mcq-option>(C) curve with vertical asymptote at x = 10, left side positive, right side negative</mcq-option>
<mcq-correct>(D) curve with vertical asymptote at x = 10, left side negative, right side positive</mcq-correct>
</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
}
},
{
"type": "functiongraph",
"params": [{"js": "20/(x-10)", "latex": "\frac{20}{x-10}"}, 10.1, 20],
"properties": {
"strokeColor": "#8C55F2",
"strokeWidth": 2.5
}
},
{
"type": "point",
"params": [[0, -2]],
"properties": {
"name": "(0, -2)",
"size": 4,
"color": "#5583F2",
"withLabel": true
}
}
],
"timestamps": [0.5, 1.0, 1.5]
}
</plot>

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