fill in each blank give the following information: given the table belo…

Question

fill in each blank give the following information:
given the table below, write a possible equation for the function. 3 points
| x | -100 | -50 | -10 | -6 | -5 | -4 | -3 | 5 | 50 | 100 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| y | 3.21 | 3.44 | 7 | 23 | error | -17 | -7 | 1 | 2.63 | 2.81 |
write an equation for a rational function which has a v.a at $x=4$ and a h.a at $y=0$. 2 points
a) ______
b) ______

Solution Steps

Understand the question
fill in each blank give the following information:
given the table below, write a possible equation for the function. 3 points
| x | -100 | -50 | -10 | -6 | -5 | -4 | -3 | 5 | 50 | 100 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| y | 3.21 | 3.44 | 7 | 23 | error | -17 | -7 | 1 | 2.63 | 2.81 |
write an equation for a rational function which has a v.a at $x=4$ and a h.a at $y=0$. 2 points
a) ______
b) ______
Explanation
Step1: Identify vertical asymptote

From the table, there is an error at $x=-5$, so the denominator has a factor of $(x+5)$.

Step2: Identify horizontal asymptote

As $x\to\pm\infty$, $y$ approaches 3. So the degrees of numerator and denominator are equal, and the leading coefficient ratio is 3. Let the function be $y=3+\frac{k}{x+5}$.

Step3: Solve for k using a point

Use $x=-4, y=-17$:
$-17=3+\frac{k}{-4+5}$
$-17=3+k$
$k=-20$

Step4: Verify with another point

Check $x=-6, y=23$:
$y=3+\frac{-20}{-6+5}=3+20=23$, which matches.

Step5: Write equation for VA/HA

For vertical asymptote $x=4$ and horizontal asymptote $y=0$, the rational function has denominator $(x-4)$ and numerator of lower degree. A simple form is $y=\frac{a}{x-4}$ (any non-zero $a$ works, e.g., $a=1$).
Final answer
a) $y=3-\frac{20}{x+5}$
b) $y=\frac{1}{x-4}$ (any non-zero numerator constant is valid)

Answer

Explanation

Step1: Identify vertical asymptote

From the table, there is an error at $x=-5$, so the denominator has a factor of $(x+5)$.

Step2: Identify horizontal asymptote

As $x\to\pm\infty$, $y$ approaches 3. So the degrees of numerator and denominator are equal, and the leading coefficient ratio is 3. Let the function be $y=3+\frac{k}{x+5}$.

Step3: Solve for k using a point

Use $x=-4, y=-17$:
$-17=3+\frac{k}{-4+5}$
$-17=3+k$
$k=-20$

Step4: Verify with another point

Check $x=-6, y=23$:
$y=3+\frac{-20}{-6+5}=3+20=23$, which matches.

Step5: Write equation for VA/HA

For vertical asymptote $x=4$ and horizontal asymptote $y=0$, the rational function has denominator $(x-4)$ and numerator of lower degree. A simple form is $y=\frac{a}{x-4}$ (any non-zero $a$ works, e.g., $a=1$).

Answer

a) $y=3-\frac{20}{x+5}$
b) $y=\frac{1}{x-4}$ (any non-zero numerator constant is valid)

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Question Analysis

Subject mathematics

Sub Subject calculus

Education Level high school

Difficulty unspecified

Question Type with chart, calculation

Multi Question No

Question Count 1

Analysis Status completed

Analyzed At 2026-02-09T20:23:09

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fill in each blank give the following information:
given the table below, write a possible equation for the function. 3 points
| x | -100 | -50 | -10 | -6 | -5 | -4 | -3 | 5 | 50 | 100 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| y | 3.21 | 3.44 | 7 | 23 | error | -17 | -7 | 1 | 2.63 | 2.81 |
write an equation for a rational function which has a v.a at $x=4$ and a h.a at $y=0$. 2 points
a) ______
b) ______

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