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

Aufgabe

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) ______

Lösungsschritte

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)

Antwort

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)

Question Image

Click to enlarge

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

OCR Text

Show OCR extraction

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) ______

Sovi AI iOS

Offizielle Website: mysovi.ai. Frage-Seiten laufen auf question-banks.mysovi.ai. Die iOS-App ist im Apple App Store verfügbar.

Im App Store laden Kategorie: Analysis

Sovi AI