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题目

解题步骤

1. 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) ______

2. 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., $a=1$).

3. Final answer

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

答案

Question Analysis

OCR Text

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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