Use the following rational function to answer the questions below.

\( f(x) = \dfrac{ \sum_{n=0}^{k} x^n }{ \sum_{n=0}^{l} x^n } = \dfrac{ 1 + x + x^2 + \ldots + x^k }{ 1 + x + x^2 + \ldots + x^l }\)

1. What do you notice about the numerator and \(f(x)\)?

- Conjecture: y-intercepts are equal
- Example: \( \dfrac{ 1 + x + x^2 }{ 1 + x }\)
- Why does this work? Will it work for all rational functions? when x = 0, function is \( \dfrac{1}{1} \); not always work but depends on constant part of polynomials

2. What do you notice about the denominator and \(f(x)\)?

- Conjecture: y-intercepts are equal
- Example: \( \dfrac{ 1 + x + x^2 }{ 1 + x }\)
- Why does this work? Will it work for all rational functions? same issues as previous

3. What do you notice when the degree of the numerator is **one** greater than the degree of the denominator? (e.g., k = l + 1)

- Conjecture: end behaviors look like y = x
- Example: \( \dfrac{ 1 + x + x^2 }{ 1 + x }\)
- Why does this work? Will it work for all rational functions? "dividing" by \(1+x\) leaves a linear function; yes

4. What do you notice when the degree of the numerator is **two** greater than the degree of the denominator? (e.g., k = l + 2)

- Conjecture: end behaviors look like \(y = x^2\)
- Example: \( \dfrac{ 1 + x + x^2 + x^3}{ 1 + x }\)
- Why does this work? Will it work for all rational functions? "dividing" by \(1+x\) leaves a quadratic function; yes

5. What do you notice when the degree of the denomonator is greater than the degree of the numerator? (e.g., k < l)

- Conjecture: end behaviors look like y = 0
- Example: \( \dfrac{ 1 + x }{ 1 + x + x^2 + x^3}\)
- Why does this work? Will it work for all rational functions? as x gets bigger, dividing by larger number; yes

Use the following rational function to answer the questions below.

\( f(x) = \dfrac{ \prod_{n=1}^{k} (x-n) }{ \prod_{n=1}^{l} (x-n) } = \dfrac{ (x - 1)(x - 2)(x - 3) \cdot \ldots \cdot (x - k) }{ (x - 1)(x - 2)(x - 3) \cdot \ldots \cdot (x - l) } \)

1. What do you notice about the numerator and \(f(x)\)?

- Conjecture: y-intercepts are not equal; sometimes zeros of numerator are zeros of rational function
- Example: \( \dfrac{ (x-1)(x-2)(x-3) }{ (x-1) }\)
- Why does this work? Will it work for all rational functions? \( \dfrac{ (x-1) }{ (x-1) } = 1\) so zeros of rational function will be zeros of numerator; yes, but needs more details

2. What do you notice about the denominator and \(f(x)\)?

- Conjecture: sometimes zeros of denominator are asymptotes of rational function
- Example: \( \dfrac{ (x-1) }{ (x-1)(x-2)(x-3) }\) has asymptote at y = 2 and y = 3, but not y = 1
- Why does this work? Will it work for all rational functions? \( \dfrac{ (x-1) }{ (x-1) } = 1\), which gives point discontinuity; else when denominator is zero rational function does not exist

3. What do you notice when the degree of the numerator is **one** greater than the degree of the denominator? (e.g., k = l + 1)

- Conjecture: end behaviors looks linear
- Example:\( \dfrac{ (x-1)(x-2) }{ (x-1) }\)
- Why does this work? Will it work for all rational functions? "dividing" by \(x-1\) leaves a linear function; yes

4. What do you notice when the degree of the numerator is **two** greater than the degree of the denominator? (e.g., k = l + 2)

- Conjecture: end behaviors quadratic
- Example:\( \dfrac{ (x-1)(x-2)(x-3) }{ (x-1) }\)
- Why does this work? Will it work for all rational functions? "dividing" by \(x-1\) leaves a quadratic function; yes

5. What do you notice when the degree of the denomonator is greater than the d egree of the numerator? (e.g., k < l)

- Conjecture: end behaviors look like y = 0
- Example:\( \dfrac{ (x-1) }{ (x-1)(x-2)(x-3) }\)
- Why does this work? Will it work for all rational functions? as x gets bigger, dividing by larger number; yes