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.
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