Question

pause the video, graph the following rational functions using other graphing technology, then fill in the chart using these graphs plus what you learned earlier in the video.
$y = \frac{x + 5}{x}$
$y = \frac{x^2 - 9}{x^2 + 4x + 4}$
$y = \frac{2x^2 - 4}{x - 3}$

	polynomial functions	rational functions
always have roots
sometimes have a y - intercept
always have a y - intercept
sometimes have a horizontal asymptote
sometimes have a vertical asymptote
sometimes have holes
have different kinds of end behavior

multiple - choice question
which functions sometimes have a vertical asymptote?

• polynomial
• rational

Explanation:

Polynomial functions are smooth and continuous everywhere, so they never have vertical asymptotes. Rational functions are of the form \(y=\frac{f(x)}{g(x)}\) where \(f(x)\) and \(g(x)\) are polynomials. A vertical asymptote occurs at \(x = a\) when \(g(a)=0\) and \(f(a)
eq0\). But if \(f(a) = 0\) and \(g(a)=0\) (i.e., there is a common factor \((x - a)\) in the numerator and denominator), then there is a hole (removable discontinuity) instead of a vertical asymptote. So rational functions sometimes have vertical asymptotes.

Answer:

rational