Practice Flashcards
Where are the vertical asymptotes of a rational function?
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All Flashcards in Topic 2.13
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2.13.18 cards
Where are the vertical asymptotes of a rational function?
Where the denominator = 0 (and the numerator isn't also 0 there).
How do you find the horizontal asymptote?
Compare degrees: top < bottom → y = 0; equal → y = (ratio of leading coefficients).
What if top degree < bottom degree?
The horizontal asymptote is y = 0.
What if top and bottom have equal degree?
y = (leading coefficient of top) ÷ (leading coefficient of bottom).
Vertical asymptotes of (2x+1)/(x² − x − 6)?
x = 3 and x = −2 (from (x − 3)(x + 2) = 0).
Horizontal asymptote of (4x − 3)/(2x + 1)?
y = 2 (equal degrees, 4/2).
What happens if numerator and denominator are both 0 at x = a?
There's a hole at x = a, not a vertical asymptote (the factor cancels).
What if the top degree is one MORE than the bottom?
There's a slant (oblique) asymptote instead of a horizontal one.
2.13.28 cards
When does a rational function have a slant (oblique) asymptote?
When the numerator's degree is exactly one more than the denominator's.
How do you find the slant asymptote?
Divide top by bottom; the quotient line y = mx + c is the asymptote (the remainder term → 0).
Slant asymptote of (x² + 1)/(x − 1)?
Divide: x + 1 + 2/(x − 1), so y = x + 1.
Steps to sketch a rational function?
x-intercepts (top = 0), y-intercept (x = 0), vertical asymptotes (bottom = 0), horizontal/slant asymptote, then fit the branches.
Can a function have both a vertical and a slant asymptote?
Yes — e.g. (x² + 1)/(x − 1) has vertical x = 1 and slant y = x + 1.
Slant asymptote of (2x² − x + 1)/(x + 1)?
y = 2x − 3 (the quotient of the division).
Does a curve ever cross its slant asymptote?
It can cross it (asymptotes are about behaviour as x → ±∞), unlike never crossing a vertical one.
What do you draw first when sketching?
The asymptotes as dashed lines, then the intercepts.
Topic 2.13 study notes
Full notes & explanations for Rational functions (HL only)
Math AA exam skills
Paper structures, command terms & tips
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