Rational functions & asymptotes
Practice Flashcards
What does the graph of y = 1/x look like?
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All Flashcards in Topic 2.8
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2.8.19 cards
What does the graph of y = 1/x look like?
A hyperbola with two branches (top-right and bottom-left), hugging both axes.
Domain and range of y = 1/x?
Domain x ≠ 0, range y ≠ 0.
Asymptotes of y = 1/x?
x = 0 (vertical) and y = 0 (horizontal).
How many intercepts does y = 1/x have?
None — it never reaches either axis.
Asymptotes of y = 1/(x − h) + k?
Vertical x = h, horizontal y = k.
Which way does 1/(x − 3) shift?
Right by 3, so the vertical asymptote is x = 3.
What does the +k do in 1/(x − h) + k?
Raises the horizontal asymptote to y = k.
How do you sketch a reciprocal graph?
Draw the asymptotes, find any intercepts, then draw the two branches hugging the asymptotes.
Why is 1/x undefined at x = 0?
Division by zero is undefined — that's the vertical asymptote.
2.8.29 cards
Where is the vertical asymptote of (ax + b)/(cx + d)?
Where the denominator is zero: cx + d = 0.
Where is the horizontal asymptote of (ax + b)/(cx + d)?
y = a/c — the ratio of the leading coefficients.
Where is the x-intercept of a rational function?
Where the numerator = 0 (a fraction is zero only when its top is zero).
Where is the y-intercept of (ax + b)/(cx + d)?
At x = 0: y = b/d.
Vertical asymptote of y = (2x + 1)/(x − 4)?
x = 4 (denominator zero).
Horizontal asymptote of y = (2x + 1)/(x − 4)?
y = 2 (leading coefficients 2/1).
Why is the horizontal asymptote a/c?
Dividing top and bottom by x, the b and d terms vanish, leaving a/c.
How many vertical asymptotes does (ax + b)/(cx + d) have?
One — the linear denominator has a single zero.
How do you sketch a rational function?
Draw the asymptotes, plot the x- and y-intercepts, then draw the two branches.
Topic 2.8 study notes
Full notes & explanations for Rational functions & asymptotes
Math AA SL exam skills
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