๐Ÿƒ Flashcards ยท Topic 1.7

Rational Functions &
End Behavior

8 printable flashcards โ€” questions on Page 1, answers on Page 2. Print double-sided, cut along the dashed lines.

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
How to use: Print this page double-sided (flip on long edge) โ†’ cut along the dashed lines โ†’ 8 flashcards ready to go!
Questions are on Page 1 ยท Answers are on Page 2
๐Ÿ“„ Page 1 โ€” Questions (print this side first) FRONT
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Topic 1.7 ยท Definition
What is a rational function?
Think: what two things make it up?
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Topic 1.7 ยท Case 1
What is the end behavior when the numerator and denominator have the same degree?
f(x) = axโฟ / bxโฟ (n = d)
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Topic 1.7 ยท Case 2
What is the end behavior when the denominator degree is greater than the numerator degree?
f(x) = axโฟ / bx^d (n < d)
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Topic 1.7 ยท Case 3
What is the end behavior when the numerator degree is greater than the denominator degree?
f(x) = axโฟ / bx^d (n > d)
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Topic 1.7 ยท Slant Asymptote
When does a rational function have a slant (oblique) asymptote?
What is the exact condition on the degrees?
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Topic 1.7 ยท Slant Direction
If a rational function has a slant asymptote, what line is it parallel to?
f(x) = (axโฟ + โ€ฆ) / (bx^d + โ€ฆ)
where n = d + 1
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Topic 1.7 ยท Limit Statements
Write the limit statements for:
f(x) = (2xยณ + 4x โˆ’ 1) / (6xยณ โˆ’ xยฒ + 4)
Identify which case first.
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Topic 1.7 ยท Limit Statements
Write the limit statements for:
h(x) = (โˆ’3xโด โˆ’ xยฒ + x) / (xยณ + 4x + 4)
Be careful about the sign of the leading term!
๐Ÿ“„ Page 2 โ€” Answers (print this side second) BACK
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โœ“ Answer โ€” Case 1 (n = d)
Horizontal asymptote: y = a/b
Divide leading coefficients.
Example: (3xยฒ)/(5xยฒ) โ†’ y = 3/5
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โœ“ Answer โ€” Definition
f(x) = p(x) / q(x)
The quotient of two polynomials, where q(x) โ‰  0.
Both p(x) and q(x) must be polynomials.
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โœ“ Answer โ€” Case 3 (n > d)
No horizontal asymptote.
End behavior like the polynomial y = (a/b)xโฟโปแตˆ.
Special: if n = d+1 exactly โ†’ slant asymptote.
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โœ“ Answer โ€” Case 2 (n < d)
Horizontal asymptote: y = 0
The denominator dominates โ€” the fraction shrinks to zero in both directions.
Example: (2x)/(xยฒ) โ†’ y = 0
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โœ“ Answer โ€” Slant Direction
Parallel to y = (a/b)x
a = leading coeff of numerator
b = leading coeff of denominator
Example: (xยฒ+3x+2)/(2x+4) โ†’ y = (1/2)x
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โœ“ Answer โ€” Slant Condition
n = d + 1 (exactly)
Numerator degree is exactly 1 more than denominator.
โš ๏ธ n = d+2 or more โ†’ NO slant asymptote.
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โœ“ Answer โ€” Limits (Case 3)
lim xโ†’โˆ’โˆž h(x) = +โˆž
lim xโ†’+โˆž h(x) = โˆ’โˆž
n=4 > d=3 โ†’ end like โˆ’3x.
xโ†’โˆ’โˆž: โˆ’3(โˆ’โˆž)=+โˆž   xโ†’+โˆž: โˆ’3(+โˆž)=โˆ’โˆž
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โœ“ Answer โ€” Limits (Case 1)
lim xโ†’โˆ’โˆž f(x) = 1/3
lim xโ†’+โˆž f(x) = 1/3
n=d=3 โ†’ Case 1. Leading ratio = 2/6 = 1/3.
Both limits equal the HA value.