AP Precalculus Flashcards: Rational Functions And End Behavior

Study Rational Functions And End Behavior in AP Precalculus with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Precalculus

Rational Functions And End Behavior

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QUESTION

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State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

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ANSWER

Vertical asymptote at $x = 2$ . Perfect square: $(x-2)^2 = 0$ gives $x = 2$ .

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What this deck covers

This deck focuses on Rational Functions And End Behavior, giving you a quick way to review the definitions, rules, and examples that matter most for AP Precalculus.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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Flashcard 1: State the vertical asymptotes for $f(x) = \frac{1}{x^2 - 4x + 4}$ .

Answer: Vertical asymptote at $x = 2$ . Perfect square: $(x-2)^2 = 0$ gives $x = 2$ .

Flashcard 2: State the vertical asymptotes for $f(x) = \frac{x^2-1}{x^2-4}$ .

Answer: Vertical asymptotes at $x = 2$ and $x = -2$ . Factor denominator: $(x-2)(x+2) = 0$ .

Flashcard 3: What is a removable discontinuity in rational functions?

Answer: A hole where $p(x)$ and $q(x)$ share a common factor. Common factors cancel, creating a hole.

Flashcard 4: What is the end behavior of $f(x) = \frac{2x^2}{x^3 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . Numerator degree less than denominator degree.

Flashcard 5: Determine the vertical asymptote of $f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 1$ . Factor denominator: $(x-3)(x-1) = 0$ .

Flashcard 6: What is the horizontal asymptote for $f(x) = \frac{2x}{x^2+3}$ ?

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 7: What is the end behavior of $f(x) = \frac{x^2 + 2x + 1}{x^2}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $1$ as $x \to \infty$ . Expand and simplify: $\frac{x^2+2x+1}{x^2} = 1 + \frac{2}{x} + \frac{1}{x^2}$ .

Flashcard 8: State the horizontal asymptote for $f(x) = \frac{4x+3}{7x+8}$ .

Answer: Horizontal asymptote at $y = \frac{4}{7}$ . Same degree: ratio of leading coefficients $\frac{4}{7}$ .

Flashcard 9: What is the end behavior of $f(x) = \frac{3x^4}{x^2 + 1}$ as $x \to \infty$ ?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 10: What is the vertical asymptote of $f(x) = \frac{2x^2}{x^2 - 9}$ ?

Answer: Vertical asymptotes at $x = 3$ and $x = -3$ . Factor denominator: $(x-3)(x+3) = 0$ .

Flashcard 11: Identify the vertical asymptote for $f(x) = \frac{x+1}{x-3}$ .

Answer: Vertical asymptote at $x = 3$ . Set denominator equal to zero: $x - 3 = 0$ .

Flashcard 12: What does the end behavior of $f(x) = \frac{x^3}{x+1}$ as $x \to \infty$ approach?

Answer: $f(x)$ approaches $\infty$ as $x \to \infty$ . Numerator degree exceeds denominator degree.

Flashcard 13: State the vertical asymptotes for $f(x) = \frac{x^2-4}{x^2-5x+6}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = 2$ . Factor denominator: $(x-3)(x-2) = 0$ .

Flashcard 14: What is the horizontal asymptote for $f(x) = \frac{5x+1}{2x-3}$ ?

Answer: Horizontal asymptote at $y = \frac{5}{2}$ . Same degree: ratio of leading coefficients $\frac{5}{2}$ .

Flashcard 15: Find the vertical asymptote of $f(x) = \frac{x^2 + 1}{x - 5}$ .

Answer: Vertical asymptote at $x = 5$ . Set denominator equal to zero: $x - 5 = 0$ .

Flashcard 16: Identify the horizontal asymptote for $f(x) = \frac{5x^3}{3x^3 + 1}$ .

Answer: Horizontal asymptote at $y = \frac{5}{3}$ . Same degree: ratio of leading coefficients $\frac{5}{3}$ .

Flashcard 17: Identify the removable discontinuity of $f(x) = \frac{x^2-1}{x^2+x}$ .

Answer: Removable discontinuity at $x = -1$ . Factor: $\frac{(x-1)(x+1)}{x(x+1)}$ cancels $(x+1)$ .

Flashcard 18: What is the end behavior of $f(x) = \frac{2x^3}{x^2+1}$ as $x \to -\infty$ ?

Answer: $f(x)$ approaches $-\infty$ as $x \to -\infty$ . Numerator degree exceeds denominator, goes to $-\infty$ .

Flashcard 19: What is a rational function?

Answer: A function of the form $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. The numerator and denominator must be polynomials.

Flashcard 20: Determine the horizontal asymptote of $f(x) = \frac{x+2}{x^2+3x+1}$ .

Answer: Horizontal asymptote at $y = 0$ . Numerator degree less than denominator degree.

Flashcard 21: Determine the vertical asymptote of $f(x) = \frac{x}{x^2-2x-3}$ .

Answer: Vertical asymptotes at $x = 3$ and $x = -1$ . Factor denominator: $(x-3)(x+1) = 0$ .

Flashcard 22: What is the domain of $f(x) = \frac{1}{x^2 - 9}$ ?

Answer: All real numbers except $x = 3$ and $x = -3$ . Exclude values where $x^2 - 9 = 0$ .

Flashcard 23: What is the domain of $f(x) = \frac{5x}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 24: Identify the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree: ratio of leading coefficients $\frac{2}{1}$ .

Flashcard 25: What is the domain of $f(x) = \frac{2}{x-4}$ ?

Answer: All real numbers except $x = 4$ . Exclude values where denominator equals zero.

Flashcard 26: What is the slant asymptote of $f(x) = \frac{x^2 + 2x + 3}{x - 1}$ ?

Answer: Slant asymptote is $y = x + 3$ . Numerator degree exceeds denominator by one.

Flashcard 27: What is the end behavior of $f(x) = \frac{1}{x}$ as $x$ approaches $\infty$ ?

Answer: $f(x)$ approaches $0$ as $x \to \infty$ . As denominator grows, fraction approaches zero.

Flashcard 28: Identify the removable discontinuity of $f(x) = \frac{(x-2)(x+3)}{x-2}$ .

Answer: Removable discontinuity at $x = 2$ . Common factor $(x-2)$ cancels out.

Flashcard 29: Does $f(x) = \frac{x^2-4}{x^2+4}$ have a horizontal asymptote?

Answer: Yes, $y = 1$ . Same degree polynomials have horizontal asymptote.

Flashcard 30: State the horizontal asymptote of $f(x) = \frac{2x^2+3}{x^2-1}$ .

Answer: Horizontal asymptote at $y = 2$ . Same degree polynomials: ratio of leading coefficients.

Flashcard 31: What is the domain of $f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}$ ?

Answer: All real numbers except $x = 2$ and $x = -2$ . Exclude values where $x^2 - 4 = 0$ .

Flashcard 32: Determine the end behavior of $f(x) = \frac{x^2}{x^3+1}$ as $x \to -\infty$ .

Answer: $f(x)$ approaches $0$ as $x \to -\infty$ . Degree of numerator less than denominator.

Flashcard 33: What is the end behavior of $f(x) = \frac{3x^3}{2x^3 + 5}$ as $x$ approaches infinity?

Answer: $f(x)$ approaches $\frac{3}{2}$ as $x \to \infty$ . Same degree: divide leading coefficients $\frac{3}{2}$ .

Flashcard 34: What is the horizontal asymptote of $f(x) = \frac{x^2+1}{2x^2-3x+5}$ ?

Answer: Horizontal asymptote at $y = \frac{1}{2}$ . Same degree: ratio of leading coefficients $\frac{1}{2}$ .

Flashcard 35: What is the horizontal asymptote of $f(x) = \frac{3x}{x + 4}$ ?

Answer: Horizontal asymptote at $y = 3$ . Same degree: ratio of leading coefficients $\frac{3}{1}$ .

AP Precalculus Flashcards: Rational Functions And End Behavior

Rational Functions And End Behavior

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: State the vertical asymptotes for f(x)=1x2−4x+4f(x) = \frac{1}{x^2 - 4x + 4}f(x)=x2−4x+41​.

Flashcard 2: State the vertical asymptotes for f(x)=x2−1x2−4f(x) = \frac{x^2-1}{x^2-4}f(x)=x2−4x2−1​.

Flashcard 3: What is a removable discontinuity in rational functions?

Flashcard 4: What is the end behavior of f(x)=2x2x3+1f(x) = \frac{2x^2}{x^3 + 1}f(x)=x3+12x2​ as x→∞x \to \inftyx→∞?

Flashcard 5: Determine the vertical asymptote of f(x)=x2+1x2−4x+3f(x) = \frac{x^2 + 1}{x^2 - 4x + 3}f(x)=x2−4x+3x2+1​.

Flashcard 6: What is the horizontal asymptote for f(x)=2xx2+3f(x) = \frac{2x}{x^2+3}f(x)=x2+32x​?

Flashcard 7: What is the end behavior of f(x)=x2+2x+1x2f(x) = \frac{x^2 + 2x + 1}{x^2}f(x)=x2x2+2x+1​ as x→∞x \to \inftyx→∞?

Flashcard 8: State the horizontal asymptote for f(x)=4x+37x+8f(x) = \frac{4x+3}{7x+8}f(x)=7x+84x+3​.

Flashcard 9: What is the end behavior of f(x)=3x4x2+1f(x) = \frac{3x^4}{x^2 + 1}f(x)=x2+13x4​ as x→∞x \to \inftyx→∞?

Flashcard 10: What is the vertical asymptote of f(x)=2x2x2−9f(x) = \frac{2x^2}{x^2 - 9}f(x)=x2−92x2​?

Flashcard 11: Identify the vertical asymptote for f(x)=x+1x−3f(x) = \frac{x+1}{x-3}f(x)=x−3x+1​.

Flashcard 12: What does the end behavior of f(x)=x3x+1f(x) = \frac{x^3}{x+1}f(x)=x+1x3​ as x→∞x \to \inftyx→∞ approach?

Flashcard 13: State the vertical asymptotes for f(x)=x2−4x2−5x+6f(x) = \frac{x^2-4}{x^2-5x+6}f(x)=x2−5x+6x2−4​.

Flashcard 14: What is the horizontal asymptote for f(x)=5x+12x−3f(x) = \frac{5x+1}{2x-3}f(x)=2x−35x+1​?

Flashcard 15: Find the vertical asymptote of f(x)=x2+1x−5f(x) = \frac{x^2 + 1}{x - 5}f(x)=x−5x2+1​.

Flashcard 16: Identify the horizontal asymptote for f(x)=5x33x3+1f(x) = \frac{5x^3}{3x^3 + 1}f(x)=3x3+15x3​.

Flashcard 17: Identify the removable discontinuity of f(x)=x2−1x2+xf(x) = \frac{x^2-1}{x^2+x}f(x)=x2+xx2−1​.

Flashcard 18: What is the end behavior of f(x)=2x3x2+1f(x) = \frac{2x^3}{x^2+1}f(x)=x2+12x3​ as x→−∞x \to -\inftyx→−∞?

Flashcard 19: What is a rational function?

Flashcard 20: Determine the horizontal asymptote of f(x)=x+2x2+3x+1f(x) = \frac{x+2}{x^2+3x+1}f(x)=x2+3x+1x+2​.

Flashcard 21: Determine the vertical asymptote of f(x)=xx2−2x−3f(x) = \frac{x}{x^2-2x-3}f(x)=x2−2x−3x​.

Flashcard 22: What is the domain of f(x)=1x2−9f(x) = \frac{1}{x^2 - 9}f(x)=x2−91​?

Flashcard 23: What is the domain of f(x)=5xx2−4f(x) = \frac{5x}{x^2 - 4}f(x)=x2−45x​?

Flashcard 24: Identify the horizontal asymptote for f(x)=2x2x2+1f(x) = \frac{2x^2}{x^2 + 1}f(x)=x2+12x2​.

Flashcard 25: What is the domain of f(x)=2x−4f(x) = \frac{2}{x-4}f(x)=x−42​?

Flashcard 26: What is the slant asymptote of f(x)=x2+2x+3x−1f(x) = \frac{x^2 + 2x + 3}{x - 1}f(x)=x−1x2+2x+3​?

Flashcard 27: What is the end behavior of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ as xxx approaches ∞\infty∞?

Flashcard 28: Identify the removable discontinuity of f(x)=(x−2)(x+3)x−2f(x) = \frac{(x-2)(x+3)}{x-2}f(x)=x−2(x−2)(x+3)​.

Flashcard 29: Does f(x)=x2−4x2+4f(x) = \frac{x^2-4}{x^2+4}f(x)=x2+4x2−4​ have a horizontal asymptote?

Flashcard 30: State the horizontal asymptote of f(x)=2x2+3x2−1f(x) = \frac{2x^2+3}{x^2-1}f(x)=x2−12x2+3​.

Flashcard 31: What is the domain of f(x)=x2+3x−4x2−4f(x) = \frac{x^2 + 3x - 4}{x^2 - 4}f(x)=x2−4x2+3x−4​?

Flashcard 32: Determine the end behavior of f(x)=x2x3+1f(x) = \frac{x^2}{x^3+1}f(x)=x3+1x2​ as x→−∞x \to -\inftyx→−∞.

Flashcard 33: What is the end behavior of f(x)=3x32x3+5f(x) = \frac{3x^3}{2x^3 + 5}f(x)=2x3+53x3​ as xxx approaches infinity?

Flashcard 34: What is the horizontal asymptote of f(x)=x2+12x2−3x+5f(x) = \frac{x^2+1}{2x^2-3x+5}f(x)=2x2−3x+5x2+1​?

Flashcard 35: What is the horizontal asymptote of f(x)=3xx+4f(x) = \frac{3x}{x + 4}f(x)=x+43x​?