AP Precalculus Flashcards: Rational Functions And Zeros

Study Rational Functions And Zeros 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 Zeros

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QUESTION

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Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ .

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ANSWER

Removable discontinuity at $x = 1$ . The factor $(x-1)$ cancels from numerator and denominator.

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What this deck covers

This deck focuses on Rational Functions And Zeros, 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: Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Answer: Removable discontinuity at $x = 1$ . The factor $(x-1)$ cancels from numerator and denominator.

Flashcard 2: How do you determine a hole in a rational function?

Answer: If $p(x)$ and $q(x)$ share a common factor. Common factors can be cancelled, creating a removable discontinuity.

Flashcard 3: Identify the hole in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Answer: Hole at $x = 1$ . Factor $x^2 - 1 = (x-1)(x+1)$ and cancel the common factor.

Flashcard 4: What is a zero of a rational function?

Answer: A value of $x$ for which $f(x) = 0$ . A zero occurs when the numerator equals zero but the denominator doesn't.

Flashcard 5: What is the behavior of $f(x) = \frac{2x}{x + 5}$ as $x \to -5$ ?

Answer: As $x \to -5$ , $f(x) \to \pm \infty$ . The function has a vertical asymptote at $x = -5$ .

Flashcard 6: What is the slant asymptote of $f(x) = \frac{x^2 + 1}{x}$ ?

Answer: The slant asymptote is $y = x$ . Divide $x^2 + 1$ by $x$ using polynomial long division.

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Answer: The horizontal asymptote is $y = \frac{7}{2}$ . Equal degrees give horizontal asymptote $y = \frac{7}{2}$ .

Flashcard 8: Identify the hole in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Answer: Hole at $x = 1$ . Factor $x^2 - 1 = (x-1)(x+1)$ and cancel the common factor.

Flashcard 9: Find the zeros of $f(x) = \frac{3x^2 - 12}{x + 2}$ .

Answer: The zeros are $x = 2$ and $x = -2$ . Factor the numerator: $3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)$ .

Flashcard 10: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Answer: The horizontal asymptote is $y = \frac{7}{2}$ . Equal degrees give horizontal asymptote $y = \frac{7}{2}$ .

Flashcard 11: What is a zero of a rational function?

Answer: A value of $x$ for which $f(x) = 0$ . A zero occurs when the numerator equals zero but the denominator doesn't.

Flashcard 12: What is the degree of the polynomial $p(x) = 3x^4 - 2x^2 + 1$ ?

Answer: The degree is 4. The highest power term determines the degree of a polynomial.

Flashcard 13: Find the vertical asymptote of $f(x) = \frac{2x + 3}{x - 4}$ .

Answer: Vertical asymptote at $x = 4$ . Set the denominator $x - 4 = 0$ to find the vertical asymptote.

Flashcard 14: Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 4}$ .

Answer: Vertical asymptotes are $x = 2$ and $x = -2$ . Factor $x^2 - 4 = (x-2)(x+2)$ to find where the denominator is zero.

Flashcard 15: What are the zeros of $f(x) = \frac{x^2 - 9}{x + 2}$ ?

Answer: The zeros are $x = 3$ and $x = -3$ . Factor the numerator: $x^2 - 9 = (x-3)(x+3)$ .

Flashcard 16: Find the vertical asymptote of $f(x) = \frac{2x + 3}{x - 4}$ .

Answer: Vertical asymptote at $x = 4$ . Set the denominator $x - 4 = 0$ to find the vertical asymptote.

Flashcard 17: What is the behavior of $f(x) = \frac{1}{x}$ as $x \to 0$ ?

Answer: As $x \to 0$ , $f(x) \to \pm \infty$ . The function has a vertical asymptote at $x = 0$ .

Flashcard 18: What is the degree of $q(x) = x^3 - 7x$ ?

Answer: The degree is 3. The highest power term $x^3$ has degree 3.

Flashcard 19: What is the behavior of $f(x) = \frac{x}{x^2 - 1}$ as $x \to 1$ ?

Answer: As $x \to 1$ , $f(x) \to \infty$ . The denominator $(x-1)$ approaches 0 while numerator approaches 1.

Flashcard 20: What is the horizontal asymptote for $f(x) = \frac{x + 1}{x + 2}$ ?

Answer: The horizontal asymptote is $y = 1$ . When degrees are equal, the horizontal asymptote is $\frac{1}{1} = 1$ .

Flashcard 21: Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Answer: Removable discontinuity at $x = 1$ . The factor $(x-1)$ cancels from numerator and denominator.

Flashcard 22: What is the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ ?

Answer: The horizontal asymptote is $y = 2$ . Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$ .

Flashcard 23: Which term denotes values not in the domain of a rational function?

Answer: These are the poles or vertical asymptotes. Points where the denominator equals zero make the function undefined.

Flashcard 24: State the horizontal asymptote for $f(x) = \frac{4x}{x + 2}$ .

Answer: The horizontal asymptote is $y = 4$ . As $x \to \infty$ , the function approaches $\frac{4}{1} = 4$ .

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

Answer: Domain: $x \neq 0, x \neq 4$ . Factor the denominator: $x^2 - 4x = x(x - 4)$ .

Flashcard 26: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{x^2 - 1}$ .

Answer: The horizontal asymptote is $y = 1$ . When degrees are equal, the horizontal asymptote equals the ratio of leading coefficients.

Flashcard 27: Identify the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 4}$ .

Answer: The horizontal asymptote is $y = 0$ . Numerator degree is less than denominator degree, so $y = 0$ .

Flashcard 28: What is the behavior of $f(x) = \frac{x}{x^2 - 1}$ as $x \to 1$ ?

Answer: As $x \to 1$ , $f(x) \to \infty$ . The denominator $(x-1)$ approaches 0 while numerator approaches 1.

Flashcard 29: State the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 1}$ .

Answer: The horizontal asymptote is $y = 2$ . Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$ .

Flashcard 30: What are the zeros of $f(x) = \frac{x^2 - 9}{x + 2}$ ?

Answer: The zeros are $x = 3$ and $x = -3$ . Factor the numerator: $x^2 - 9 = (x-3)(x+3)$ .

Flashcard 31: State the condition for a vertical asymptote in a rational function.

Answer: Occurs where $q(x) = 0$ and $p(x) \neq 0$ . The denominator is zero while the numerator is non-zero.

Flashcard 32: What is the behavior of $f(x) = \frac{1}{x}$ as $x \to 0$ ?

Answer: As $x \to 0$ , $f(x) \to \pm \infty$ . The function has a vertical asymptote at $x = 0$ .

Flashcard 33: Which term describes $x$ values making $f(x)$ undefined?

Answer: These are the excluded values or domain restrictions. Values that make the denominator zero are excluded from the domain.

Flashcard 34: Which term describes $x$ values making $f(x)$ undefined?

Answer: These are the excluded values or domain restrictions. Values that make the denominator zero are excluded from the domain.

Flashcard 35: Find the zero of $f(x) = \frac{2x - 4}{x + 3}$ .

Answer: The zero is $x = 2$ . Set $2x - 4 = 0$ to find where the numerator equals zero.

Flashcard 36: Find the domain of $f(x) = \frac{4x + 1}{x^2 - 9}$ .

Answer: Domain: $x \neq 3, x \neq -3$ . Set $x^2 - 9 = 0$ to find where the function is undefined.

Flashcard 37: Identify the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 4}$ .

Answer: The horizontal asymptote is $y = 0$ . Numerator degree is less than denominator degree, so $y = 0$ .

Flashcard 38: Which term denotes values not in the domain of a rational function?

Answer: These are the poles or vertical asymptotes. Points where the denominator equals zero make the function undefined.

Flashcard 39: What is the degree of $q(x) = x^3 - 7x$ ?

Answer: The degree is 3. The highest power term $x^3$ has degree 3.

Flashcard 40: What is the horizontal asymptote for $f(x) = \frac{x + 1}{x + 2}$ ?

Answer: The horizontal asymptote is $y = 1$ . When degrees are equal, the horizontal asymptote is $\frac{1}{1} = 1$ .

Flashcard 41: Identify the vertical asymptote of $f(x) = \frac{5}{x^2 - 25}$ .

Answer: Vertical asymptotes at $x = 5$ and $x = -5$ . Factor the denominator: $x^2 - 25 = (x-5)(x+5)$ .

Flashcard 42: Find the zeros of $f(x) = \frac{3x^2 - 12}{x + 2}$ .

Answer: The zeros are $x = 2$ and $x = -2$ . Factor the numerator: $3x^2 - 12 = 3(x^2 - 4) = 3(x-2)(x+2)$ .

Flashcard 43: What is the domain of $f(x) = \frac{x}{x^2 - 4x}$ ?

Answer: Domain: $x \neq 0, x \neq 4$ . Factor the denominator: $x^2 - 4x = x(x - 4)$ .

Flashcard 44: State the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 1}$ .

Answer: The horizontal asymptote is $y = 2$ . Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$ .

Flashcard 45: State the condition for a vertical asymptote in a rational function.

Answer: Occurs where $q(x) = 0$ and $p(x) \neq 0$ . The denominator is zero while the numerator is non-zero.

Flashcard 46: Find the zero of $f(x) = \frac{x - 4}{x + 1}$ .

Answer: The zero is $x = 4$ . Set the numerator $x - 4 = 0$ to find where $f(x) = 0$ .

Flashcard 47: For $f(x) = \frac{x^2 - 4}{x - 2}$ , what is the discontinuity?

Answer: Discontinuity (hole) at $x = 2$ . Factor and cancel: $(x-2)$ appears in both numerator and denominator.

Flashcard 48: Find the domain of $f(x) = \frac{4x + 1}{x^2 - 9}$ .

Answer: Domain: $x \neq 3, x \neq -3$ . Set $x^2 - 9 = 0$ to find where the function is undefined.

Flashcard 49: State the vertical asymptote of $f(x) = \frac{2x + 1}{x^2 - 16}$ .

Answer: Vertical asymptotes at $x = 4$ and $x = -4$ . Factor the denominator: $x^2 - 16 = (x-4)(x+4)$ .

Flashcard 50: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{x^2 - 1}$ .

Answer: The horizontal asymptote is $y = 1$ . When degrees are equal, the horizontal asymptote equals the ratio of leading coefficients.

Flashcard 51: What is the domain of $f(x) = \frac{1}{x(x + 3)}$ ?

Answer: Domain: $x \neq 0, x \neq -3$ . Set $x(x + 3) = 0$ to find where the function is undefined.

Flashcard 52: What is the end behavior of $f(x) = \frac{3x^3 + 4}{5x^3 + 2}$ ?

Answer: As $x \to \pm \infty$ , $f(x) \to \frac{3}{5}$ . When degrees are equal, divide leading coefficients: $\frac{3}{5}$ .

Flashcard 53: What is the behavior of $f(x) = \frac{2x}{x + 5}$ as $x \to -5$ ?

Answer: As $x \to -5$ , $f(x) \to \pm \infty$ . The function has a vertical asymptote at $x = -5$ .

Flashcard 54: Identify the vertical asymptote of $f(x) = \frac{5}{x^2 - 25}$ .

Answer: Vertical asymptotes at $x = 5$ and $x = -5$ . Factor the denominator: $x^2 - 25 = (x-5)(x+5)$ .

Flashcard 55: For $f(x) = \frac{x^2 - 4}{x - 2}$ , what is the discontinuity?

Answer: Discontinuity (hole) at $x = 2$ . Factor and cancel: $(x-2)$ appears in both numerator and denominator.

Flashcard 56: What is the intercept of $f(x) = \frac{3}{x-2}$ ?

Answer: Vertical intercept at $(0, -\frac{3}{2})$ . Substitute $x = 0$ into the function to find the y-intercept.

Flashcard 57: What is the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ ?

Answer: The horizontal asymptote is $y = 2$ . Equal degrees give horizontal asymptote $y = \frac{2}{1} = 2$ .

Flashcard 58: Describe when a rational function has no horizontal asymptote.

Answer: When the degree of $p(x)$ is greater than $q(x)$ . When numerator degree exceeds denominator degree, no horizontal asymptote exists.

Flashcard 59: How do you determine a hole in a rational function?

Answer: If $p(x)$ and $q(x)$ share a common factor. Common factors can be cancelled, creating a removable discontinuity.

Flashcard 60: Find the zero of $f(x) = \frac{2x - 4}{x + 3}$ .

Answer: The zero is $x = 2$ . Set $2x - 4 = 0$ to find where the numerator equals zero.

Flashcard 61: Describe when a rational function has no horizontal asymptote.

Answer: When the degree of $p(x)$ is greater than $q(x)$ . When numerator degree exceeds denominator degree, no horizontal asymptote exists.

Flashcard 62: What is the degree of the polynomial $p(x) = 3x^4 - 2x^2 + 1$ ?

Answer: The degree is 4. The highest power term determines the degree of a polynomial.

Flashcard 63: What happens to $f(x) = \frac{1}{x-3}$ as $x \to 3$ ?

Answer: $f(x) \to \pm \infty$ , vertical asymptote at $x = 3$ . As $x$ approaches 3, the denominator approaches 0 while numerator stays 1.

Flashcard 64: What is the general form of a rational function?

Answer: $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. This defines a rational function as a ratio of two polynomial functions.

Flashcard 65: State the vertical asymptote of $f(x) = \frac{2x + 1}{x^2 - 16}$ .

Answer: Vertical asymptotes at $x = 4$ and $x = -4$ . Factor the denominator: $x^2 - 16 = (x-4)(x+4)$ .

Flashcard 66: What is the slant asymptote of $f(x) = \frac{x^2 + 1}{x}$ ?

Answer: The slant asymptote is $y = x$ . Divide $x^2 + 1$ by $x$ using polynomial long division.

Flashcard 67: State the horizontal asymptote for $f(x) = \frac{4x}{x + 2}$ .

Answer: The horizontal asymptote is $y = 4$ . As $x \to \infty$ , the function approaches $\frac{4}{1} = 4$ .

Flashcard 68: What is the y-intercept of $f(x) = \frac{2x - 5}{x + 3}$ ?

Answer: The y-intercept is $(0, -\frac{5}{3})$ . Substitute $x = 0$ to get $f(0) = \frac{-5}{3}$ .

Flashcard 69: Find the zero of $f(x) = \frac{x - 4}{x + 1}$ .

Answer: The zero is $x = 4$ . Set the numerator $x - 4 = 0$ to find where $f(x) = 0$ .

Flashcard 70: What is the domain of $f(x) = \frac{1}{x(x + 3)}$ ?

Answer: Domain: $x \neq 0, x \neq -3$ . Set $x(x + 3) = 0$ to find where the function is undefined.

Flashcard 71: What happens to $f(x) = \frac{1}{x-3}$ as $x \to 3$ ?

Answer: $f(x) \to \pm \infty$ , vertical asymptote at $x = 3$ . As $x$ approaches 3, the denominator approaches 0 while numerator stays 1.

Flashcard 72: What is the general form of a rational function?

Answer: $f(x) = \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials. This defines a rational function as a ratio of two polynomial functions.

AP Precalculus Flashcards: Rational Functions And Zeros

Rational Functions And Zeros

What this deck covers

How to use these flashcards

All flashcards

Flashcard 1: Identify a removable discontinuity in f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1​.

Flashcard 2: How do you determine a hole in a rational function?

Flashcard 3: Identify the hole in f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1​.

Flashcard 4: What is a zero of a rational function?

Flashcard 5: What is the behavior of f(x)=2xx+5f(x) = \frac{2x}{x + 5}f(x)=x+52x​ as x→−5x \to -5x→−5?

Flashcard 6: What is the slant asymptote of f(x)=x2+1xf(x) = \frac{x^2 + 1}{x}f(x)=xx2+1​?

Flashcard 7: What is the horizontal asymptote of f(x)=7x2x+3f(x) = \frac{7x}{2x + 3}f(x)=2x+37x​?

Flashcard 8: Identify the hole in f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1​.

Flashcard 9: Find the zeros of f(x)=3x2−12x+2f(x) = \frac{3x^2 - 12}{x + 2}f(x)=x+23x2−12​.

Flashcard 10: What is the horizontal asymptote of f(x)=7x2x+3f(x) = \frac{7x}{2x + 3}f(x)=2x+37x​?

Flashcard 11: What is a zero of a rational function?

Flashcard 12: What is the degree of the polynomial p(x)=3x4−2x2+1p(x) = 3x^4 - 2x^2 + 1p(x)=3x4−2x2+1?

Flashcard 13: Find the vertical asymptote of f(x)=2x+3x−4f(x) = \frac{2x + 3}{x - 4}f(x)=x−42x+3​.

Flashcard 14: Identify the vertical asymptote of f(x)=2xx2−4f(x) = \frac{2x}{x^2 - 4}f(x)=x2−42x​.

Flashcard 15: What are the zeros of f(x)=x2−9x+2f(x) = \frac{x^2 - 9}{x + 2}f(x)=x+2x2−9​?

Flashcard 16: Find the vertical asymptote of f(x)=2x+3x−4f(x) = \frac{2x + 3}{x - 4}f(x)=x−42x+3​.

Flashcard 17: What is the behavior of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ as x→0x \to 0x→0?

Flashcard 18: What is the degree of q(x)=x3−7xq(x) = x^3 - 7xq(x)=x3−7x?

Flashcard 19: What is the behavior of f(x)=xx2−1f(x) = \frac{x}{x^2 - 1}f(x)=x2−1x​ as x→1x \to 1x→1?

Flashcard 20: What is the horizontal asymptote for f(x)=x+1x+2f(x) = \frac{x + 1}{x + 2}f(x)=x+2x+1​?

Flashcard 21: Identify a removable discontinuity in f(x)=x2−1x−1f(x) = \frac{x^2 - 1}{x - 1}f(x)=x−1x2−1​.

Flashcard 22: What is the horizontal asymptote for f(x)=2x2x2+1f(x) = \frac{2x^2}{x^2 + 1}f(x)=x2+12x2​?

Flashcard 23: Which term denotes values not in the domain of a rational function?

Flashcard 24: State the horizontal asymptote for f(x)=4xx+2f(x) = \frac{4x}{x + 2}f(x)=x+24x​.

Flashcard 25: What is the domain of f(x)=xx2−4xf(x) = \frac{x}{x^2 - 4x}f(x)=x2−4xx​?

Flashcard 26: Identify the horizontal asymptote of f(x)=x2+3x2−1f(x) = \frac{x^2 + 3}{x^2 - 1}f(x)=x2−1x2+3​.

Flashcard 27: Identify the horizontal asymptote of f(x)=5xx2+4f(x) = \frac{5x}{x^2 + 4}f(x)=x2+45x​.

Flashcard 28: What is the behavior of f(x)=xx2−1f(x) = \frac{x}{x^2 - 1}f(x)=x2−1x​ as x→1x \to 1x→1?

Flashcard 29: State the horizontal asymptote of f(x)=2x2+3x2+1f(x) = \frac{2x^2 + 3}{x^2 + 1}f(x)=x2+12x2+3​.

Flashcard 30: What are the zeros of f(x)=x2−9x+2f(x) = \frac{x^2 - 9}{x + 2}f(x)=x+2x2−9​?

Flashcard 31: State the condition for a vertical asymptote in a rational function.

Flashcard 32: What is the behavior of f(x)=1xf(x) = \frac{1}{x}f(x)=x1​ as x→0x \to 0x→0?

Flashcard 33: Which term describes xxx values making f(x)f(x)f(x) undefined?

Flashcard 34: Which term describes xxx values making f(x)f(x)f(x) undefined?

Flashcard 35: Find the zero of f(x)=2x−4x+3f(x) = \frac{2x - 4}{x + 3}f(x)=x+32x−4​.

Flashcard 36: Find the domain of f(x)=4x+1x2−9f(x) = \frac{4x + 1}{x^2 - 9}f(x)=x2−94x+1​.

Flashcard 37: Identify the horizontal asymptote of f(x)=5xx2+4f(x) = \frac{5x}{x^2 + 4}f(x)=x2+45x​.

Flashcard 38: Which term denotes values not in the domain of a rational function?

Flashcard 39: What is the degree of q(x)=x3−7xq(x) = x^3 - 7xq(x)=x3−7x?

Flashcard 40: What is the horizontal asymptote for f(x)=x+1x+2f(x) = \frac{x + 1}{x + 2}f(x)=x+2x+1​?

Flashcard 41: Identify the vertical asymptote of f(x)=5x2−25f(x) = \frac{5}{x^2 - 25}f(x)=x2−255​.

Flashcard 42: Find the zeros of f(x)=3x2−12x+2f(x) = \frac{3x^2 - 12}{x + 2}f(x)=x+23x2−12​.

Flashcard 43: What is the domain of f(x)=xx2−4xf(x) = \frac{x}{x^2 - 4x}f(x)=x2−4xx​?

Flashcard 44: State the horizontal asymptote of f(x)=2x2+3x2+1f(x) = \frac{2x^2 + 3}{x^2 + 1}f(x)=x2+12x2+3​.

Flashcard 45: State the condition for a vertical asymptote in a rational function.

Flashcard 46: Find the zero of f(x)=x−4x+1f(x) = \frac{x - 4}{x + 1}f(x)=x+1x−4​.

Flashcard 47: For f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2}f(x)=x−2x2−4​, what is the discontinuity?

Flashcard 48: Find the domain of f(x)=4x+1x2−9f(x) = \frac{4x + 1}{x^2 - 9}f(x)=x2−94x+1​.

Flashcard 49: State the vertical asymptote of f(x)=2x+1x2−16f(x) = \frac{2x + 1}{x^2 - 16}f(x)=x2−162x+1​.

Flashcard 50: Identify the horizontal asymptote of f(x)=x2+3x2−1f(x) = \frac{x^2 + 3}{x^2 - 1}f(x)=x2−1x2+3​.

Flashcard 51: What is the domain of f(x)=1x(x+3)f(x) = \frac{1}{x(x + 3)}f(x)=x(x+3)1​?

Flashcard 52: What is the end behavior of f(x)=3x3+45x3+2f(x) = \frac{3x^3 + 4}{5x^3 + 2}f(x)=5x3+23x3+4​?

Flashcard 53: What is the behavior of f(x)=2xx+5f(x) = \frac{2x}{x + 5}f(x)=x+52x​ as x→−5x \to -5x→−5?

Flashcard 54: Identify the vertical asymptote of f(x)=5x2−25f(x) = \frac{5}{x^2 - 25}f(x)=x2−255​.

Flashcard 55: For f(x)=x2−4x−2f(x) = \frac{x^2 - 4}{x - 2}f(x)=x−2x2−4​, what is the discontinuity?

Flashcard 56: What is the intercept of f(x)=3x−2f(x) = \frac{3}{x-2}f(x)=x−23​?

Flashcard 57: What is the horizontal asymptote for f(x)=2x2x2+1f(x) = \frac{2x^2}{x^2 + 1}f(x)=x2+12x2​?

Flashcard 58: Describe when a rational function has no horizontal asymptote.

Flashcard 59: How do you determine a hole in a rational function?

Flashcard 60: Find the zero of f(x)=2x−4x+3f(x) = \frac{2x - 4}{x + 3}f(x)=x+32x−4​.

Flashcard 61: Describe when a rational function has no horizontal asymptote.

Flashcard 62: What is the degree of the polynomial p(x)=3x4−2x2+1p(x) = 3x^4 - 2x^2 + 1p(x)=3x4−2x2+1?

Flashcard 63: What happens to f(x)=1x−3f(x) = \frac{1}{x-3}f(x)=x−31​ as x→3x \to 3x→3?

Flashcard 64: What is the general form of a rational function?

Flashcard 65: State the vertical asymptote of f(x)=2x+1x2−16f(x) = \frac{2x + 1}{x^2 - 16}f(x)=x2−162x+1​.

Flashcard 66: What is the slant asymptote of f(x)=x2+1xf(x) = \frac{x^2 + 1}{x}f(x)=xx2+1​?

Flashcard 67: State the horizontal asymptote for f(x)=4xx+2f(x) = \frac{4x}{x + 2}f(x)=x+24x​.

Flashcard 68: What is the y-intercept of f(x)=2x−5x+3f(x) = \frac{2x - 5}{x + 3}f(x)=x+32x−5​?

Flashcard 69: Find the zero of f(x)=x−4x+1f(x) = \frac{x - 4}{x + 1}f(x)=x+1x−4​.

Flashcard 70: What is the domain of f(x)=1x(x+3)f(x) = \frac{1}{x(x + 3)}f(x)=x(x+3)1​?

Flashcard 71: What happens to f(x)=1x−3f(x) = \frac{1}{x-3}f(x)=x−31​ as x→3x \to 3x→3?

Flashcard 72: What is the general form of a rational function?

Flashcard 1: Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Flashcard 3: Identify the hole in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Flashcard 5: What is the behavior of $f(x) = \frac{2x}{x + 5}$ as $x \to -5$ ?

Flashcard 6: What is the slant asymptote of $f(x) = \frac{x^2 + 1}{x}$ ?

Flashcard 7: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Flashcard 8: Identify the hole in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Flashcard 9: Find the zeros of $f(x) = \frac{3x^2 - 12}{x + 2}$ .

Flashcard 10: What is the horizontal asymptote of $f(x) = \frac{7x}{2x + 3}$ ?

Flashcard 12: What is the degree of the polynomial $p(x) = 3x^4 - 2x^2 + 1$ ?

Flashcard 13: Find the vertical asymptote of $f(x) = \frac{2x + 3}{x - 4}$ .

Flashcard 14: Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 4}$ .

Flashcard 15: What are the zeros of $f(x) = \frac{x^2 - 9}{x + 2}$ ?

Flashcard 16: Find the vertical asymptote of $f(x) = \frac{2x + 3}{x - 4}$ .

Flashcard 17: What is the behavior of $f(x) = \frac{1}{x}$ as $x \to 0$ ?

Flashcard 18: What is the degree of $q(x) = x^3 - 7x$ ?

Flashcard 19: What is the behavior of $f(x) = \frac{x}{x^2 - 1}$ as $x \to 1$ ?

Flashcard 20: What is the horizontal asymptote for $f(x) = \frac{x + 1}{x + 2}$ ?

Flashcard 21: Identify a removable discontinuity in $f(x) = \frac{x^2 - 1}{x - 1}$ .

Flashcard 22: What is the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ ?

Flashcard 24: State the horizontal asymptote for $f(x) = \frac{4x}{x + 2}$ .

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

Flashcard 26: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{x^2 - 1}$ .

Flashcard 27: Identify the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 4}$ .

Flashcard 28: What is the behavior of $f(x) = \frac{x}{x^2 - 1}$ as $x \to 1$ ?

Flashcard 29: State the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 1}$ .

Flashcard 30: What are the zeros of $f(x) = \frac{x^2 - 9}{x + 2}$ ?

Flashcard 32: What is the behavior of $f(x) = \frac{1}{x}$ as $x \to 0$ ?

Flashcard 33: Which term describes $x$ values making $f(x)$ undefined?

Flashcard 34: Which term describes $x$ values making $f(x)$ undefined?

Flashcard 35: Find the zero of $f(x) = \frac{2x - 4}{x + 3}$ .

Flashcard 36: Find the domain of $f(x) = \frac{4x + 1}{x^2 - 9}$ .

Flashcard 37: Identify the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 4}$ .

Flashcard 39: What is the degree of $q(x) = x^3 - 7x$ ?

Flashcard 40: What is the horizontal asymptote for $f(x) = \frac{x + 1}{x + 2}$ ?

Flashcard 41: Identify the vertical asymptote of $f(x) = \frac{5}{x^2 - 25}$ .

Flashcard 42: Find the zeros of $f(x) = \frac{3x^2 - 12}{x + 2}$ .

Flashcard 43: What is the domain of $f(x) = \frac{x}{x^2 - 4x}$ ?

Flashcard 44: State the horizontal asymptote of $f(x) = \frac{2x^2 + 3}{x^2 + 1}$ .

Flashcard 46: Find the zero of $f(x) = \frac{x - 4}{x + 1}$ .

Flashcard 47: For $f(x) = \frac{x^2 - 4}{x - 2}$ , what is the discontinuity?

Flashcard 48: Find the domain of $f(x) = \frac{4x + 1}{x^2 - 9}$ .

Flashcard 49: State the vertical asymptote of $f(x) = \frac{2x + 1}{x^2 - 16}$ .

Flashcard 50: Identify the horizontal asymptote of $f(x) = \frac{x^2 + 3}{x^2 - 1}$ .

Flashcard 51: What is the domain of $f(x) = \frac{1}{x(x + 3)}$ ?

Flashcard 52: What is the end behavior of $f(x) = \frac{3x^3 + 4}{5x^3 + 2}$ ?

Flashcard 53: What is the behavior of $f(x) = \frac{2x}{x + 5}$ as $x \to -5$ ?

Flashcard 54: Identify the vertical asymptote of $f(x) = \frac{5}{x^2 - 25}$ .

Flashcard 55: For $f(x) = \frac{x^2 - 4}{x - 2}$ , what is the discontinuity?

Flashcard 56: What is the intercept of $f(x) = \frac{3}{x-2}$ ?

Flashcard 57: What is the horizontal asymptote for $f(x) = \frac{2x^2}{x^2 + 1}$ ?

Flashcard 60: Find the zero of $f(x) = \frac{2x - 4}{x + 3}$ .

Flashcard 62: What is the degree of the polynomial $p(x) = 3x^4 - 2x^2 + 1$ ?

Flashcard 63: What happens to $f(x) = \frac{1}{x-3}$ as $x \to 3$ ?

Flashcard 65: State the vertical asymptote of $f(x) = \frac{2x + 1}{x^2 - 16}$ .

Flashcard 66: What is the slant asymptote of $f(x) = \frac{x^2 + 1}{x}$ ?

Flashcard 67: State the horizontal asymptote for $f(x) = \frac{4x}{x + 2}$ .

Flashcard 68: What is the y-intercept of $f(x) = \frac{2x - 5}{x + 3}$ ?

Flashcard 69: Find the zero of $f(x) = \frac{x - 4}{x + 1}$ .

Flashcard 70: What is the domain of $f(x) = \frac{1}{x(x + 3)}$ ?

Flashcard 71: What happens to $f(x) = \frac{1}{x-3}$ as $x \to 3$ ?