AP Precalculus Flashcards: Rational Functions And Holes

What this deck covers

This deck focuses on Rational Functions And Holes, 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.

Flashcard 1: Perform long division on $f(x) = \frac{x^2 + 3x + 2}{x + 1}$.

Answer: Quotient is $x + 2$, remainder is 0. $x^2 + 3x + 2 = (x+1)(x+2)$ divides evenly.

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

Answer: Removable discontinuity at $x = 3$. $\frac{(x-3)(x+3)}{x-3}$ cancels at $x = 3$.

Flashcard 3: What is a horizontal asymptote for a rational function?

Answer: Occurs as $x \to \infty$, based on the degrees of $p(x)$ and $q(x)$. Determined by comparing degrees of numerator and denominator.

Flashcard 4: What role do intercepts play in graphing rational functions?

Answer: Determine where the graph crosses axes. Show where function crosses or touches coordinate axes.

Flashcard 5: State the behavior near a hole in a rational function.

Answer: Function approaches a limit but is not defined at the hole. Limiting value exists despite the discontinuity.

Flashcard 6: State the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$.

Answer: Horizontal asymptote at $y = 0$. Denominator degree exceeds numerator, so limit is zero.

Flashcard 7: What defines a vertical asymptote in a rational function?

Answer: Occurs at $x$ where $q(x) = 0$ and $p(x)$ does not cancel. Denominator zero with no cancellation creates vertical line.

Flashcard 8: Define a slant (oblique) asymptote in a rational function.

Answer: Occurs when degree of $p(x)$ is one more than $q(x)$. Creates diagonal asymptote from polynomial long division.

Flashcard 9: What is the end behavior of a rational function?

Answer: Describes $f(x)$ as $x \to \pm\infty$, often related to asymptotes. Behavior of function values as $x$ approaches infinity.

Flashcard 10: What role do intercepts play in graphing rational functions?

Answer: Determine where the graph crosses axes. Show where function crosses or touches coordinate axes.

Flashcard 11: Identify the hole in $f(x) = \frac{x^2 - 25}{x^2 - 5x}$.

Answer: Hole at $x = 5$. $\frac{(x-5)(x+5)}{x(x-5)}$ has $(x-5)$ common factor.

Flashcard 12: What is the significance of the leading coefficients in asymptotes?

Answer: They determine the horizontal asymptote when degrees are equal. When degrees are equal, their ratio gives horizontal asymptote.

Flashcard 13: What happens to $f(x) = \frac{x^2 - 1}{x - 1}$ near $x = 1$?

Answer: Approaches $x + 1$, but undefined at $x = 1$. $\frac{(x-1)(x+1)}{x-1}$ approaches $x+1 = 2$ as $x \to 1$.

Flashcard 14: Define the term 'non-removable discontinuity.'

Answer: A vertical asymptote in a rational function. Infinite discontinuity that cannot be removed by cancellation.

Flashcard 15: What are the steps to identify holes in a rational function?

Answer: Factor $p(x)$ and $q(x)$, find common factors. Cancel common factors to reveal removable discontinuities.

Flashcard 16: Simplify $f(x) = \frac{x^2 - 4x + 4}{x - 2}$.

Answer: $f(x) = x - 2$ for $x \neq 2$. $\frac{(x-2)^2}{x-2}$ simplifies to $(x-2)$ with hole.

Flashcard 17: What is the effect of a higher-degree $p(x)$ on end behavior?

Answer: Dominates $q(x)$, leading to polynomial-like behavior. Function grows without bound as $x$ increases.

Flashcard 18: State the behavior near a hole in a rational function.

Answer: Function approaches a limit but is not defined at the hole. Limiting value exists despite the discontinuity.

Flashcard 19: Identify the slant asymptote of $f(x) = \frac{x^2 + 1}{x - 1}$.

Answer: Slant asymptote is $y = x + 1$. Divide $x^2 + 1$ by $x - 1$ using long division.

Flashcard 20: Identify the condition for a hole in a rational function.

Answer: A hole occurs if $p(x)$ and $q(x)$ have a common factor. Common factors cancel, creating removable discontinuities.

Flashcard 21: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 1}{2x^2 + 5}$.

Answer: Horizontal asymptote at $y = \frac{3}{2}$. Equal degrees: ratio of leading coefficients is $\frac{3}{2}$.

Flashcard 22: What is the $y$-intercept of a rational function $f(x) = \frac{p(x)}{q(x)}$?

Answer: $f(0) = \frac{p(0)}{q(0)}$, if defined. Evaluate function at $x = 0$ if denominator nonzero.

Flashcard 23: Which values of $x$ cause holes in $f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$?

Answer: Hole at $x = 1$, as $(x-1)$ is a common factor. $\frac{(x-1)(x+1)}{(x-1)^2}$ has $(x-1)$ in common.

Flashcard 24: What is the result of long division of $p(x)$ by $q(x)$ in rational functions?

Answer: Quotient determines slant asymptote if degrees differ by 1. Provides slant asymptote when numerator degree exceeds by one.

Flashcard 25: What defines a vertical asymptote in a rational function?

Answer: Occurs at $x$ where $q(x) = 0$ and $p(x)$ does not cancel. Denominator zero with no cancellation creates vertical line.

Flashcard 26: Find the hole of $f(x) = \frac{x^2 - 4}{x - 2}$.

Answer: Hole at $x = 2$, as $x - 2$ is a common factor. Factor: $\frac{(x-2)(x+2)}{x-2}$, so $(x-2)$ cancels.

Flashcard 27: What does it mean for a rational function to be simplified?

Answer: No common factors between $p(x)$ and $q(x)$. All common factors have been canceled from the fraction.

Flashcard 28: When does a rational function have no horizontal asymptote?

Answer: When degree of $p(x)$ is greater than $q(x)$. Numerator degree exceeds denominator degree.

Flashcard 29: Describe the end behavior of $f(x) = \frac{x^3}{x^2}$.

Answer: As $x \to \pm\infty$, $f(x) \to \pm\infty$. $\frac{x^3}{x^2} = x$ grows linearly to infinity.

Flashcard 30: Find the removable discontinuity of $f(x) = \frac{x^2 - 9}{x - 3}$.

Answer: Removable discontinuity at $x = 3$. $\frac{(x-3)(x+3)}{x-3}$ cancels at $x = 3$.

Flashcard 31: What is the impact of a common factor on the graph of a rational function?

Answer: Creates a hole in the graph at the factor's root. Removable discontinuity where factors cancel out.

Flashcard 32: What is the end behavior of a rational function?

Answer: Describes $f(x)$ as $x \to \pm\infty$, often related to asymptotes. Behavior of function values as $x$ approaches infinity.

Flashcard 33: Identify the hole in $f(x) = \frac{x^2 - 25}{x^2 - 5x}$.

Answer: Hole at $x = 5$. $\frac{(x-5)(x+5)}{x(x-5)}$ has $(x-5)$ common factor.

Flashcard 34: Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 9}$.

Answer: Vertical asymptotes at $x = 3$ and $x = -3$. $x^2 - 9 = (x-3)(x+3) = 0$ when $x = \pm 3$.

Flashcard 35: 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. Standard notation where both numerator and denominator are polynomials.

Flashcard 36: Define the term 'non-removable discontinuity.'

Answer: A vertical asymptote in a rational function. Infinite discontinuity that cannot be removed by cancellation.

Flashcard 37: Describe the end behavior of $f(x) = \frac{x^3}{x^2}$.

Answer: As $x \to \pm\infty$, $f(x) \to \pm\infty$. $\frac{x^3}{x^2} = x$ grows linearly to infinity.

Flashcard 38: What are the steps to identify holes in a rational function?

Answer: Factor $p(x)$ and $q(x)$, find common factors. Cancel common factors to reveal removable discontinuities.

Flashcard 39: Which values of $x$ cause holes in $f(x) = \frac{x^2 - 1}{x^2 - 2x + 1}$?

Answer: Hole at $x = 1$, as $(x-1)$ is a common factor. $\frac{(x-1)(x+1)}{(x-1)^2}$ has $(x-1)$ in common.

Flashcard 40: Find the $x$-intercept of $f(x) = \frac{x^2 - 9}{x + 3}$.

Answer: $x$-intercept at $x = 3$. $\frac{(x-3)(x+3)}{x+3}$ has zero at $x = 3$.

Flashcard 41: What is the impact of a common factor on the graph of a rational function?

Answer: Creates a hole in the graph at the factor's root. Removable discontinuity where factors cancel out.

Flashcard 42: State the removable discontinuity in a rational function.

Answer: A point where the function is not defined due to a hole. Gap in graph that can be 'filled' by canceling factors.

Flashcard 43: Identify the non-removable discontinuity of $f(x) = \frac{1}{x^2 - 1}$.

Answer: Vertical asymptotes at $x = 1$ and $x = -1$. $x^2 - 1 = (x-1)(x+1) = 0$ when $x = \pm 1$.

Flashcard 44: What is the $y$-intercept of a rational function $f(x) = \frac{p(x)}{q(x)}$?

Answer: $f(0) = \frac{p(0)}{q(0)}$, if defined. Evaluate function at $x = 0$ if denominator nonzero.

Flashcard 45: Identify the slant asymptote of $f(x) = \frac{x^2 + 1}{x - 1}$.

Answer: Slant asymptote is $y = x + 1$. Divide $x^2 + 1$ by $x - 1$ using long division.

Flashcard 46: Identify the end behavior of $f(x) = \frac{x^3}{x^2 + 1}$.

Answer: As $x \to \pm\infty$, $f(x) \to \pm\infty$. Numerator degree exceeds denominator, so no horizontal limit.

Flashcard 47: When does a rational function have no horizontal asymptote?

Answer: When degree of $p(x)$ is greater than $q(x)$. Numerator degree exceeds denominator degree.

Flashcard 48: What is the effect of a higher-degree $p(x)$ on end behavior?

Answer: Dominates $q(x)$, leading to polynomial-like behavior. Function grows without bound as $x$ increases.

Flashcard 49: Find the common factor of $f(x) = \frac{x^2 - 4x}{x(x-4)}$.

Answer: Common factor is $x$. Factor $x$ appears in both numerator and denominator.

Flashcard 50: Determine the horizontal asymptote of $f(x) = \frac{3x^2 + 1}{2x^2 + 5}$.

Answer: Horizontal asymptote at $y = \frac{3}{2}$. Equal degrees: ratio of leading coefficients is $\frac{3}{2}$.

Flashcard 51: Simplify $f(x) = \frac{x^2 - 4x + 4}{x - 2}$.

Answer: $f(x) = x - 2$ for $x \neq 2$. $\frac{(x-2)^2}{x-2}$ simplifies to $(x-2)$ with hole.

Flashcard 52: Find the $x$-intercept of $f(x) = \frac{x^2 - 9}{x + 3}$.

Answer: $x$-intercept at $x = 3$. $\frac{(x-3)(x+3)}{x+3}$ has zero at $x = 3$.

Flashcard 53: Find the common factor of $f(x) = \frac{x^2 - 4x}{x(x-4)}$.

Answer: Common factor is $x$. Factor $x$ appears in both numerator and denominator.

Flashcard 54: Find the hole of $f(x) = \frac{x^2 - 4}{x - 2}$.

Answer: Hole at $x = 2$, as $x - 2$ is a common factor. Factor: $\frac{(x-2)(x+2)}{x-2}$, so $(x-2)$ cancels.

Flashcard 55: State the domain of a rational function.

Answer: All real numbers except where $q(x) = 0$. Excludes values that make the denominator zero.

Flashcard 56: Identify the vertical asymptote of $f(x) = \frac{2x}{x^2 - 9}$.

Answer: Vertical asymptotes at $x = 3$ and $x = -3$. $x^2 - 9 = (x-3)(x+3) = 0$ when $x = \pm 3$.

Flashcard 57: Identify the end behavior of $f(x) = \frac{x^3}{x^2 + 1}$.

Answer: As $x \to \pm\infty$, $f(x) \to \pm\infty$. Numerator degree exceeds denominator, so no horizontal limit.

Flashcard 58: Identify the condition for a hole in a rational function.

Answer: A hole occurs if $p(x)$ and $q(x)$ have a common factor. Common factors cancel, creating removable discontinuities.

Flashcard 59: State the domain of a rational function.

Answer: All real numbers except where $q(x) = 0$. Excludes values that make the denominator zero.

Flashcard 60: State the horizontal asymptote of $f(x) = \frac{2x}{x^2 + 1}$.

Answer: Horizontal asymptote at $y = 0$. Denominator degree exceeds numerator, so limit is zero.

Flashcard 61: Perform long division on $f(x) = \frac{x^2 + 3x + 2}{x + 1}$.

Answer: Quotient is $x + 2$, remainder is 0. $x^2 + 3x + 2 = (x+1)(x+2)$ divides evenly.

Flashcard 62: What is a horizontal asymptote for a rational function?

Answer: Occurs as $x \to \infty$, based on the degrees of $p(x)$ and $q(x)$. Determined by comparing degrees of numerator and denominator.

Flashcard 63: What is the significance of the leading coefficients in asymptotes?

Answer: They determine the horizontal asymptote when degrees are equal. When degrees are equal, their ratio gives horizontal asymptote.

Flashcard 64: Determine the horizontal asymptote of $f(x) = \frac{4x^3}{2x^3 + 1}$.

Answer: Horizontal asymptote at $y = 2$. Equal degrees: $\frac{4}{2} = 2$ gives horizontal asymptote.

Flashcard 65: What does it mean for a rational function to be simplified?

Answer: No common factors between $p(x)$ and $q(x)$. All common factors have been canceled from the fraction.

Flashcard 66: Define a slant (oblique) asymptote in a rational function.

Answer: Occurs when degree of $p(x)$ is one more than $q(x)$. Creates diagonal asymptote from polynomial long division.

Flashcard 67: 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. Standard notation where both numerator and denominator are polynomials.

Flashcard 68: Identify the non-removable discontinuity of $f(x) = \frac{1}{x^2 - 1}$.

Answer: Vertical asymptotes at $x = 1$ and $x = -1$. $x^2 - 1 = (x-1)(x+1) = 0$ when $x = \pm 1$.

Flashcard 69: Determine the horizontal asymptote of $f(x) = \frac{4x^3}{2x^3 + 1}$.

Answer: Horizontal asymptote at $y = 2$. Equal degrees: $\frac{4}{2} = 2$ gives horizontal asymptote.

Flashcard 70: State the removable discontinuity in a rational function.

Answer: A point where the function is not defined due to a hole. Gap in graph that can be 'filled' by canceling factors.