Graphs of Rational Functions: Answers & Solutions

These are the solutions to the Graphs of Rational Functions Questions. Each solution includes the function's key characteristics that determine its graph.

1. Question 1:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{1}{x} \]

   Answer:

   b

   Key Features: Hyperbola with vertical asymptote at \( x = 0 \) and horizontal asymptote at \( y = 0 \).

2. Question 2:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{1}{x - 1} \]

   Answer:

   a

   Key Features: Translation of \( \frac{1}{x} \) one unit right (vertical asymptote at \( x = 1 \)).

3. Question 3:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{x + 1}{x - 1} \]

   Answer:

   c

   Key Features: Vertical asymptote at \( x = 1 \), horizontal asymptote at \( y = 1 \), x-intercept at \( (-1, 0) \).

4. Question 4:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{1}{x^2 + x - 2} \]

   Answer:

   a

   Key Features: Denominator factors as \( (x+2)(x-1) \), giving vertical asymptotes at \( x = -2 \) and \( x = 1 \).

5. Question 5:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{x + 2}{x^2 - 1} \]

   Answer:

   d

   Key Features: Vertical asymptotes at \( x = \pm 1 \), x-intercept at \( (-2, 0) \), horizontal asymptote at \( y = 0 \).

6. Question 6:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{x + 2}{x^2 - x - 2} \]

   Answer:

   a

   Key Features: Note that \( x^2 - x - 2 = (x-2)(x+1) \), but \( x+2 \) in numerator doesn't cancel. Vertical asymptotes at \( x = 2 \) and \( x = -1 \).

7. Question 7:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{x - 1}{x^2 + x + 1} \]

   Answer:

   c

   Key Features: Denominator has no real zeros (discriminant \( 1^2 - 4(1)(1) = -3 < 0 \)), so no vertical asymptotes. Horizontal asymptote at \( y = 0 \).

8. Question 8:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{x^2 + x - 2}{x^2 - x - 2} \]

   Answer:

   a

   Key Features: Factor both: numerator \( = (x+2)(x-1) \), denominator \( = (x-2)(x+1) \). No cancellation. Vertical asymptotes at \( x = 2 \) and \( x = -1 \), horizontal asymptote at \( y = 1 \).

9. Question 9:

   Identify the graph of the rational function \( f \). \[ f(x) = \frac{1}{x^2 + x + 1} \]

   Answer:

   d

   Key Features: No vertical asymptotes (denominator always positive), horizontal asymptote at \( y = 0 \), graph always positive.

10. Question 10:

    Identify the graph of the rational function \( f \). \[ f(x) = \frac{x^2 - x}{x^2 - x - 2} \]

    Answer:

    b

    Key Features: Factor: \( \frac{x(x-1)}{(x-2)(x+1)} \). Vertical asymptotes at \( x = 2 \) and \( x = -1 \), horizontal asymptote at \( y = 1 \), x-intercepts at \( x = 0 \) and \( x = 1 \).

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