(2426) Graphing Rational Functions With Vertical, Horizontal & Slant Asymptotes, Holes, Domain & Range

Introduction to Graphing Rational Functions

• Overview of what will be covered:

  • Graphing asymptotes: horizontal, vertical, slant/oblique

  • Finding holes in functions

  • Writing the domain and range of functions

Parent Function: ( y = \frac{1}{x} )

• Vertical Asymptote:

  • At ( x = 0 ) (y-axis)

• Horizontal Asymptote:

  • At ( y = 0 ) (x-axis)

• Graph Description:

  • Positioned in the upper right and lower left quadrants

• Domain:

  • All x-values except where the function is undefined

  • Domain: ( (-\infty, 0) \cup (0, \infty) )

• Range:

  • All y-values except where the function is undefined

  • Range: ( (-\infty, 0) \cup (0, \infty) )

Function: ( y = \frac{1}{x^2} )

• Asymptotes:

  • Vertical Asymptote: ( x = 0 )

  • Horizontal Asymptote: ( y = 0 )

• Graph Description:

  • Looks similar to ( y = \frac{1}{x} ) but lies entirely above the x-axis due to squaring the numerator

  • Shows symmetry about the y-axis

• Domain:

  • Same as ( y = \frac{1}{x} )

  • Domain: ( (-\infty, 0) \cup (0, \infty) )

• Range:

  • All y-values from ( 0 ) to ( \infty )

  • Range: ( (0, \infty) )

Function: ( f(x) = \frac{1}{x - 1} )

• Finding Asymptotes:

  • Vertical Asymptote:

    • Set denominator to zero: ( x - 1 = 0 ) gives ( x = 1 )

  • Horizontal Asymptote:

    • Function is bottom-heavy, so horizontal asymptote: ( y = 0 )

• Graphing Steps:

  • Identify vertical asymptote at ( x = 1 )

  • Identify horizontal asymptote at ( y = 0 )

• Points for Graphing:

  • For accurate graphing:

    • When ( x = 2 ), ( y = 1 ) ( (2, 1) )

    • When ( x = 3 ), ( y = \frac{1}{2} ) ( (3, \frac{1}{2}) )

    • When ( x = 1.5 ), ( y = 2 ) ( (1.5, 2) )

    • For the left of the asymptote, when ( x = 0 ), ( y = -1 ) ( (0, -1) )

    • When ( x = -1 ), ( y = -\frac{1}{2} ) ( (-1, -\frac{1}{2}) )

    • When ( x = -0.5 ), ( y = -\frac{2}{3} ) ( (-0.5, -\frac{2}{3}) )

• Domain:

  • Domain: ( (-\infty, 1) \cup (1, \infty) )

• Range:

  • Range: ( (-\infty, 0) \cup (0, \infty) )

Function: ( y = \frac{1}{x + 2} - 3 )

• Finding Asymptotes:

  • Vertical Asymptote:

    • Set denominator to zero: ( x + 2 = 0 ) implies ( x = -2 )

  • Horizontal Shift:

    • Shifted left by 2 units.

  • Vertical Shift:

    • The function is shifted down 3 units.

Introduction to Graphing Rational Functions

Overview of what will be covered:

• Graphing asymptotes: horizontal, vertical, slant/oblique

• Finding holes in functions and understanding their graphical implications

• Writing the domain and range of functions properly, including intervals

Parent Function: ( y = \frac{1}{x} )

Vertical Asymptote:

• Located at ( x = 0 ) which corresponds to the y-axis. This means the function approaches infinity as x approaches 0 from the right and negative infinity as x approaches 0 from the left.

Horizontal Asymptote:

• Found at ( y = 0 ) which is the x-axis. This indicates that as x approaches positive or negative infinity, the value of the function approaches 0.

Graph Description:

• The graph is positioned in the upper right (Quadrant I) and lower left (Quadrant III) quadrants of the Cartesian plane, showcasing its characteristics as a hyperbola.

Domain:

• The function is defined for all x-values except at the point where it is undefined (x = 0), expressed as:Domain: ( (-\infty, 0) \cup (0, \infty) )

Range:

• Similarly, y-values are accepted except where the function becomes undefined, noted as:Range: ( (-\infty, 0) \cup (0, \infty) )

Function: ( y = \frac{1}{x^2} )

• Asymptotes:

  • Vertical Asymptote: ( x = 0 )

  • Horizontal Asymptote: ( y = 0 )

Graph Description:

• Resembles the graph of ( y = \frac{1}{x} ) but lies entirely above the x-axis, due to squaring the denominator.

• The graph exhibits symmetry about the y-axis, indicating that f(x) = f(-x).

Domain:

• Remains the same as for ( y = \frac{1}{x} ): Domain: ( (-\infty, 0) \cup (0, \infty) )

Range:

• The output y-values range from ( 0 ) to ( \infty ), denoted as:Range: ( (0, \infty) )

Function: ( f(x) = \frac{1}{x - 1} )

Finding Asymptotes:

• Vertical Asymptote:

  • Set the denominator to zero:

    • ( x - 1 = 0 ) gives the vertical asymptote at ( x = 1 )

• Horizontal Asymptote:

  • Since the degree of the numerator is less than that of the denominator (bottom-heavy), the horizontal asymptote is:

    • ( y = 0 )

Graphing Steps:

1. Identify the vertical asymptote at ( x = 1 )

2. Identify the horizontal asymptote at ( y = 0 )

3. Plot points to provide a clearer picture of the function's behavior

Points for Graphing:

• To accurately depict the function, consider the following points:

  • When ( x = 2 ), ( y = 1 ) → ( (2, 1) )

  • When ( x = 3 ), ( y = \frac{1}{2} ) → ( (3, \frac{1}{2}) )

  • When ( x = 1.5 ), ( y = 2 ) → ( (1.5, 2) )

  • For points on the left of the vertical asymptote:

    • When ( x = 0 ), ( y = -1 ) → ( (0, -1) )

    • When ( x = -1 ), ( y = -\frac{1}{2} ) → ( (-1, -\frac{1}{2}) )

    • When ( x = -0.5 ), ( y = -\frac{2}{3} ) → ( (-0.5, -\frac{2}{3}) )

Domain:

• Domain:( (-\infty, 1) \cup (1, \infty) )

Range:

• Range:( (-\infty, 0) \cup (0, \infty) )

Function: ( y = \frac{1}{x + 2} - 3 )

Finding Asymptotes:

• Vertical Asymptote:

  • Set the denominator to zero:

    • ( x + 2 = 0 ) implies ( x = -2 ) is a vertical asymptote.

Transformations:

• Horizontal Shift:

  • The function experiences a shift left by 2 units, moving the entire graph to the left on the x-axis.

• Vertical Shift:

  • The function undergoes a downward shift by 3 units, moving the value of the entire function downwards on the y-axis.