Reciprocal graphs

Reciprocal Functions

• Definition: A function defined as (y = \frac{a}{x}) where (a) is a constant and (x) is a variable.

• Key characteristic: Recognizable by having a variable in the denominator.

Graphing Reciprocal Functions

• Positive Constant (a > 0):

  • Functions like (y = \frac{2}{x}) or (y = \frac{10}{x}) appear in the first and third quadrants.

• Negative Constant (a < 0):

  • Functions like (y = -\frac{2}{x}) or (y = -\frac{10}{x}) appear in the second and fourth quadrants.

• Graph Characteristics:

  • The curves do not touch the x-axis or the y-axis.

  • The graph approaches but never touches both axes.

Sketching the Basic Function (y = \frac{1}{x})

• Quadrants: For (a = 1), the graph is in the first and third quadrants.

• Table of Values Examples:

  • (x = -1): (y = -1)

  • (x = -0.5): (y = -2)

• Plotting Points:

  • Plot the points on the x-y axis and connect them with a smooth curve, not a straight line.

Undefined Values

• No values exist for (x = 0) or (y = 0):

  • When (x = 0), (y) is undefined.

  • When (y = 0), (x) is also undefined.

  • This results in the graph never touching the origin (0,0).

Key Features of the Graph

• Symmetry: The graph is symmetrical about the origin.

• Asymptotes:

  • Horizontal Asymptote: x-axis (y = 0)

  • Vertical Asymptote: y-axis (x = 0)

Graphing Variations

• Effect of Constant (a):

  • Increasing (a) (e.g., from 1 to 4 in (y = \frac{4}{x})) moves the graph further away from the axes.

Transforming Reciprocal Functions with Vertical Shifts

• When applying a shift, for example: (y = \frac{1}{x} + 5):

  • The graph of (y = \frac{1}{x}) moves up by 5 units.

  • New Horizontal Asymptote shifts from y = 0 to y = 5.

  • The vertical asymptote remains as the y-axis (x = 0).

• Sketch Example:

  • Original curve is shifted vertically up, retaining the general shape but adjusting to new asymptotes.