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a) Describe how you know that a graph is proportional.

b) Describe how you know an equation is proportional. Give an example of an equation that is proportional and a graph that is not proportional.

c) Describe how you know that a table is proportional.

1 answer

a) How to Determine if a Graph is Proportional

A graph is proportional if it represents a linear relationship between two variables that passes through the origin (0, 0). Specifically, you can determine if a graph is proportional by looking for the following characteristics:

  1. Straight Line: The graph must be a straight line.
  2. Passes Through the Origin: The line must intersect the origin (0, 0). This indicates that when one variable is zero, the other variable is also zero.
  3. Constant Rate of Change: The slope of the line, which represents the ratio of change between the two variables, should remain constant throughout the graph.

b) How to Determine if an Equation is Proportional

An equation is considered proportional if it can be represented in the form \( y = kx \), where \( k \) is a constant (the rate of change or the constant of proportionality). Key characteristics of a proportional equation include:

  1. Directly Related Variables: The variables are directly related with no added constants or intercepts.
  2. No Constant Term: The equation does not include a constant term (i.e., no "+ b" term).
  3. Linear Relationship Through the Origin: The equation's graph would be a straight line that passes through the origin.

Example of a Proportional Equation:

  • \( y = 3x \) (This represents a proportional relationship, where \( k = 3 \)).

Example of a Non-Proportional Equation:

  • \( y = 2x + 1 \) (This is not proportional because it has a constant term, and its graph will not pass through the origin).

c) How to Determine if a Table is Proportional

A table is proportional if the ratio of the y-values to the corresponding x-values is constant for all pairs of values. To identify a proportional table, you can check the following:

  1. Calculate Ratios: For each pair of values (x, y), calculate the ratio \( \frac{y}{x} \) (assuming \( x \) is not zero).
  2. Constant Ratios: All calculated ratios should be the same (a constant).
  3. Origin Inclusion: If one of the entries in the table corresponds to \( x = 0 \), then the corresponding \( y \) value should also be zero.

Example of a Proportional Table: | x | y | |----|-----| | 1 | 3 | | 2 | 6 | | 3 | 9 |

  • Ratios: \( \frac{3}{1} = 3 \), \( \frac{6}{2} = 3 \), \( \frac{9}{3} = 3 \) (constant ratio of 3).

Example of a Non-Proportional Table: | x | y | |----|-----| | 1 | 2 | | 2 | 5 | | 3 | 8 |

  • Ratios: \( \frac{2}{1} = 2 \), \( \frac{5}{2} = 2.5 \), \( \frac{8}{3} \approx 2.67 \) (not constant).