What is a family? Generally, a family is a group of people that are related either by birth, marriage, or adoption. Graphs can also form families. A family of graphs includes graphs and equations of graphs that have at least one characteristic in common. That characteristic differentiaties the group of graphs from other groups.

Families of linear graphs often fall into two categories - those with the same slope of those with the same intercept. The parent graph is the simplest of the graphs in the family. For many linear functions, the parent graph is of the form y = mx, where m is any number.

A graphing calculator is a useful tool in studying a group of graphs to determine if they form a family.

Example 1: Graph y = x, y = 2x, and y = 4x in the standard viewing window. Describe any similarities and differences among the graphs. Write a descroption of the family.

Clear the Y= list of all other equations.

ENTER: Y= 2 4 6

These three graphs form a family in which the slope of each graph is positive and each graph goes through the origin. However, each graph has a different slope. The parent graph is the graph of y = x. The graph of y = 4x is the steepest, and the graph of y = x is the least steep.

You might describe this family of graphs as lines that pass through the origin. Another description might be lines that have 0 as their y-intercept.

 

 

You can use braces to enter equations that have a common characteristic. Since all of the equations in Example 1 are of the form y = mx, you can use braces to enter the different values of m using one step as follows.

ENTER: Y= { 1 , 2 , 4 }

This tells the calculator to graph equations for which the coefficients of x are 1, 2, and 4 and the y-intercept is 0.

Example 2: Graph y = x, y = x +3, and y = x - 3 in the standard viewing window. Describe any similarities and the differences among the graphs. Write a description of the family.

A function that is closely related to linear functions is the absolute value function.

Example 3: Graph y = |x|, y = |x + 2|, and y = |x| + 4 on the same screen. Describe any similarities and differences among the graphs.

ENTER: Y= ( + 2 ) + 4 6

Each graph is shaped like the letter v. The graph of y = |x + 2| is shaped like the graph of y = |x|, but is shifted 2 units to the left. The graph of y = |x| +4 is shaped like the graph of y = |x|, but is shifted 4 units up.

For more practice do the Exercises 1 - 10

Go back to Chapter 6