Key Features of Graphs of Lines
Let us look again at the graphs we made for Desmond and Charles.
Remember: The equation for Desmond's graph is y = 2x and the equation for Charles' graph is y = 20,000 - 1,000x.
Finding the Y - intercept
Take a minute to look at the two graphs. With a partner, find the point where each graph intersects the y axis. The y-intercept is the y value of the point where the graph crosses the y axis. Discuss the following: Do either of the y-intercepts relate to the equations for the graphs? One does, while the other appears not to. What might be the explanation of why Desmond's equation does not contain the y-intercept?
Activity: Class Discussion
What is the relationship between the equation and the y-intercept?
Finding the Slope
Look again at the two graphs. With a partner, find the slopes of the two graphs. Slope is change in y value divided by change in x value. Now look at the equations that represent each of the graphs. Where does the slope show up in these equations?
Activity: Class Discussion
What is the relationship between the slope and the equation of a line?
The General Equation for a Line
We are used to dealing with the variables x and y. Variables can "vary" or change. We can use letters to represent a different kind of thing - a constant. Constants only have one value in an equation - for example in the equation y = 3x + 2, 3 stays the same no matter what happens to x and y. The number 3 can be represented by the letter n, as in y = nx + 2 and different equations might have a different number for n instead of 3. If n = 4, then we have the equation y = 4x + 2. Let us examine using constants so we can come up with an equation that represents different line equations.
Activity: Group Discussion followed by Class Discussion
Using "m" as a constant for slope and "b" as a constant for y-intercept, write the general form of the equation for any line. Where is the slope located in a line equation and where is the y-intercept located?
Finding the Roots
Activity: Class Discussion
Locate the points on the graphs that have a y value of zero. What are the ordered pairs of these points? Is there a way to find the x values of these points algebraicly?
Reflection
What are some features of linear graphs? What have you learned?