Modifying the slope and y-intercept of a linear function.

Kim Seay

EMAT6680

A linear function (a function that graphs as a line) can be written in the form

f(x) = mx+b where m and b are real numbers.

Modifying the slope

Since m and b can be any real numbers, let's suppose that m = 3 and b = 0.

The graph of the equation f(x) = 3x + 0 would look like this:

If x is fixed, what happens to the value of f(x) as m changes?

If we increase m by one, the equation becomes f(x) = 4x + 0 and the graph becomes

The graph appears to have a steeper slope.

Let's try using a smaller m value such as 1.

In order to compare the changes we are seeing in m, let's graph all three functions on the same graph.

It seems as if the graph is getting steeper when m increases and flatter as m decreases. In order to investigate this further (and try out some negative m values), let's see what happens as m changes from -10 to 10.

In the function f(x) = mx +b , the change in m value (while x is constant) affects the slope of the graph. While m is positive, the graph of the function will become steeper as m increases in value. As m decreases, the graph becomes flatter as it approaches zero and forms a horizontal line (through b) when m=0. When m takes on negative values, the graph starts to fall from the opposite direction. Hence, the graph is now going through quadrants 2 and 4 as opposed to 1 and 3. The greater the negative values for m, the steeper the decline of the graph.

Modifying the y-intercept

Now let's look at changes in b. Going back to our original equation f(x)=3x+0 whose graph looks like this:

If we increase b by one to form the equation f(x)=3x+1, the y-intercept of the line changes from the point (0,0) to (0,1).

Let's try decreasing the b value to -2.

As you might have guessed, the y-intercept takes on a value of -2; therefore, the graph of the function crosses the y axis at -2. We can try this with a slider value for b ranging from -10 to 10.

You can see that as n (or b) changes values, the y-intercept changes accordingly.

We can try a greater range for b to see if the same thing happens.

The same thing is happening. The b value corresponds with the y-intercept of the function..

Conclusion:

I think Graphing Calculator is a wonderful tool for students to use to explore changes in the slope and y-intercept of a linear equation. It is much more effective for students to discover what happens to the values of "m" and "b" as they vary on their own, than to read about it in a textbook. I think some students go through an entire course of Algebra without having a clear concept of what slope and y-intercept are. I think this is an excellent way to help with that.

Return