Linear Functions: Slope-Intercept Form
Use this page to explore how the constants m and b affect the appearance of a line whose equation is written in Slope-Intercept form:
A series of challenges below the graph provide an opportunity to test your understanding of the uses for each parameter.
There are two ways you can use the sliders above. You may either drag the point/button on the slider with your mouse, our click on the point then use the arrow keys on your keyboard to move the point in small increments.
Change the values of m and b using sliders and see what happens. Can you:
- Make the blue line go through the origin?
- Make the line go through the blue point (-3,-1)?
- Make the line move down as it goes to the right?
- Make the line horizontal (parallel to the x-axis)?
- Make the line trace each of the railroad tracks in the center of the background?
- Make the line vertical (parallel to the y-axis)?
Which of the above could NOT be achieved? Why not?
Looking at the equation
y = mx + b
Why does b have the effect it has?
When x is 0, what does this equation simplify to?
y = m(0) + b
y = 0 + b
y = b
This is a horizontal line passing through the point (0,b), the y-intercept. So, when x=0, only the b term remains. But as x takes on values other than 0, the line rises or falls from b.
What happens to the line as m is changed? Why does m have this effect?
Is there a difference in the way a small change in m affects the line when it is large (say 10) compared to when it is closer to zero (say 1)? Why?
Watch what happens to the distance between the points where x = 0 and x = 1 as m is changed. What is it about the nature of the way m is connected to the equation that causes it to have this effect?
To summarize Slope-Intercept form:
- (0,b) are the coordinates of a point the line passes through:
The line y = mx, which passes through the origin, has been translated (shifted) vertically by b, causing the point that used to be at the origin to move to b.
- m is the slope of the line:
Every change in x is "scaled" (multiplied) by the value of m, causing y to change by m when x increases by one.
Note that a "scaling factor" stretches or shrinks things, as you can see by observing the distance
between the points at x = 0 and x = 1 as m is changed.
If you wish to use other applets similar to this, you may find an index of all my applets here: https://mathmaine.com/2010/04/27/geogebra/