Linear Functions: Standard Form
Use this page to explore how the constants a, b, and c in the Standard Form of the equation of a line
affect its graph. A series of challenges below the graph help you better understand the role that each parameter plays on the graph.
Start by playing around with various values for a, b, and c by using the green sliders towards the bottom of the graph, and see if you can develop a sense of what role each slider plays in the graph's appearance.
As you may have realized by now, Standard Form is not very useful when seeking to graph a linear function. It is most useful when solving a system of linear equations. However, that does not prevent you from being able to analyze Standard Form almost as readily as you can Slope-Intercept or Point-Slope Form.
Using the sliders, see if you can:
- Make the line go through the origin?
- Make the line go through the point (-3,-1)?
- Make the line have a negative slope?
- Make the line horizontal (parallel to the x-axis)?
- Make the line vertical (parallel to the y-axis)?
Looking at the equation
it can be challenging to figure out what effect each constant is likely to have because this form differs so much from the other forms you are probably more used to working with by now. Therefore it is useful to solve this equation for y:
Now that the equation is in Slope-Intercept form, notice that both the slope and the y-intercept are ratios of two constants from Standard Form.
Using the last equation above,
- what happens to the slope as the value of b is increased?
- what happens to the slope as the value of b is decreased?
- what happens to the y-intercept as b is increased or decreased? Verify your answers by changing b in the applet above.
- what happens to the slope as the value of a is changed? Verify your answer by changing a in the applet above.
- what happens to the y-intercept as the value of c is changed? Verify your answer by changing c in the applet above.
To summarize Standard form:
(0, c/b) must be the coordinates of the y-intercept, because the line , which passes through the origin, has been translated vertically by c/b, causing the point that used to be at the origin to shift vertically to (c/b).
(-a/b) is the slope of the line because every change in x is scaled by -a/b, causing y to change by -a/b when x increases by one.
Standard form is less useful than Slope-Intercept form or Point-Slope form if you wish to graph a line quickly, since you will need to do either some calculations or some algebra to determine both the slope and a point that the line passes through.
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/