Slope-intercept form is taught in nearly every classroom in the United States, yet there are some countries that never teach it at all.

(One exchange student at our school was only familiar with method #9 below.)

1. Slope-intercept form 

The slope is the change in y divided by the change in x, often known by students as “rise of run.”

There is a direct geometric interpretation, so it is easy to transfer from graph to equation and back again. Extensive use of this form also allows an intimate understanding of slope, which translates directly into the difference quotient from calculus. However, the equation is restricted to 2-space (the plane), and doesn’t include every possible line (vertical lines) like:

2. Intercept form

This form has the advantage that it is directly related to the usual form of conic sections (parabolas, circles, ellipses, and hyperbolas). For instance, substituting x² and y² for x and y gives the formula for an ellipse.

3. Point-slope form

alternately written as

This is useful if a problem gives you a slope and a point on the line but not the intercept. The “alternate” form matches the function notation, and also can be typed directly into a graphing calculator.

4. Two point form

Given two points on the line, (x1,y1) and (x2,y2):

This is just point-slope form with the formula for slope substituted in. It has the advantage of expressing in only one equation for a process that normally takes two (find the slope, then plug into point-slope form).

5. Function form 

 
This is a generalized form that works for any function, linear or not. The “scale factor” in the case of lines is equal to the slope. While working for any kind of function is a definite advantage, this form is also difficult for students to understand (and equivalent to point-slope form in the case of lines anyway).

6. Standard form and general form

This is standard form:

Where the slope is -a/b, the intercept is c/a, and the y-intercept is c/b.

General form is the closely related:

Where the slope is –a/b, the intercept is –c/a, and the y-intercept is –c/b.

This form is a linear combination, linking it directly with linear algebra. It’s also the usual form for when solving lines as a system of equations. While the direct geometric intuition is removed, but it is at least still possible to derive the slope and intercepts from the formulas given above. It also isn’t hard to extend the formula into 3-space:

7. Modified standard form 

(I have not been able to discover an “official” name for this form.)

Here the slope is –a/b.

8. Parametric

Where v is the velocity of an object, q its angle, and x0 and y0 the point the line is at when t = 0 (that is, the starting time).

While this form can be used to describe lines in general (by letting t be any real number) it’s often used to describe the motion of an object moving in a straight line, since t can be used to note its location at any given time.

9. Vector

A line can be composed of a position vector and a direction vector.

x0 and y0 are the starting point of the line. a indicates the x portion of the direction vector (equivalent to the “run”) and b the y portion (equivalent to the “rise”).

t is any real number, so the formula generates a line infinite in both directions.

Commentary: This is equivalent to the parametric form, that is the two formulas:

An additional advantage of this form is the easy generalization into other dimensions, for example 3 space:

This form can also be turned back into an equation by solving for t, then substituting:

10. Matrix determinant form 

Commentary: Working out the determinant of this matrix will eventually lead to two point form.

(And if that looks confusing, you may be interested in my entry on visual clarity in variable names.)

Conclusion

It’s easy after enough teaching the same curriculum for a long time to get locked into a particular orthodoxy, and forget that there are many approaches to any topic in mathematics. Just like learning new languages, learning new forms can increase our flexibility of thought and put into place concepts that were previously obscured.

Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.

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