COMPARING LINES

By

Kiisha Gibbs

 

Problem:

 

Let f(x) = ax + b, and g(x) = cx + d, where a, b, c, and d, are any real numbers.

If f(x) and g(x) are graphed, what can you conclude about a, b, c and/or d, if:

 

a. f(x) and g(x) are parallel?

b. f(x) and g(x) are perpendicular?

c. f(x) does not cross the x-axis?

d. g(x) is horizontal?

e. f(x) and g(x) have the same y-intercept?

 

Solution:

 

In solving this problem, I used Graphing Calculator to explore the different lines.   In general, a and c change the steepness of the lines and c and d changed the location of the line, vertically speaking.  Let’s discuss each case.

 

I.             f(x) and g(x) are parallel.


 

 


When the lines are parallel, a and c have the same value.  The values of b and d did not affect whether or not the lines were parallel or not; hence, the coefficient of x determines the slope of the line, that is whether the lines are parallel or not.

 

II.         f(x) and g(x) are perpendicular.

 

 

Recall that the coefficient of x determines the slope of a line and hence whether or not the lines are parallel; that coefficient also determines whether or not f and g are perpendicular.  In the above graph, you can see that equations 1 (purple) and 2 (red) are perpendicular; likewise, equations 3 (blue) and 4 (green) are perpendicular.  The slopes are not the same, but rather are “negative reciprocals” of each other.  That is, (-3)*(1/3) = -1 and (2/3) *(-3/2) = -1.

 

 

III.     f(x) does not cross the x-axis.

 

If line f does not cross the x-axis, then line f is parallel to the x-axis, whose equation is y = 0.  Normally, when we talk of the equation of a line, we write it in slope-intercept form: y = mx + b.  M equals the slope of the line and b equals the y-intercept.  So, in this case, y = 0x + 0, where m = 0 and b = 0.  From my previous observation about parallel lines, all lines of the form “y = n”, where n is a real number, are parallel to the x-axis and do not cross the x-axis.  In terms of the values of a, and b, a equals zero and b is any real number.

 

 

 

 

 

IV.        g(x) is horizontal.

If line g is horizontal then it is oriented like the x-axis; that is, parallel to the x-axis.  This case boils down to case III. 

 

 

 

 

V.            f(x) and g(x) have the same y-intercept.

 

This time I investigated the different graphs in GSP.  In GSP, you could set the y-intercept and create different lines by simply chosing another point.  After chosing 2 points, create the line and get the equation of the line from the measure feature.  In doing this, we see that in each equation that “3.95” occurs.  That is, b and d are the same in lines f and g.  This just further shows the “slope-intercept” form of the equation, where the constant is the y-intercept. 

 

In conclusion, when your line is in slope-intercept form, you know an extreme amount of information about the graph of that line just at first glance:

  1. how steep the line is and
  2. where it crosses the y-axis.

This form is simply based on the definition of slope,m.

Links and further exploration:

Click here for a link to Graphing Calculator.

Click here for a link to GSP.

 


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