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?
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:
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.