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Title
Comparing Lines 

Problem Statement
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[change to vertical]?
e. f(x) and g(x) have the same y-intercept?

 

Problem setup

Using the TI-82 calculator, I set out to graph coordinates and plot graphs in order to determine certain truths about the coordinate plane.    

Plans to Solve/Investigate the Problem

First, I made sure I knew what the key terms meant.  Parallel, perpendicular, the x-axis, y-intercept.  Parallel, perpendicular, and x-axis were easy, but recalling y-intercept took some doing.  I figured out after some graphing that the y-intercept is where the graph crosses the y-axis.  I planned to plot some graphs and see if I could plot others that are parallel to that one, as well as perpendicular.  I also plotted a graph that didn’t cross the x-axis, another one that was vertical and two that had the same y-intercept.  I knew that if I could plot the graphs and if I knew what I was trying to create (parallel lines, perpendicular lines, etc.) that I could find some properties. 

Investigation/Exploration of the Problem

For parallel lines, I came up with y = 2x – 1.  I plugged in a value of zero for x and came up with a line sloping from quadrant one to quadrant three. I graphed y = 2x + 4 and came up with a line parallel to the first function.  I began to see that I was going to find my answers by plugging in values for parts of the function and producing graphs.  I learned that in parallel lines, a (see problem statement) and c are the same and as long as b and d are NOT the same, it doesn’t matter what they are.   For perpendicular lines, I tried opposite values for a and c (+2 and -2) and that didn’t work.  I tried reciprocal values (1/2 and 2) and that didn’t work but soon it became apparent that the opposite reciprocal was the answer.  If a is 4 then c is -1/4.  I thought of the different quadrants and how if a perpendicular line crossed another that the signs would be opposite and that helped.  For a line that doesn’t cross the x axis, I knew that it had to be a horizontal line.  In order for the line to be horizontal, the y value had to be zero, so the function of x is zero.  To be vertical, y is undefined because the x value would be zero and when presenting rise over run, as a division problem, you cannot have a divisor of zero.  Using the function I could see that if the y intercept has the same number they would cross in the same place in the y axis. 

 

Extensions of the Problem

Parabolas are the logical next step in linear functions. 

Author & Contact
Kevin Smith
Email me!

Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2

 

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