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