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 vertical?
e. f(x) and g(x) have the same y-intercept?
To do this problem I first had
to find out how to graph a linear equation. I’ll probably have to play
around with the variables to see how they graph out on the graphing
Solve/Investigate the Problem
I had to have our teacher show
me how to interpret the equations, and how to use a graphing
calculator. I learned that in
the equation y+mx + b the “m” stands for the slope of a line,
which is rise/run, or how high on x axis versus how far on the y axis. We also reviewed how to create a
table on paper of x and y values, and plot them out, then how to use the
TI-82 graphing calculator to do the same. Someone told us that “b”
is the y-intercept, or in language I understand, it’s where “x”
will be equal to zero.
of the Problem
In the problem, we use the
equations f(x) = ax + b and
g(x) = cx + d.
a. If f(x) and g(x) are
parallel, then I deduced that the slope has to be the same on both. Therefore a and c will need to be
equal. However b and d do not
have to be equal.
In the following example, the two red lines (k and l) are parallel
lines.Parallel lines are written using the symbol | |. In this case k |
b. If f(x) and g(x) are perpendicular,
the slope will change. By a lot
of trial and error on the TI-82, we were able to determine that a and c are
negative reciprocals and b & d remained the same.
Perpendicular Lines: Two lines are
perpendicular if they intersect at a right angle.
f(x) does not cross the x axis, then it would have a slope of
zero. Therefore, “a”
would be 0 and “b” could be anything but zero.
g(x) is vertical, then the
slope would be rise/0; therefore the slope would be undefined,
because no number can be divided by 0.
f(x) and g(x) have the same y-intercept, then “b” and “d”
would have to be the same number, and the slopes, “a”
and “c” could be anything. They could even be the same
Extensions of the Problem
One of the other students
talked about how we could use linear equations to solve story
problems. I think I’d
like to try that out.
Author & Contact
Link(s) to resources, references, lesson plans, and/or other
Important Note: You should compose your write-up
targeting an audience in mind rather than just the instructor for the
You are creating a page to publish it on the web.