Download Comparing Lines

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?
e. f(x) and g(x) have the same y-intercept?
Problem setup
I will use the graphing tool of Geometer’s Sketchpad to explore the graphs of functions
when a, b, c and d are changed in the functions f(x) = ax + b, and g(x) = cx + d.
Plans to Solve/Investigate the Problem
Using the Geometer’s Sketchpad, I will plot the graphs of various functions to demonstrate
what happens to a, b, c and d when changes are made in their values. I will use positives
and negatives of the same number and I will use fractions and zero. I will keep the values
of a and c the same and see the effect of using different values for b and d. I will keep b
and d the same and see the effect of using different values for a and c.
Investigation/Exploration of the Problem
I plotted various graphs and color coded the graph with its equation:
8
hx = 1x+0
qx = 1x+1
6
vx = 0x+6
ux = -1x+0
4
sx =
tx =
-10
 
-3
2

2
3
x+0
2
x-8
-5
5
-2
kx = 0x-4
-4
wx = 3x+3
-6
mx = -5x+3
10
I noticed that:
a. f(x) and g(x) are parallel when a=c. These are the black (h(x)) and red (q(x)) graphs.
Notice the a = c = 1. Notice that these numbers are the coefficients of x in both
equations, and they are the same number for parallel lines.
b. f(x) and g(x) are perpendicular when c = - 1/a (or, as we learned to say in my high
school days, when one slope is “negative the reciprocal” of the other). These are the green
(s(x)) and the blue (t(x)) graphs, where a = 2/3 and c = -3/2. Notice that – 3/2 is
“negative the reciprocal” of 2/3. Notice that these numbers are the coefficients of x
in both equations, and they are “negative the reciprocal” of each other” for
perpendicular lines.
c. f(x) does not cross the x axis when a=0 and when b ≠ 0. An example of this is the dark
green line (k(x)). Notice that this line is horizontal. If b = 0, the horizontal line would
be the x-axis itself.
d. g(x) is horizontal when c=0. An example or this is the brown line (v(x)). Notice that
the horizontal line occurs when the coefficient of x is 0.
e. f(x) and g(x) have the same y intercept when b = d. The orange line (w(x)) and the
turquoise line (m(x)) are examples of this, because both have the same constant term,
+3.
.
Author & Contact
Clare S. Deaver
[email protected]