Download Comparing Lines

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Elementary mathematics wikipedia, lookup

Addition wikipedia, lookup

Mathematics of radio engineering wikipedia, lookup

Line (geometry) wikipedia, lookup

System of polynomial equations wikipedia, lookup

Signal-flow graph wikipedia, lookup

Halting problem wikipedia, lookup

Transcript
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
bdeaver3@aol.com