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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 vertical to the x-axis?
e. f(x) and g(x) have the same y-intercept?


Problem setup

When comparing the lines of the following functions:  f(x) = ax + b and g(x) = cx +d, what are the relationships between a, b, c and d.

 

Plans to Solve/Investigate the Problem

1.      Write an equation.  Graph it.  Draw a line that is parallel to it, write the equation for the new line and compare the a, b, c and d values.

2.      Write an equation.  Graph it.  Draw a line that is perpendicular to it, write the equation for the new line and compare the a, b, c and d values.

3.      Draw a line that does not cross the x-axis, write the equation for the new line and discover possible values for a and b.

4.      Draw a line that is vertical to the x-axis, write the equation for the new line and compare the values of a and b.

5.      Draw 2 linesWrite an equation.  Graph it.  Draw a line that is parallel to it, write the equation for the new line and compare the a, b, c and d values.

 

Investigation/Exploration of the Problem

a.       If f(x) and g(x) are parallel, they must have the same slope.  Therefore a = c and b ≠ d.

b.      If f(x) and g(x) are perpendicular, then the slopes are opposite, reciprocals of each other.  Therefore        a = -1/c.

c.       If f(x) does not cross the x-axis that means that it is a horizontal line that has a slope with a value of 0.  Therefore a = 0, and b ≠ 0. 

d.      If g(x) is vertical then x = c.

e.       If f(x) and g(x) have the same y-intercept then b = d. 

 

Extensions of the Problem

Investigate the graph of problems such as f(x) = ax2 + bx + c.

Author & Contact
Insert name and contact information.
lparsons@rockdale.k12.ga.us

Link(s) to resources, references, lesson plans, and/or other materials
http://nlvm.usu.edu/en/nav/frames_asid_109_g_3_t_2.html?open=activities
http://www.shodor.org/interactivate/activities/slopeslider/index.html

 

 

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