Intermath | Workshop Support


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

The problem is asking us to find the relationship between two different equations each with different variables.   


Plans to Solve/Investigate the Problem

My initial plan to solve the problem was to substitute values for the variables and manipulate the numbers in order to draw inferences.   


Investigation/Exploration of the Problem

a.       Variables a and c have to be the same numbers in order for the two equations to be parallel, but we know that b is not equal to d.

b.      The two equations have to be opposite reciprocals (i.e. the numbers 2 and -1/2) in order to be perpendicular.

c.       In order for f(x) not to cross the x-axis, a has to be 0, and b can be any real number.  There must be no slope.

d.   If g(x) is horizontal, then c has to be 0 and d can be any real number. 

e.   In order for f(x) and g(x) to have the same y-intercept, b and d have to be the same, causing the lines to intersect.  The slopes can vary.   



Extensions of the Problem


Author & Contact
Mesha Rainey

Link(s) to resources, references, lesson plans, and/or other materials


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