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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 hx = 1x+0 qx = 1x+1 6 vx = 0x+6 ux = -1x+0 4 sx = tx = -10 -3 2 2 3 x+0 2 x-8 -5 5 -2 kx = 0x-4 -4 wx = 3x+3 -6 mx = -5x+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