Given the function p(x)=3x,a) Write the equation for the function q(x) obtained by dilating p(x) vertically by a factor of ଵଶ.b) Determine the y intercept domain and range q(x).c) Now, let r(x)=2⋅3x. Describe the transformation applied to p(x) to obtain r(x).d) Sketch the graphs of p(x), q(x), and r(x) on the same set of axes.
Question
Given the function p(x)=3x,a) Write the equation for the function q(x) obtained by dilating p(x) vertically by a factor of ଵଶ.b) Determine the y intercept domain and range q(x).c) Now, let r(x)=2⋅3x. Describe the transformation applied to p(x) to obtain r(x).d) Sketch the graphs of p(x), q(x), and r(x) on the same set of axes.
Solution
a) The function q(x) obtained by dilating p(x) vertically by a factor of 5 is q(x) = 5 * p(x) = 5 * 3x = 15x.
b) The y-intercept of q(x) is the value of q at x = 0, which is q(0) = 15 * 0 = 0. The domain of q(x) is all real numbers, because you can substitute any real number for x and get a real number for q(x). The range of q(x) is also all real numbers, because for any real number y, you can find a real number x such that q(x) = y (just take x = y/15).
c) The function r(x) = 2 * 3x is obtained from p(x) = 3x by multiplying the output by 2. This is a vertical dilation by a factor of 2.
d) To sketch the graphs of p(x), q(x), and r(x), you would draw three lines through the origin. The line for p(x) would have a slope of 3, the line for q(x) would have a slope of 15, and the line for r(x) would have a slope of 6. The line for q(x) would be steeper than the line for p(x), and the line for r(x) would be somewhere in between.
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