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The graph of the inverse trigonometric function arctan and its properties are explored using an applet. You may want to work through an interactive tutorial on Inverse Trigonometric Functions before you work through the present tutorial.
The exploration is carried out by analyzing the effects of the parameters a, b, c and d included in the definition of arctan as follows
f(x) = a*Arctan(b*x + c) + d
Interactive Tutorial
click on the button above "click here to start" and MAXIMIZE the window obtained. The graph of arctan is in blue and the red horizontal lines are the asymptotes of arctan.
- Set the parameters to a = 1, b = 1, c = 0 and d = 0 to obtain
f(x) = arctan(x)
Check that the domain of arctan(x) is the set of all real numbers and the range is given by the interval (-pi/2 , +pi/2). Check also that arctan(x) has horizontal asymptotes at y = -pi/2 and y = +pi/2.
- Change parameter a and note how the graph of arctan changes (vertical compression, stretching, reflection). How does it affect the range? asymptotes?
Does a change in parameter a affect the domain of arctan?
- Change parameter b and note how the graph of arctan changes (horizontal compression, stretching). Does a change in b affect the domain of arctan? range? asymptotes?
- Change parameter c and note how the graph of arctan changes (horizontal shift). Does a change in c affect the domain of arctan? range? asymptotes?
- Change parameter d and note how the graph of arctan changes (vertical shift). Does a change in d affect the range of arctan? domain? asymptotes?
- If the range of arctan(x) is given by the interval (-pi/2 , +pi/2) what is the range of a*arctan(x)? What is the range of a*arctan(x)+ d?
- If the horizontal asymptots of arctan(x) are given by the horizontal lines y = -pi/2 and y = +pi/2) what are the horizontal asymptotes of a*arctan(x)?
- What are the horizontal asymptotes of a*arctan(x) +d?
More on Inverse Trigonometric Functions
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