chapter2ProblemsAndSolutions

chapter2ProblemsAndSolutions - Chapter 2 Review of Simple...

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Unformatted text preview: Chapter 2 Review of Simple Functions 2.1 (a) On the same set of axes, sketch the functions y = x , y = x 2 , and y = x 3 for values of x greater than 0. Pay particular attention to the shapes of these graphs in the ranges [0,1] on the x axis and for x > 1. Which of these graphs is steeper over the interval 0 < x < 1? Over the interval x > 1? (Note: Your sketch need not be too accurate, but should reflect the shapes of the graphs and their relationships.) (b) On one set of axes, for both positive and negative values of x sketch a few functions of the form y = x 2 , y = x 4 , (even powers of x ). On a second set of axes, sketch functions of the form y = x , y = x 3 (odd powers of x ) for- 1 . 5 < x < 1 . 5. (c) Consider the functions y = ax 2 and y = bx 3 where a, b are positive constants. At what points do the graphs of these functions intersect? Detailed Solution: (a) The functions y = x , y = x 2 , and y = x 3 are shown in Figure 2.1(a). For values of x in [0,1], the higher the power, the shallower the graph, but for values of x above 1, the higher the power, the steeper the graph. (b) In Figure 2.1(b) we see the even power functions, y = x 2 , y = x 4 (and higher even powers), are symmetric about the y axis. The odd power functions, y = x , y = x 3 (and higher odd powers), are symmetric about the origin, as seen in Figure 2.1(c) (c) The functions y = ax 2 and y = bx 3 intersect when ax 2 = bx 3 . This happens when x = 0 and when x = a/b . 2.2 Simple transformations Consider the graphs of the simple functions y = x , y = x 2 , and y = x 3 . What happens to each of these graphs when the functions are transformed as follows: (a) y = Ax , y = Ax 2 , and y = Ax 3 where A > 1 is some constant? v.2005.1 - September 4, 2009 1 Math 102 Problems Chapter 2 1 y 1 x Power Functions 1 2 y-1 1 x Even Power Functions-1 1 y-1 1 x Odd Power Functions (a) (b) (c) Figure 2.1: Figures for example 2.1 (b) y = x + a , y = x 2 + a , and y = x 3 + a where a > 0 is some constant? (c) y = ( x- b ) 2 , and y = ( x- b ) 3 where b > 0 is some constant? Detailed Solution: (a) The graphs are all stretched in the y direction by the magnification factor A . (b) The graphs are all shifted up the y axis by an amount a . (c) The graphs are all shifted along the x axis in the positive direction by an amount b . 2.3 Simple sketches Sketch the graphs of the following functions: (a) y = x 2 , (b) y = ( x + 4) 2 (c) y = a ( x- b ) 2 + c for the case a > 0, b > 0, c > 0. (d) Comment on the effects of the constants a , b , c on the properties of the graph of y = a ( x- b ) 2 + c . Detailed Solution: (a)-(c): See Figure 2.2. (d) In this function, the constant a scales the height of the parabola, b is a shift in the positive x direction, and c is a shift in the positive y direction....
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chapter2ProblemsAndSolutions - Chapter 2 Review of Simple...

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