Lesson 2.1 Exercises, pages 90 96

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1 Lesson.1 Eercises, pages 9 96 A. a) Complete the table of values b) For each function in part a, sketch its graph then state its domain and range. For : the domain is ; and the range is. For ç ç : the domain is ; and the range is». c) How are the graphs in part b related? Both graphs pass through ( 1, 1) and (, ). The graph of is above the graph of between these points. Ever value of for is the square root (if it eists) of the corresponding -value on the graph of. 5. For each graph of = g() below: i) Mark points where = or. ii) Sketch the graph of = = 1 g(). iii) Identif the domain and range of = g(). a) b) g() g() g() g() ii) Choose the point (3, ) on g(). The corresponding point on g() is (3, ). Draw a curve through this point and the marked points. iii) Domain is:» 1 ii) Choose the point (5, ) on g(). The corresponding point on g() is (5, ). Draw a curve through this point and the marked points. iii) Domain is:» 1.1 Properties of Radical Functions Solutions 1

2 c) d) g() g() g() g() ii) Choose the point ( 5, ) on g(). The corresponding point on g() is ( 5, ). Draw a curve through this point and the marked points. iii) Domain is: 1 ii) Choose the point ( 3, ) on g(). The corresponding point on g() is ( 3, ). Draw a curve through this point and the marked points. iii) Domain is: 1 B 6. a) Use technolog to graph each function. Sketch each graph. i) = + 3 ii) = iii) = iv) =.5 + v) = vi) =.5 - b) Given a linear function of the form f() a b, a, b i) For which values of a does the graph of = Z f() open to the right? Use eamples to support our answer. The graph opens to the right when a>, as in part a, i, iv, and vi..1 Properties of Radical Functions Solutions DO NOT COPY. P

3 ii) For which values of a does the graph of = f() open to the left? Use eamples to support our answer. The graph opens to the left when a<, as in part a, ii, iii, and v. 7. a) Complete this table of values for and =, then graph the functions on the same grid b) What other function describes the graph of =? Eplain wh. The graph of is the same as the graph of. For an number, ( ) ; both and ( ) are equal to a positive root, which is.. a) For the graph of each quadratic function f()below: Sketch the graph of = f(). State the domain and range of = f(). i) ii) 16 f() f() 6 1 f() f() Mark points where or 1. Choose, then mark another point on the graph of f(). f() f() 5 Join all points with a smooth curve. Domain is: 7 3 Range is: There are no points where or 1. Choose, then mark other points on the graph of f(). f() f() Join all points with a smooth curve. Domain is: ç Range is:».1 Properties of Radical Functions Solutions 3

4 b) Choose one pair of functions = f() and = f() from part a. If the domains are different and the ranges are different, eplain wh. Sample response: For part a, i, the domains are different because the radical function onl eists for those values of where» ; while the quadratic function eists for all real values of. The ranges are different because the value of for the radical function can onl be or a positive number; while the range of the quadratic function is all real numbers less than or equal to, which includes all negative real numbers. 9. Solve each radical equation b graphing. Give the solution to the nearest tenth where necessar. a) - 5 = + 3 b) = f() 5 3 The zero is: 13 So, the root is: 13 f() The zero is: 3 So, the root is: 3 c) = + d) = f() 3 1 The approimate zeros are: and 9.67 So, the roots are:. and f() The approimate zero is:.71 So, the root is:..1 Properties of Radical Functions Solutions DO NOT COPY. P

5 1. For the graph of each cubic function below: Sketch the graph of = = g() g(). State the domain and range of = g(). a) b) g() 1 1 g() g() g() Mark points where or 1. Identif and mark the coordinates of other points on the graph of g(). Mark points where or 1. Identif and mark the coordinates of other points on the graph of g(). g() g() g() g() Join the points with smooth curves. Domain is: 1 or 1 3 Join the points with smooth curves. Domain is: or» 11. a) Sketch the graph of a linear function for which = = g() is not defined. Eplain how ou know that = g() g() is not defined. Sample response: For g() to be undefined, the value of g() must alwas be negative; for eample, a function is 3. O 3 b) Sketch the graph of a quadratic function for which = = f() is not defined. Eplain how ou know that = f() f() is not defined. Sample response: For f() to be undefined, the value of f() must alwas be negative; so the graph of f() must alwas lie beneath the -ais; for eample, a function is ( ). ( ) 6.1 Properties of Radical Functions Solutions 5

6 c) For ever cubic function = g(), the function = g() eists. Eplain wh. Since the graph of ever cubic function either begins in Quadrant and ends in Quadrant, or begins in Quadrant 3 and ends in Quadrant 1, there is alwas part of the graph above the -ais; that is, the function has positive values, so its square root eists. 1. For each graph of = g(), sketch the graph of = g(). a) b) 6 g() g() g() 6 g() Mark points where or 1. Identif and mark the coordinates of other points on the graph of g(). Mark points where and 1. Identif and mark the coordinates of other points on the graph of g(). g() g() g() g() Join the points with smooth curves. Join the points with smooth curves, and a line segment. 6.1 Properties of Radical Functions Solutions DO NOT COPY. P

7 13. When a satellite is h kilometres above Earth, the time for one complete orbit, t minutes, can be calculated using this formula: t = 1.66 * 1 - (h + 637) 3 A communications satellite is to be positioned so that it is alwas above the same point on Earth s surface. It takes h for this satellite to complete one orbit. What should the height of the satellite be? Substitute t ()(6), or (h 637) 3 Write the equation with all the terms on one side (h 637) 3 Write a related function. f(h) (h 637) 3 Graph the function, then determine the approimate zero, which is So, to the nearest kilometre, the satellite should be 35 9 km high. C 1. Given the graph of = f(), sketch the graph of = 3 f() without using graphing technolog. What are the invariant points on the graph of = 3 f()? Since 3 1 1, 3, and 3 1 1, the invariant points occur where:, f() label these points A and B; 1, label these points C and D; and 1, 3 C D f() label these points E and F. Since the cube root of a number between 3 A B 3 and 1 is greater than the number, E F the graph of 3 f() lies above the graph of f() between A and C, and between B and D. Since the cube root of a number between and 1 is less than the number, the graph of 3 f() lies below the graph of f() between A and E, and between B and F. Identif and mark the coordinates of other points. f() 3 f() Join the points with a smooth curve..1 Properties of Radical Functions Solutions 7