SECTION 1.2 The Distance and Midpoint Formulas 13 68. Challenge Problem Use a graphing utility that does not require the equation to be written in the form y x expression in { } = to find the intercepts of the graph of y y x x x 4 6 2 2 2 2 ( ) ( ) − = − − (a) How far was Caleb from home after 10 minutes? (b) How far was Caleb from home after 20 minutes? (c) Identify and interpret the intercepts. 67. Challenge Problem Use a graphing utility that does not require the equation to be written in the form y x expression in { } = to find the intercepts of the graph of x y x y 1 0 2 2 3 2 3 ( ) + − − = Explaining Concepts 71. Draw a graph that contains the points 2, 1 , 0, 1 , 1, 3 , ( ) ( ) ( ) − − and 3, 5 . ( ) Compare your graph with those of other students. Are most of the graphs almost straight lines? How many are “curved”? Discuss the various ways that these points might be connected. 72. Explain what is meant by a complete graph. 73. Write a paragraph that describes a Cartesian plane. Then write a second paragraph that describes how to plot points in the Cartesian plane. Your paragraphs should include the terms “coordinate axes,” “ordered pair,” “coordinates,” “plot,” “ x -coordinate,” and “ y -coordinate.” 69. Make up an equation satisfied by the ordered pairs 2,0 , 4,0 , ( ) ( ) and 0,1 . ( ) Compare your equation with a friend’s equation. Comment on any similarities. In Problem 70, you may use a graphing utility, but it is not required. 70. (a) Graph = = = y x y x y x , , , 2 and y x , 2 ( ) = noting which graphs are the same. (b) Explain why the graphs of y x2 = and = y x are the same. (c) Explain why the graphs of y x = and y x 2 ( ) = are not the same. (d) Explain why the graphs of y x2 = and y x = are not the same. ‘Are You Prepared?’ Answers 1. 0 2. 5 1.2 The Distance and Midpoint Formulas Now Work the ‘Are You Prepared?’ problems on page 17. • Algebra Essentials (Section A.1, pp. A1–A10) • Rectangular Coordinates (Section 1.1, p. 2) • Geometry Essentials (Section A.2, pp. A14–A19) PREPARING FOR THIS SECTION Before getting started, review the following: Finding the Distance between Two Points Find the distance d between the points 1, 3 ( ) and 5,6 . ( ) Solution EXAMPLE 1 First plot the points 1, 3 ( ) and 5, 6 ( ) and connect them with a line segment. See Figure 23(a) on the next page.To find the length d, begin by drawing a horizontal line segment from 1, 3 ( ) to 5, 3 ( ) and a vertical line segment from 5, 3 ( ) to 5, 6 , ( ) forming a right triangle, as shown in Figure 23(b) on the next page. One leg of the triangle is OBJECTIVES 1 Use the Distance Formula (p. 13) 2 Use the Midpoint Formula (p. 16) 1 Use the Distance Formula If the same units of measurement (such as inches, centimeters, and so on) are used for both the x -axis and y -axis, then all distances in the xy -plane can be measured using this unit of measurement. (continued)
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