By Hille E.

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8. 2 Graphs of Equations in Two Variables; Circles plotting points. Shortly, we shall investigate various techniques that will enable us to graph an equation without plotting so many points. Two techniques that sometimes reduce the number of points required to graph an equation involve finding intercepts and checking for symmetry. Figure 11 Graph crosses y-axis y Graph crosses x-axis 2 Find Intercepts from a Graph x Graph touches x-axis 11 Intercepts EXAMPLE 4 The points, if any, at which a graph crosses or touches the coordinate axes are called the intercepts.

54. Geometry Verify that the points (0, 0), (a, 0), and a 23 a a , b are the vertices of an equilateral triangle. Then 2 2 show that the midpoints of the three sides are the vertices of a second equilateral triangle (refer to Problem 52). In Problems 55–58, find the length of each side of the triangle determined by the three points P1 , P2 , and P3. State whether the triangle is an isosceles triangle, a right triangle, neither of these, or both. ) 55. P1 = 12, 12; P2 = 1-4, 12; P3 = 1 -4, -32 56.

63. Distance between Moving Objects A Dodge Neon and a Mack truck leave an intersection at the same time. The Neon heads east at an average speed of 30 miles per hour, while the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours. 3) x (a) Find an estimate for the desired intersection point. (b) Find the length of the median for the midpoint found in part (a). See Problem 51. 66. , have grown from 2002 through 2008.

### A General Type of Singular Point by Hille E.

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