313 CHAPTER 2 Test Prep Concepts Examples Find the center-radius form of the equation of the circle with center 1-2, 32 and radius 4. 3x - 1-2242 + 1 y - 322 = 42 1x + 222 + 1 y - 322 = 16 The general form of the equation of the preceding circle is x2 + y2 + 4x - 6y - 3 = 0. 2.2 Circles Center-Radius Form of the Equation of a Circle The equation of a circle with center 1h, k2 and radius r is given by the following. 1 x −h22 + 1 y −k22 =r2 General Form of the Equation of a Circle x2 +y2 +Dx +Ey +F =0 2.3 Functions A relation is a set of ordered pairs. A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component. The set of first components is the domain. The set of second components is the range. The relation y = x2 defines a function because each choice of a number for x corresponds to one and only one number for y. The domain is 1-∞, ∞2, and the range is 30, ∞2. The relation x = y2 does not define a function because a number x may correspond to two numbers for y. The domain is 30, ∞2, and the range is 1-∞, ∞2. Vertical Line Test If every vertical line intersects the graph of a relation in no more than one point, then the relation is a function. Determine whether each graph is that of a function. x y 0 A. x y 0 B. By the vertical line test, graph A is the graph of a function, but graph B is not. x y f (x1) f (x2) x1 x2 0 f is increasing on (x1, x2). y x f (x2) f (x1) x1 x2 0 f is decreasing on (x1, x2). y x f(x1) = f(x2) y = f(x) y = f(x) y = f(x) x1 x2 0 f is constant on (x1, x2). Increasing, Decreasing, and Constant Functions Discuss the function in graph A above in terms of whether it is increasing, decreasing, or constant. The function in graph A is decreasing over the open interval 1-∞, 02 and increasing over the open interval 10, ∞2. The equation y = ƒ1x2 = 1 2 x - 4 defines y as a linear function ƒ of x. Find the slope of the line through the points 12, 42 and 1-1, 72. m= 7 - 4 -1 - 2 = 3 -3 = -1 2.4 Linear Functions A function ƒ is a linear function if, for real numbers a and b, ƒ1x2 =ax +b. The graph of a linear function is a line. Slope Formula The slope m of the line through the points 1x1, y12 and 1x2, y22 is defined as follows. m= rise run = y x = y2 −y1 x2 −x1 , where x 30
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