OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Graphs of Equations Sketch a graph by plotting points. Find the intercepts of a graph. Find the symmetries in a graph. Find the equation of a circle. SECTION

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions An ordered pair (a, b) is said to satisfy an equation with variables a and b if, when a is substituted for x and b is substituted for y in the equation, the resulting statement is true. An ordered pair that satisfies an equation is called a solution of the equation. Frequently, the numerical values of the variable y can be determined by assigning appropriate values to the variable x. For this reason, y is sometimes referred to as the dependent variable and x as the independent variable.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GRAPH OF AN EQUATION The graph of an equation in two variables, such as x and y, is the set of all ordered pairs (a, b) in the coordinate plane that satisfy the equation.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Sketching a Graph by Plotting Points Sketch the graph of Solution (3, 6)y = 3 2 – 3 = 9 – 3 = 63 (2, 1)y = 2 2 – 3 = 4 – 3 = 12 (1, –2)y = 1 2 – 3 = 1 – 3 = –21 (0, –3)y = 0 2 – 3 = 0 – 3 = –30 (–1, –2)y = (–1) 2 – 3 = 1 – 3 = –2–1 (–2, –1)y = (–2) 2 – 3 = 4 – 3 = 1–2 (–3, 6)y = (–3) 2 – 3 = 9 – 3 = 6–3 (x, y)y = x 2 – 3x

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Sketching a Graph by Plotting Points Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SKETCHING A GRAPH BY PLOTTING POINTS Step1.Make a representative table of solutions of the equation. Step 2.Plot the solutions as ordered pairs in the Cartesian coordinate plane. Step 3.Connect the solutions in Step 2 by a smooth curve.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions The points where a graph intersects (crosses or touches) the coordinate axes are of special interest in many problems. Since all points on the x-axis have a y-coordinate of 0, any point where a graph intersects the x-axis has the form (a, 0). The number a is called an x-intercept of the graph. Similarly, any point where a graph intersects the y-axis has the form (0, b), and the number b is called a y-intercept of the graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley PROCEDURE FOR FINDING THE INTERCEPTS OF A GRAPH Step1To find the x-intercepts of an equation, set y = 0 in the equation and solve for x. Step 2To find the y-intercepts of an equation, set x = 0 in the equation and solve for y.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Finding Intercepts Find the x- and y-intercepts of the graph of the equation y = x 2 – x – 2. Solution Step 1To find the x-intercepts, set y = 0, solve for x. The x-intercepts are –1 and 2.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Finding Intercepts Solution continued Step 2To find the y-intercepts, set x = 0, solve for y. The y-intercept is –2. This is the graph of the equation y = x 2 – x –2.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Definitions A useful tool for sketching the graph of an equation is the concept of symmetry, which means that one portion of the graph is a mirror image of another portion. The mirror line is usually called the axis of symmetry or line of symmetry.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TESTS FOR SYMMETRY 1.A graph is symmetric with respect to the y-axis if, for every point (x, y) on the graph, the point (–x, y) is also on the graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TESTS FOR SYMMETRY 2.A graph is symmetric with respect to the x-axis if, for every point (x, y) on the graph, the point (x, –y) is also on the graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley TESTS FOR SYMMETRY 3.A graph is symmetric with respect to the origin if, for every point (x, y) on the graph, the point (–x, –y) is also on the graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Determine whether the graph of the equation Checking for Symmetry Solution Replace x with –x in the original equation. is symmetric with respect to the y-axis. When we replace x with –x in the equation, we obtain the original equation. The graph is symmetric with respect to the y-axis.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Show that the graph of Checking for Symmetry Solution Replace y with –y in the original equation. is symmetric with respect to the x-axis. When we replace y with –y in the equation, we obtain the original equation. The graph is symmetric with respect to the x-axis.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Show that the graph of Checking for Symmetry with Respect to the Origin Solution Replace x with –x and y with –y. with respect to the origin, but not with respect to either axis. Replacing x with –x and y with –y in the equation, we obtain the original equation. The graph is symmetric with respect to the origin. is symmetric

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Checking for Symmetry with Respect to the Origin Solution continued The graph of y 5 = x 3 is not symmetric with respect to the y-axis because the point (1, 1) is on the graph, but the point (–1, 1) is not on the graph. Similarly, the graph of y 5 = x 3 is not symmetric with respect to the x-axis since the point (1, 1) is on the graph, but the point (1, –1) is not on the graph.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Initially, there are 400 deer on Dooms island. The number y of deer on the island after t years is described by the equation a. Sketch the graph of the equation b. Adjust the graph in part (a) to account for only the physical aspects of the problem. c. When does the population of deer become extinct on Dooms?

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued a. Find all intercepts. Set y = 0. The t-intercepts are –10 and 10. y-intercept is 400. Set t = 0.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued Check for symmetry (t replaces x). Symmetry in the t-axis: replace y with –y (0, – 400) is a solution of but not of The graph is not symmetric in the t-axis.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued Symmetry in the y-axis: replace t with –t This is the original equation. Thus, (–t, y) also satisfies the equation and the graph is symmetric in the y-axis.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued Symmetry in the origin: replace t with –t and y with –y (0, 400) is a solution but (–0, –400) is not a solution, the graph is not symmetric with respect to the origin.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued Plot points for t ≥ 0 and then using symmetry in the y-axis. ty = – t 4 +96t (t, y) 0400(0, 400) 1495(1, 495) 52175(5, 2175) 72703(7, 2703) 91615(9, 1615) 100(10, 0) 11–2625(11, –2625)

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Sketching a Graph by Using Intercepts and Symmetry Solution continued b.The graph pertaining to the physical aspects of the problem is the blue portion. c.The positive t-intercept, which is 10, gives the time in years when the deer population of Dooms is 0.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CIRCLE A circle is a set of points in a Cartesian coordinate plane that are at a fixed distance r from a specified point (h, k). The fixed distance r is called the radius of the circle, and the specified point (h, k) is called the center of the circle.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CIRCLE The graph of a circle with center (h, k) and radius r.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CIRCLE The equation of a circle with center (h, k) and radius r is This equation is also called the standard form of an equation of a circle with radius r and center (h, k).

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Finding the Equation of a Circle Find the center–radius form of the equation of the circle with center (–3, 4) and radius 7. Solution

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Graphing a Circle Graph each equation. Solution Center: (0, 0) Radius: 1 Called the unit circle

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Graphing a Circle Solution continued Center: (–2, 3) Radius: 5

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EQUATION CIRCLE Note that stating that the equation represents the circle of radius 5 with center (–3, 4) means two things: (i)If the values of x and y are a pair of numbers that satisfy the equation, then they are the coordinates of a point on the circle with radius 5 and center (–3, 4). (ii)If a point is on the circle, then its coordinates satisfy the equation.

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley GENERAL FORM OF THE EQUATION OF A CIRCLE The general form of the equation of a circle is

Slide Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Converting the General Form to Center-Radius Form Find the center and radius of the circle with equation Solution Complete the squares on both x and y. Center: (3, – 4) Radius: