6 CHAPTER 1 Graphs * These input values depend on the values of Xmin and Xmax. For example, if = − Xmin 10 and = Xmax 10, then the first input value will be −10 and the next input value will be ( ) ( ) − + − − = − 10 10 10 264 9.9242, and so on. Section 1.5 that if an equation is of the form y mx b, = + then its graph is a line. In this case, only two points are needed to obtain the complete graph . One purpose of this text is to investigate the properties of equations in order to decide whether a graph is complete. Sometimes we shall graph equations by plotting points on the graph until a pattern becomes evident and then connect these points with a smooth curve, following the suggested pattern. (Shortly, we shall investigate various techniques that will enable us to graph an equation without plotting so many points.) Other times we shall graph equations using a graphing utility. 2 Graph Equations Using a Graphing Utility From Examples 2 and 3, we see that a graph can be obtained by plotting points in a rectangular coordinate system and connecting them. Graphing utilities perform these same steps when graphing an equation. For example, the TI-84 Plus CE determines 265 evenly spaced input values (starting at Xmin and ending at Xmax),* uses the equation to determine the output values, plots these points on the screen, and finally (if in the connected mode) draws a line between consecutive points. To graph an equation in two variables x and y using a graphing utility often requires that the equation be written in the form y x expression in . { } = If the original equation is not in this form, rewrite it using equivalent equations until the form y x expression in { } = is obtained. Procedures That Result in Equivalent Equations • Interchange the two sides of the equation: x y y x 3 5 is equivalent to 3 5 + = = + • Simplify the sides of the equation by combining like terms, eliminating parentheses, and so on: y x x y x 2 2 6 2 5 1 is equivalent to 2 8 7 5 ( ) + + = + + + = + • Add or subtract the same expression on both sides of the equation: y x y x 3 5 4 is equivalent to 3 5 5 4 5 + − = + − + = + • Multiply or divide both sides of the equation by the same nonzero expression: y x y x 3 6 2 is equivalent to 1 3 3 1 3 6 2 ( ) = − ⋅ = − Solution Replace the original equation by a succession of equivalent equations. + − = + − + = + + = + − = − = − = − = − y x y x y x y x x x y x y x y x 2 3 5 4 2 3 5 5 4 5 2 3 9 2 3 3 9 3 2 9 3 2 2 9 3 2 9 3 2 CAUTION Be careful when entering the expression − x 9 3 2 . Use a fraction template or use parentheses as follows: ( ) − x 9 3 2. j Add 5 to both sides. Simplify. Subtract x3 from both sides. Simplify. Divide both sides by 2. Simplify. Expressing an Equation in the Form y x expression in { } = Solve for y: y x 2 3 5 4 + − = EXAMPLE 4
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