SECTION 1.4 Solving Equations Using a Graphing Utility 29 Figure 43 NOTE Most graphing utilities use a process in which they search for a solution until the answer is found within a certain tolerance level (such as within 0.0001). Therefore, the y-coordinate may sometimes be a nonzero value such as 1.1527E-8, which is × − 1.1527 10 ,8 very close to zero. j 1 Solve Equations Using a Graphing Utility When a graphing utility is used to solve an equation, usually approximate solutions are obtained. Unless otherwise stated, we shall follow the practice of giving approximate solutions as decimals rounded to two decimal places. The ZERO (or ROOT) feature of a graphing utility can be used to find the solutions of an equation when one side of the equation is 0. In using this feature to solve equations, make use of the fact that when the graph of an equation in two variables, x and y, crosses or touches the x-axis then = y 0. For this reason, any value of x for which = y 0 will be a solution to the equation. That is, solving an equation for x when one side of the equation is 0 is equivalent to finding where the graph of the corresponding equation in two variables crosses or touches the x-axis. Solution The solutions of the equation − + = x x 1 0 3 are the same as the x-intercepts of the graph of = − + Y x x 1. 1 3 Begin by graphing Y . 1 Figure 42(a) shows the graph. From the graph there appears to be one x-intercept (solution to the equation) between −2 and −1. Using the ZERO (or ROOT) feature of a graphing utility, determine that the x-intercept, and thus the solution to the equation, is −1.32 rounded to two decimal places. See Figure 42(b). Using ZERO (or ROOT) to Approximate Solutions of an Equation Find the solution(s) of the equation − + = x x 1 0. 3 Round answers to two decimal places. EXAMPLE 1 (b) 10 25 23 3 (a) = − + Y x x 1 1 3 10 25 23 3 Figure 42 Now Work PROBLEM 5 Some graphing utilities, such as Geogebra or Desmos, do not require the use of a command such as ROOT or ZERO. Instead, the user graphs the non-zero side of the equation (as we did in Example 1). Clicking on the graph prompts the graphing utility to highlight points of interest such as the intercepts. See Figure 43 for the graph of = − + y x x 1 3 using Geogebra. Notice the intercept ( ) −1.3247,0 is labeled Root. Therefore, the solution to the equation − + = x x 1 0 3 is −1.32 rounded to two decimal places.
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