Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
Copyright © 2013 Pearson Education, Inc. All rights reserved x axis y axis origin Rectangular or Cartesian Coordinate System (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate)
Copyright © 2013 Pearson Education, Inc. All rights reserved Let's plot the point (6,4) (-3,-5) (0,7) Let's plot the point (-6,0) (6,4) (-6,0) Let's plot the point (-3,-5)Let's plot the point (0,7)
Copyright © 2013 Pearson Education, Inc. All rights reserved Quadrant I x > 0, y > 0 Quadrant II x 0 Quadrant III x < 0, y < 0 Quadrant IV x > 0, y < 0
Copyright © 2013 Pearson Education, Inc. All rights reserved All graphing utilities (graphing calculators and computer software graphing packages) graph equations by plotting points on a screen. The screen of a graphing utility will display the coordinate axes of a rectangular coordinate system.
Copyright © 2013 Pearson Education, Inc. All rights reserved You must set the scale on each axis. You must also include the smallest and largest values of x and y that you want included in the graph. This is called setting the viewing rectangle or viewing window.
Copyright © 2013 Pearson Education, Inc. All rights reserved Here are these settings and their relation to the Cartesian coordinate system.
Copyright © 2013 Pearson Education, Inc. All rights reserved Finding the Coordinates of a Point Shown on a Graphing Utility Screen Find the coordinates of the point shown. Assume the coordinates are integers. Viewing Window 2 ticks to the left on the horizontal axis (scale = 1) and 1 tick up on the vertical axis (scale = 2), point is (–2, 2)
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Horizontal or Vertical Segments
Copyright © 2013 Pearson Education, Inc. All rights reserved Find the distance d between the points (2, – 4) and (–1, 3).
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Find the midpoint of the line segment from P 1 = (4, –2) to P 2 = (2, –5). Plot the points and their midpoint. P1P1 P2P2 M
Copyright © 2013 Pearson Education, Inc. All rights reserved Graph Equations by Hand by Plotting Points
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Determine if the following points are on the graph of the equation –3x +y = 6 (b) (–2, 0)(a) (0, 4)(c) (–1, 3)
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Graph Equations Using a Graphing Utility
Copyright © 2013 Pearson Education, Inc. All rights reserved To graph an equation in two variables x and y using a graphing utility requires that the equation be written in the form y = {expression in x}. If the original equation is not in this form, rewrite it using equivalent equations until the form y = {expression in x} is obtained. In general, there are four ways to obtain equivalent equations.
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Solve for y: 2y + 3x – 5 = 4 Expressing an Equation in the Form y = {expression in x} We replace the original equation by a succession of equivalent equations.
Copyright © 2013 Pearson Education, Inc. All rights reserved Use a graphing utility to graph the equation: 6x 2 + 2y = 36 Graphing an Equation Using a Graphing Utility Step 1: Solve for y.
Copyright © 2013 Pearson Education, Inc. All rights reserved Step 2: Enter the equation into the graphing utility. Graphing an Equation Using a Graphing Utility Step 3: Choose an initial viewing window.
Copyright © 2013 Pearson Education, Inc. All rights reserved Step 4: Graph the equation. Graphing an Equation Using a Graphing Utility Step 5: Adjust the viewing window.
Copyright © 2013 Pearson Education, Inc. All rights reserved Use a Graphing Utility to Create Tables
Copyright © 2013 Pearson Education, Inc. All rights reserved Create a table that displays the points on the graph of 6x 2 + 3y = 36 for x = –3, –2, –1, 0, 1, 2, and 3. Create a Table Using a Graphing Utility Step 1: Solve for y: y = –2x Step 2: Enter the equation into the graphing utility.
Copyright © 2013 Pearson Education, Inc. All rights reserved Step 3: Set up a table using AUTO mode Create a Table Using a Graphing Utility Step 4: Create the table.
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Use a Graphing Utility to Approximate Intercepts
Copyright © 2013 Pearson Education, Inc. All rights reserved Use a graphing utility to approximate the intercepts of the equation y = x 3 – 16. Approximating Intercepts Using a Graphing Utility Here’s the graph of y = x 3 – 16.
Copyright © 2013 Pearson Education, Inc. All rights reserved The eVALUEate feature of a TI-84 Plus graphing calculator accepts as input a value of x and determines the value of y. If we let x = 0, the y-intercept is found to be –16. Approximating Intercepts Using a Graphing Utility
Copyright © 2013 Pearson Education, Inc. All rights reserved The ZERO feature of a TI-84 Plus is used to find the x-intercept(s). Rounded to two decimal places, the x-intercept is Approximating Intercepts Using a Graphing Utility