Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations

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1 Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations

2 origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or Cartesian Coordinate System

3 Let's plot the point (6,4) (-6,0) (0,7) Let's plot the point (-6,0) (6,4) (-3,-5) 6 7 Let's plot the point (-3,-5) Let's plot the point (0,7)

4 Quadrant II x < 0, y > 0 Quadrant I x > 0, y > 0 Quadrant III x < 0, y < 0 Quadrant IV x > 0, y < 0

5 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.

6 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.

7 Here are these settings and their relation to the Cartesian coordinate system.

8 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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13 Horizontal or Vertical Segments

14 Find the distance d between the points (2, 4) and ( 1, 3). 2 ( ) 2 d = ( ) ( ) d 2 = ( 3) ( ) 2 = + =

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21 Find the midpoint of the line segment from P 1 = (4, 2) to P 2 = (2, 5). Plot the points and their midpoint. y x y 4+ 2 = = 3 2 = = 7 2 M 7 = 3, M P 1 x P 2 5

22 Graph Equations by Hand by Plotting Points

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24 Determine if the following points are on the graph of the equation 3x +y = 6 (a) (0, 4) (b) ( 2, 0) (c) ( 1, 3) 3( 0) + 4= 4 6 3( 2) + 0= 6 ( ) = 3 + 3=

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27 Graph Equations Using a Graphing Utility

28 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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30 Expressing an Equation in the Form y = {expression in x} Solve for y: 2y + 3x 5 = 4 We replace the original equation by a succession of equivalent equations.

31 Graphing an Equation Using a Graphing Utility Use a graphing utility to graph the equation: 6x 2 + 2y = 36 Step 1: Solve for y. 6x 2 + 3y = 36 3y = 6x y = 2x 2 +12

32 Graphing an Equation Using a Graphing Utility Step 2: Enter the equation into the graphing utility. Step 3: Choose an initial viewing window.

33 Graphing an Equation Using a Graphing Utility Step 4: Graph the equation. Step 5: Adjust the viewing window.

34 Use a Graphing Utility to Create Tables

35 Create a Table Using a Graphing Utility 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. Step 1: Solve for y: y = 2x Step 2: Enter the equation into the graphing utility.

36 Create a Table Using a Graphing Utility Step 3: Set up a table using AUTO mode Step 4: Create the table.

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40 Use a Graphing Utility to Approximate Intercepts

41 Approximating Intercepts Using a Graphing Utility Use a graphing utility to approximate the intercepts of the equation y = x Here s the graph of y = x 3 16.

42 Approximating Intercepts Using a Graphing Utility 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.

43 Approximating Intercepts Using a Graphing Utility 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 2.52.