Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing Equations
origin (x, y) Ordered pair (x-coordinate, y-coordinate) (abscissa, ordinate) x axis Rectangular or Cartesian Coordinate System
Let's plot the point (6,4) (-6,0) (0,7) 7 6 5 4 3 2 1 Let's plot the point (-6,0) (6,4) 8 7 6 5 4 3 2 1 1 1 2 3 4 5 6 7 8 2 3 4 5 (-3,-5) 6 7 Let's plot the point (-3,-5) Let's plot the point (0,7)
Quadrant II x < 0, y > 0 Quadrant I x > 0, y > 0 Quadrant III x < 0, y < 0 Quadrant IV x > 0, y < 0
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.
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.
Here are these settings and their relation to the Cartesian coordinate system.
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)
Horizontal or Vertical Segments
Find the distance d between the points (2, 4) and ( 1, 3). 2 ( ) 2 d = 1 2 + 3 4 ( ) ( ) d 2 = ( 3) + 7 9 49 ( ) 2 = + = 58 7.62
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 = 2 5 2 = 7 2 M 7 = 3, 2 1 2 3 4 1 2 3 4 5 M P 1 x P 2 5
Graph Equations by Hand by Plotting Points
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 1 + 3 = 3 + 3= 6 4 3 2 1 4 3 2 1 1 2 1 2
Graph Equations Using a Graphing Utility
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.
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.
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 2 + 36 y = 2x 2 +12
Graphing an Equation Using a Graphing Utility Step 2: Enter the equation into the graphing utility. Step 3: Choose an initial viewing window.
Graphing an Equation Using a Graphing Utility Step 4: Graph the equation. Step 5: Adjust the viewing window.
Use a Graphing Utility to Create Tables
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 2 + 12 Step 2: Enter the equation into the graphing utility.
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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Use a Graphing Utility to Approximate Intercepts
Approximating Intercepts Using a Graphing Utility Use a graphing utility to approximate the intercepts of the equation y = x 3 16. Here s the graph of y = x 3 16.
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.
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.