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

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Page 1: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Section 1.1

The Distance and Midpoint Formulas; Graphing Utilities;

Introduction to Graphing Equations

Page 2: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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)

Page 3: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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)

Page 4: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Quadrant I x > 0, y > 0

Quadrant II x < 0, y > 0

Quadrant III x < 0, y < 0

Quadrant IVx > 0, y < 0

Page 5: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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.

Page 6: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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.

Page 7: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

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

Page 8: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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)

Page 9: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 10: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 11: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 12: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Horizontal or Vertical Segments

Page 13: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the distance d between the points (2, – 4) and (–1, 3).

221 2 3 4d

22( 3) 7d 9 49 58 7.62

Page 14: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Try This

Find the distance between the points (4, -7) and (0, -2)

Page 15: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 16: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 17: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Find the midpoint of the line segment from P1 = (4, –2) to P2 = (2, –5). Plot the points and their midpoint.

4 2

2x

3

2 5

2y

7

2

73,

2M

x

y

P1

P2

M

Page 18: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Try this

Find the midpoint of the line segment from (2, -3) and (10, 3).

Page 19: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Graph Equations by Hand by Plotting Points

Page 20: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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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)

0 43 4 6 3 62 0 3 3 61 33

Page 21: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 22: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 23: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Graph Equations Using a Graphing Utility

Page 24: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

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.

Page 25: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Use a graphing utility to graph the equation:

6x2 + 2y = 36

Graphing an Equation Using a Graphing Utility

Step 1: Solve for y.

6x2 3y36

3y 6x2 36

y 2x2 12

Page 26: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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.

Page 27: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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.

Page 28: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Use a Graphing Utility to Create Tables

Page 29: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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Create a table that displays the points on the graph of 6x2

+ 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 = –2x2 + 12

Step 2: Enter the equation into the graphing utility.

Page 30: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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.

Page 31: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Try this

Graph the equation. List 3 values of the table.

12 xy

Page 32: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 33: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Page 34: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

.

Page 35: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Use a Graphing Utility to Approximate Intercepts

Page 36: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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Use a graphing utility to approximate the intercepts of the equation y = x3 – 16.

Approximating Intercepts Using a Graphing Utility

Here’s the graph of y = x3 – 16.

Page 37: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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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, they-intercept is found to be –16.

Approximating Intercepts Using a Graphing Utility

Page 38: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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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.

Approximating Intercepts Using a Graphing Utility

Page 39: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

Copyright © 2013 Pearson Education, Inc. All rights reserved

Try this

Find the intercepts of the equation

193 2 xy

Page 40: Copyright © 2013 Pearson Education, Inc. All rights reserved Section 1.1 The Distance and Midpoint Formulas; Graphing Utilities; Introduction to Graphing

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Homework

P. 94 – 96 Pencil Problems (skip 49), 40, 60, 72, 82, 90