Section 1.1 The Distance and Midpoint Formulas; Graphing...

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Section 1.1

The Distance and Midpoint Formulas; Graphing Utilities;

Introduction to Graphing Equations


x axis origin

Rectangular or Cartesian Coordinate System

(x, y) Ordered pair

(x-coordinate, y-coordinate) (abscissa, ordinate)

•


−8−7−6−5−4−3−2−1 1 2 3 4 5 6 7 8

−7−6−5−4−3−2−1

1234567

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)


Quadrant I x > 0, y > 0

Quadrant II 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).

( ) ( )( )221 2 3 4d = − − + − −

( )22( 3) 7d −= + 9 49= + 58= 7.62≈


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

4 22

x += 3=

2 52

y − −=

72

= −

73,2

M = −

1 2 3 4 5

−5

−4

−3

−2

−1

x

y

P1

P2

M


Graph Equations by Hand by Plotting Points


Determine if the following points are on the graph of the equation –3x +y = 6

(b) (–2, 0) (a) (0, 4) (c) (–1, 3)

−4 −3 −2 −1 1 2

2

−1

1

2

3

4

( )0 43 4 6− + = ≠ ( )3 62 0− − + = ( )3 3 61 33− + = + =−


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.


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.


Use a graphing utility to graph the equation:

6x2 + 2y = 36

Graphing an Equation Using a Graphing Utility

Step 1: Solve for y.

6x2 + 3y = 363y = −6x2 + 36y = −2x2 +12


Step 2: Enter the equation into the graphing utility.


Step 3: Choose an initial viewing window.


Step 4: Graph the equation.


Step 5: Adjust the viewing window.


Use a Graphing Utility to Create Tables


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.


Step 3: Set up a table using AUTO mode

Create a Table Using a Graphing Utility

Step 4: Create the table.


.


Use a Graphing Utility to Approximate Intercepts


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.


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.



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.


Section 1.1 The Distance and Midpoint Formulas; Graphing...

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