Unit 1.1 Lines Equations of a line - mrsk.ca

Unit 1.1 Lines

A line is defined by any two points that are one the line. That is, given two points that are on the line we can graph the line and/or write down the equation of the line.

The most important idea that we’ll be discussing in this section is that of slope. The slope of a line is a measure of the steepness of a line and it can also be used to measure whether a line is increasing or decreasing as we move from left to right.

 Any two parallel lines have same slope.

 Any two perpendicular lines have slopes that are negative reciprocal to each other.

If m₁is the slope of one line, and m₂is the slope of the other line, then when the lines are perpendicular we have:

1

2

m 1

 m

Equations of a line

Slope intercept form :

y   mx b

Standard form:

Ax    By C 0

Slope point fom:

y   y

1

m x   x

1



Linear:

Definition: Slope

Let

P x y

1



1

,

1



and

P x

2



2

, y

2



be points on a non vertical line L. The slope is:

2 1

2 1

y y rise y

m run x x x

   







Properties:

Perpendicular lines have slopes that are negative reciprocal to each other.

Vertical lines have equations that are of the form

x  a

Horizontal lines have equations that are of the form

y  a

Example:

Let L be a line with slope

5

6

. Then any line with slope

6

 5

will be perpendicular to L.

Example

Find the equations of the vertical and horizontal lines that pass through the point

  ^{3, 4}

^.

Solution:

The vertical line has the equation

x  3

, and the horizontal line has the equations

4 y 

.

Definition

Equations of lines

Point-Slope Equation:

y   y

1

m x   x

1



Slope-Intercept Equation:

y   mx b

Standard Equation:

Ax    By C 0

Example:

a) Write an equation of the line through

  ^{3, 4}

^{with slope}

¹

 2

b) Write an equation of the line that passes through

  ^{ 3, 1}

^and

  ^{1, 2}

c) Find the slope and x-intercept of the line

6 x    5 y 10 0

Solution:

a)use point-slope equation of the line

 

4 1 3

2

1 3

2 2 4

1 11

2 2

y x

y x

x

  

  

 

b)Find slope:

2 1  

1 3

1 4 m  

 



Now use point-slope equation with one of these points.

1  

2 1

4

1 1

4 4 2

1 7

4 4

y x

y x

x

  

  

 

c)place in slope intercept form

6 5 10 0

5 6 10

6 2

5 x y

y x

y x

  

 

 

The slope is

6

5

and the x-intercept is 2 Regression Analysis

It is sometimes very difficult to see trends of patterns in lists of paired numbers. For this reason we plot the list of ordered pairs (such a plot is called a scatter plot). We then examining the points to see if there is a curve that passes through (or very close) the set of points. The process of finding a curve to fit data is called regression analysis and the actual curve itself is called a regression curve.

Using your graphing calculator (or excel), find the equation of the line that best matches the data below. Use the equation (or model) to determine the value is 2000.

Year Number

1987 4940

1988 5050

1989 5105

1990 5200

1991 5335

1992 5410

1. Clear the calculator by pressing 2nd, +, 7, 1, 2

2. Press Stat, 1

3. Enter the Years into column 1 (L1)

4. Press the ►key and enter the numbers into column 2 (L2)

5. Press 2nd, y=, 1 and turn on plot 1 as Scatter using direction and Enter keys

6. Press Zoom, 9 to see scatter plot

7. To do a linear regression

Press Stat,►to highlight CALC, Press 4, ENTER

8. Enter the equation

y  94.2857 x  182408.0952

into

Y

₁



(press Y=)

9. Press GRAPH to see graph

10. plug 2000 for x to determine y value. Change the window scale on the calculator to determine the approximate y value when x=2000 (TRACE and ▼,► keys)

y=6163.3048