Do Now:

• What do you know about the following:– Slope– x-intercept, y-intercept– Linear equations– Calculus

Lines

Increments

• Calculus has proven to be useful for relating the rate of change of a quantity to the graph of the quantity.

• In order to begin explaining this relationship we must begin with the slopes of lines.

Increments

• When a particle in the plane moves from one point to another we must use the starting point and stopping point to discuss change.

Increments

• If a particle moves from point (x1,y1) to the point (x2,y2), the increments in its coordinates are:

x = x2 – x1 and y = y2 – y1

Increments

Let a particle move from P1(x1, y1) to P2(x2, y2). The increments in its coordinates are x = (x2 – x1) and y= (y2 – y1).

Ex. The coordinate increments from (4, -3) to (2, 5) are:

Solutionx = (x2 – x1) = 2 – 4 = -2 and y = (y2 – y1) = 5- -3 = 8.

Ex. The coordinate increments from (5, 6) to (5, 1) are:

Solution

x = (x2 – x1) = 5 – 5 = 0 and y = (y2 – y1) = 1- 6 = -5.

Slope

• Each nonvertical line has a slope, and we can use increments to calculate our slope.

Slope

• Let P1(x1, y1) and P2(x2, y2) be points on a nonvertical line, L. The slope of L is

12

12

xx

yy

x

y

run

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PARALLEL PERPENDICULAR

Parallel and Perpendicular lines

m1 = m2

Equations of vertical & horizontal lines

Ex. Find the equations of the vertical and horizontal lines that pass through the point (2, 3)

Solution

(2, 3)Y = 3

x = 2

Point-slope Equation

The equation y - y1 = m(x –x1) is the point slope formula with m = slope and (x1, y1) is a pt on line

Ex. Find the equation of the line that passes through (-1, 2) and is

a)Parallel to y = 3x – 4

b)Perpendicular to y = 3x - 4

Solutiona)y – 2 = 3(x - -1) => y = 3x + 5

b) Y – 2 = (-1/3)*(x - -1) => y = -x/3 + 5/3

Slope-intercept Equation

The equation y = mx + b is the slope-intercept formula with m = slope and b = y-intercept

Ex. Find the equation of the line through (-1, 2) that passes through (0, 5).

Solution

m = (5 – 2)/(0 - –1) => m = 3. Since (0, 5) is the y-intercept, we get y = 3x + 5.

General Linear Equation

The equation Ax + By = C (A and B not both 0) is the general linear equation.

Solution

Sketch the line 8x + 5y = 40

Substitute x = 0 => y = 8; substitute y = 0 => x = 5.

X

Y

X

5

X8

Using the best fitting line to predict future trends

Linear Regression

Year Population (mil)

1986 4936

1987 5023

1988 5111

1989 5201

1990 5329

1991 5422

Ex Use a linear model of the data in the table to predict the population in the year 2010.

Method 1

Draw a scatter plot by hand and overlap the best fitting straight line.

4900

5000

5100

5200

5300

5400

5500

1984 1986 1988 1990 1992

Best fit

Method 1, continued

Find the equation of the line and use its equation to predict population in 2010.

Eqn: Use point slope formula. Use 2 pts on best line.

(1986, 4936mil), (1990, 5300mil)

m = (5300-4936)/(1990-1986) = 364/4 = 91 (mil/yr)

y – 4936mil = 91mil(x – 1986)

y = 91mil*x - 175790mil.

In 2010, the population will be 91(2010) – 175790

Pop = 7120 mil <= INACCURATE

Method 2

Use the capabilities of the TI-89 calculator.

To simplify, let x = 0 represent 1986, x = 1 represent 1987 etc.

{0, 1, 2, 3, 4, 5} -> L1 ENTER

2nd { 0, 1, 2, 3, 4, 5 2nd }

STO alpha L 1

ENTER(Upper case L used for clarity.)

Method 2, continued

{4936, 5023, 5111, 5201, 5329, 5422} -> L2 ENTER

2nd { 4936, 5023, 5111, 5201, 5329, 5422 2nd }

STO alpha L 2

ENTER(Upper case L used for clarity.)

2nd MATH 6 3 2

LinReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

LinReg L1, L2 ENTER

Statistics Regressions

Method 2, continued

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants:

y = ax + b

a = 98.228571, b = 4924.761905, corr = 0.997826

R2 = 0.995658

Method 2, continued

Use calculator - plot new curve & the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 ENTER

ENTER

WINDOW

Use alpha etc. Type in L1 and L2

Method 2, continued

WINDOW Xmin = 0

Xmax = 5

Xsc = 1

Ymin = 4936

Ymax = 5422

Ysc = 1

Xres = 1

GRAPH produces the graph

Method 2, continued

Go to the homescreen

2nd QUIT

To get the population in 2010, note that 2010 is 24 years after 1986. So we enter y1(24) at homescreen and obtain 7282.25 (mil).

Compare this value with the value obtained using the “by hand” method