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• 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 – x 1
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:
Solution
x = (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
rise y y2  y1
m


run x x2  x1
Parallel and Perpendicular lines
m 1 = m2
PARALLEL
PERPENDICULAR

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
Solution
a)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 yintercept, we get y = 3x + 5.

General Linear Equation
The equation Ax + By = C (A and B not both 0) is
the general linear equation.
Sketch the line 8x + 5y = 40
Solution
Substitute x = 0 => y = 8; substitute y = 0 => x = 5.
Y
8
X
X
5
X

Linear Regression
Using the best fitting line to predict future trends
Ex Use a linear model of the data in the table to
predict the population in the year 2010.
Year
Population (mil)
1986
4936
1987
5023
1988
5111
1989
5201
1990
5329
1991
5422

Method 1
Draw a scatter plot by hand and overlap the
best fitting straight line.
5500
5400
5300
5200
5100
5000
Best fit
4900
1984
1986
1988
1990
1992

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
STO
(Upper case L
used for clarity.)
2nd
alpha
}
L 1
ENTER

Method 2, continued
{4936, 5023, 5111, 5201, 5329, 5422} -> L2 ENTER
2nd
{
4936, 5023, 5111, 5201, 5329, 5422
STO
alpha
(Upper case L
used for clarity.)
2nd
}
L 2
ENTER
LinReg L1, L2 ENTER
2nd
MATH
6
3
2
alpha
LinReg
Statistics
Regressions
L 1
,
alpha
L 2
The calculator
should return:
ENTER
Done 

Method 2, continued
ShowStat ENTER
2nd
MATH
6
Statistics
8
ENTER
ShowStat
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=
2nd
Plot 1
y1=regeq(x)
VAR-LINK
x
)
regeq
ENTER
Use alpha etc. Type in
L1 and L2
ENTER
WINDOW


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
p