CHAPTER 1:PREREQUISITES FOR CALCULUS

SECTION 1.1 - LINES

AP CALCULUS AB

What you’ll learn about…

…and why.

Linear equations are used extensively in business and economic applications.

Increments

1 1 2 2

2 1 2 1

If a particle moves from the point ( , ) to the point ( , ), the

in its coordinates are

and

x y x y

x x x y y y increments

The symbols and are read delta and delta .

The letter is a Greek capital for difference.

Neither nor denotes multiplication;

is not delta times nor is delta times .

x y x y

d

x y

x x y y

D

D

D

D D

Example Increments

The coordinate increments from (8, 3) to (-6, 1) are:

6 8 14, 1 3 2x y

Slope of a Line

1 1 2 2

1

2 1

Let ( , ) and ( , ) be points on a nonvertical line, . The

of is

rise

run

1 2

2

P x y P x y L

L

y ym =

x x

-=

-

slope

A line that goes uphill as x increases has a positive slope. A line that goes downhill as x increases has a negative slope.

Slope of a Line

A horizontal line has slope zero since all of its points have the same

-coordinate, making 0.

For vertical lines, 0 and the ratio is undefined.

We express this by saying that vertical li

y y

yx

x

D =

DD =

D

nes have no slope.

Parallel and Perpendicular Lines

1 2

Parallel lines form equal angles with the -axis. Hence, nonvertical

parallel lines have the same slope.

=

If two nonvertical lines and are perpendicular, their slopes

and satisfy

1 2

1 2

x

m m

L L

m m 1 2

1 22 1

1, so each slope is the negative reciprocal

1 1of the other: ,

m m

m mm m

=-

=- =-

Section 1.1 – Lines

Parallel and Perpendicular Lines:1. Two distinct non-vertical lines are parallel

if and only if their slopes are equal.So

1

1

1 2

2

2

2 5 4 2 8

2 5 2 4 8

2 2 4

2

so the lines are parallel.

.

x y x y

y - x y x

m y x

m

m

m

m

m

Section 1.1 – LinesParallel and Perpendicular Lines:2. Two non-vertical lines are

perpendicular if and only if their slopes are negative reciprocals of each other. That is,

1

2

,2

2 5 2 4 12

2 5 4 2 12

12 3

21

2

1 12 , so the lines are perpendicular.

12

x y x y

y x y x

m y x

m

mm

1 1 22

1 or 1.m m m

m

Section 1.1 – Lines

You try: Write an equation of the line passing through the point (1, 3) that is (a) parallel, and (b) perpendicular to each given line L.

1. : 4

2. : 4 5

L y x

L y x

Equations of Lines

( )

( )

The vertical line through the point , has equation

since every -coordinate on the line has the same value .

Similarly, the horizontal line through , has equation .

a b x=a

x a

a b y b=

Example Equations of Lines

Write the equations of the vertical and horizontal lines through

the point ( 3,8).-

3 and 8x y =- =

Section 1.1Forms of Equations of Lines

Point-slope Equation: The equation

is the point-slope equation of the line through the point with slope m.

Example: Find the equation of the line that contains the points (1, 2) and (3, 5).

11 xxmyy

11, yx

5

3 2

2

1

3m

3

22 1y x

Point Slope Equation

1

1

The equation ) is the of the line

through the point ( , ) with slope .1

1

y=m(x x y

x y m

point - slope equation- +

Example: Point Slope Equation

( ) ( )Write the point-slope equation for the line through 7, -2 and -5, 8 .

1 1

8 ( 2) 10 10 5The line's slope is = = =

5 (7) 12 12 6

We can use this slope with either of the two given points in the point-slope

equation. For ( , ) = (7, 2) we obtain

5= ( )

6

=

7 2

m

x y

y x

y

- -= - -

- - -

-

- - +-

5 35 2

6 65 23

= 6 6

x

y x

- + -

- +

Section 1.1 – Lines

You try: Write the point-slope equation for the line that passes through the given point with the given slope.

21. 3, 2

3

2. -2, 11 5

33. 0, 2

11

m

m

m

Equations of Lines

The -coordinate of the point where a non-vertical line

intersects the -axis is the -intercept of the line.

Similarly, the -coordinate of the point where a non-horizontal

line intersects the -axis

y

y y

x

x

( )

is the -intercept of the line.

A line with slope and -intercept passes through 0, so( 0) , or

x

m y b by m x b y m x b= - + = +

Section 1.1Forms of Equations of Lines

Slope-intercept Equation: The equationy = mx + b

is the slope-intercept equation of the line with slope m and y-intercept b.

Example: Find the equation of the line with slope 2 and y-intercept (0, 5).

y = 2x + 5

Slope-Intercept Equation

The equation is the of the line

with slope and -intercept

y=m x + b

m y b.

slope - intercept equation

Section 1.1 – Lines

You try: Write the slope-intercept equation of the line passing through each pair of points.

1. 2, 15 and 0, 9

2. 1,6 and 3,16

General Linear Equation

The equation ( and not both 0)

is a in and .

Ax By = C A B

x y

+

general linear equation

Although the general linear form helps in the quick identification of lines, the slope-intercept form is the one to enter into a calculator for graphing.

Section 1.1Forms of Equation of Lines

Standard Linear Equation: The equationAx + By + C = 0

is a standard linear equation in x and y.

Section 1.1Forms of Equations of Lines

Equation of a Vertical line: The equation of a vertical line going through the point (a, b) is

x = a.

Equation of an Horizontal line: The equation of an horizontal line going through the point (a, b) is

y = b.

Intercept Equation: The equation of the line with x-intercept (a, 0) and y-intercept (0, b) is

1b

y

a

x

Example Analyzing and Graphing a General Linear Equation

Find the slope and y-intercept of the line 2 3 15. Graph the line. x y = -

Solve the equation for to put the equation in slope-intercept form:

3 = 2 15

2 15 =

3 32

= 53

y

y x

y x

y x

- - +

-+

- -

-

[-10, 10] by [-10, 10]

Example Determining a Function

( ) ( ) ( )The graph of is a line. We know from the table that the following points are on

the line: 1, 1 , 1,5 , 3,11

11 5 6Using the last two points the slope is: = = = 3

3 1 2So = 3 Because f

f

m m

f(x) x b.

- -

--

+ (1) = 5, we have

( ) 3( )

5 = 3

= 2 Thus, 3, 2 and (

1 1

) 3 2

f b

b

b m b f x x

= +

+

= = = +

The following table gives values for the linear function ( ) .

Determine and .

f x mx b

m b

= +

x f(x)

-1 -1

1 5

3 11

Section 1.1 – Lines

You try: The table below gives the values for a linear function. Determine m and b.

2 6

3 3

5 3

x f x

Section 1.1 – Lines

You try: The table below gives the values for a linear function. Determine m and b.

3 8

2 7

1 4

x f x

Example Reimbursed Expenses

Because we know that the relationship is linear, we know that it conforms to the

equation ( ) .34 150.

If a sales representative drives 137 miles, then 137. Thus,

( ) .34( ) 150

( ) 46.58

137 137

13 17

C x x

x

C

C

= +

=

= +

= + 50

( ) 196.58

It will cost the company $196.58 for a sales representative to drive 137 miles a .

1

y

37

da

C =

A company reimburses its sales representatives $150 per day for lodging and

meals plus $0.34 per mile driven.

Write a linear equation giving the daily cost to the company in terms of ,

the number o

C x

f miles driven.

How much does it cost the company if a sales representative drives 137 miles on

a given day?

Section 1.1Regression Analysis

Regression analysis is the process of finding an equation to fit a set of data. This allows us to:

1. Summarize the data with a simple expression, and

2. Predict values of y for other values of x.

Section 1.1Regression Analysis

On the TI-83/TI-83 Plus/TI-84 Calculator1. STAT, EDIT enter x’s in List1 and y’s in List2.2. STAT PLOT turn Plot1 ON, choose scatter plot.3. Choose an appropriate window to match your data.4. GRAPH and look at the data points.5. Decide what type of equation best fits your data

(linear, quadratic, exponential, trigonometric, power, etc.).

6. STAT, CALC, type of regression equation, L1, L2, VARS, Y-VARS, FUNCTION, Y1 (this pastes the regression equation into y1).

7. GRAPH and see how close the regression curve is to your data points. If it appears too far off, choose a different type of equation and repeat steps 5-7.

Section 1.1Regression Analysis

On the TI-89 Calculator:1. { x values separated by commas } STO ALPHA L 1.2. { y values separated by commas } STO ALPHA L 2.3. Y= arrow up to Plot1, ENTER4. Choose: Scatter, Box, x … L1, y … L2 ENTER5. Choose appropriate window for your data.6. Graph and look at data points.7. Decide what type of equation best fits your data. QUIT8. CATALOG type of regression equation, L1, L2 ENTER9. CATALOG SHOW STAT10. Enter equation shown in Y111. GRAPH and see how close the regression curve is to your

data points. If it appears too far off, choose a different type of equation and repeat steps 7-11.