Pre-Calculus Chapter 1

Functions and Their Graphs

1.1 Lines in the PlaneObjectives:

Find the slopes of lines. Write linear equations given points on

lines and their slopes. Use slope-intercept forms of linear

equations to sketch lines. Use slope to identify parallel and

perpendicular lines.

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Warm Up 1.1 – Use “ZOOM SQUARE”

For each set of

equations, use your

calculator to compare

the slopes of the

lines.

What do you

observe?

Set 1 Set 2

y = 0.5x

y = –0.5x

y = x y = –x

y = 2x y = – 2x

y = 4x y = – 4x

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VocabularySlope of a LinePoint-Slope Form of a LineLinear FunctionSlope-Intercept Form a LineGeneral Form of the Equation of a LineParallel LinesPerpendicular LinesLinear ExtrapolationLinear Interpolation

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Find the Slope and Equation of the Line

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(5, 19)

(10, 6)

-4 -2 2 4 6 8 10 12 14

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5

10

15

20

25

30

x

y

Definition of the Slope of a Line

Slope = Change in y Change in x

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Slope of a LineDraw a diagram of each:

A line with positive slope (m > 0)

______________ from left to right.

A line with negative slope (m < 0)

______________ from left to right.

A line with zero slope (m = 0) is

________________.

A line with undefined slope is

_________________.

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Point-Slope Form of Line

Let (x1, y1) be a point on

the line whose slope is m.

If (x, y) is any other point

on the line, then

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Point-Slope Equation

The point-slope form of the equation of the

line that passes through the point (x1, y1) and

has a slope of m is

y – y1 = m (x – x1)

This equation is used frequently in

calculus.9

Example 1: Application Problem

During 2000, Nike’s net sales were $9.0

billion, and in 2001 net sales were $9.5

billion. Write a linear equation giving the net

sales y in terms of the year x. Then use the

equation to predict net sales for 2002.

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Slope-Intercept Form

The graph of the equation y = mx + b is a

line whose slope is m and whose y-

intercept is (0, b).

This can be written as a Linear Function

f (x) = mx + b.

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Example 2 Determine the slope and y-intercept of each

linear equation. Then describe the graph of

each equation.

a. 3x + 4y = 1

b. y = 12

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General Form of Linear EquationA horizontal line (m = 0) has an equation

of the form y = b.

A vertical line has an equation of the form

x = a. Can this equation be written in slope-

intercept form? Why?

Any line can be written in General Form:

Ax + By + C = 0 or Ax + By = c13

Summary of Equations of Lines

1. General Form Ax + By + C = 0

2. Vertical Line x = a

3. Horizontal Line y = b

4. Slope-Intercept y = mx + b

5. Point-Slope y – y1 = m(x – x1)

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Parallel & Perpendicular LinesParallel Lines

Never intersect.

Slopes are equal: m1 = m2

Perpendicular Lines

Intersect at a right angle.

Slopes are negative reciprocals:

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Example 3Find the slope-intercept form of the

equation of the line that passes through

the point (2, –1) and is parallel to the line

2x – 3y = 5.

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Example 4Find the slope-intercept form of the

equation of the line that passes through

the point (2, –1) and is perependicular to

the line 2x – 3y = 5.

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Extrapolation vs. InterpolationLinear Extrapolation

Use the equation of a line to estimate a point outside the two given points.

Linear InterpolationUse the equation of a line to estimate a point between the two given points.

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Additional Example

The number of gallons of gas left in your

gas tank can be approximated by a linear

function of the number of miles you have

driven. Suppose you have driven 123 mi

since your last fill-up and you have 13 gal

left in your tank. Your car uses 0.05

gal/mi (or 20 mi/gal).19

Additional Example (cont.)a. Write an equation in point-slope form

relating the number of miles driven since you filled the tank and the number of gallons left in the tank. Rewrite the equation to express gallons as a function of miles.

b. Use the answer found in part “a” to determine how many gallons your car’s tank holds.

c. You want to fill up before the amount left drops below 1 gallon. How much farther can you drive before you must fill up again?

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Homework 1.1Worksheet 1.1

# 1, 19, 25, 29, 35, 43, 51,

55, 59, 65, 68, 71, 75 – 78

(matching), 83, 95, 96, 97

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