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Chapter 1: Linear Models

V. J. MottoM110 Modeling with Elementary Functions

1.3 Two Important Questions (Revisited)

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Overview

HousekeepingWho’s here today?Who needs materials?

Homework ProblemsQuestionsHomework

Two Important Questions (Revisited)Homework Assignment

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Example 1 --- Intercepts

Where does the graph cross the x-axis?

Where does the graph cross the y-axis?

These points are called the x-intercept and y-intercept.

Problems Set 1:

Let’s look at the following linear equations using inspection and our calculator to find x- and y-intercepts.

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. 3 5

. 0.4 3

3. 2

5

a y x

b y x

c y x

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Example 2 – Non-linear

This is a non-linear function.

X-intercepts: (-2, 0) (3, 0)

Y-intercept is: (0, -6)

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Finding x- and y-intercepts

To find the x-intercept, let y = 0 and solve for x.

To find the y-intercept, let x = 0 and solve for y.

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Example 3: Graphing x – 3y = 6

3 6

0.

3 6

3(0) 6

0 6

6

x y

Let y

x y

x

x

x

3 6

0.

3 6

0 3 6

3 6

2

x y

Let x

x y

y

y

y

Thus, (6, 0) is the x-intercept. Thus, (0, -2) is the y-intercept. We can use these two points to make a sketch of the graph.

Sketch the graph of x – 3y = 6 by plotting the x-interceptand the y-intercept.

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Example 3 (continued)

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Special Lines

Vertical LinesThe graph x = c, where c is a real number,

is a vertical line with x-intercept (c, 0).Horizontal Lines

The graph of y = c, where c is a real number, is a horizontal line with y-intercept (0, c)

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Example 4

y = 1 x = 1y = 2.5 x = 4y = 3 x = 5.5

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Slope

Lines can be slanted in many different ways.

We use slope to measure a lines slant. The green line has a big slope, because it is slanted sharply. Because the red line is close to flat, it has a small slope. These lines have positive slope.

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Negative Slope

Lines with negative slope point down instead of up. A line of negative slope is pictured at the right.

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Putting Slope in perspective

We can tell whether a line’s slope is big or small, and whether the slope is positive or negative.

How can we quantify this idea of slope so that we can compare slopes?

Let’s start with a definition of slope.

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The Definition of Slope --- 1

Slope is defined as the change in the y-coordinates divided by the change in the x-coordinates. People often remember this definition as “rise/run”

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The Definition of Slope --- 2

2 1

2 1

'

'

risem

runchangein y s

mchangein x s

y ym

x x

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Example 5

2 1

2 1

'

'

4 22

2 1

risem

runchangein y s

mchangein x s

y ym

x x

m

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Problem Set 2:

Find the slopes for the following pairs of points: (3, 2) and (-2, 5) (-4, 3) and ( -2, -4) (0, -4) and ( -3, -5)

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Slope

Which line has negative slope?

Which line has positive slope?

Which line has the greater slope?

Which line has no slope?

Which line has 0 slope?

Which line has the smaller slope?

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Example 6

Find the slope of the line that passes through the points (0, 1) and ( 3, 4).

2 1

2 1

'

'

4 1

3 03

31

11

rise difference in y sm

run difference in x s

y y

x x

or

Often it is best to think of slope as a fraction

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Comments on Slope

Vertical lines Points on these types of lines have the same x-

coordinate. Hence, their slope is undefined because we would

be dividing by 0 when we calculate it.

Horizontal lines Points on these types of lines have the same y-

coordinate. Hence, their slope is 0 because the numerator is 0.

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Parallel and Perpendicular Lines

Parallel lines have the same slope

Perpendicular lines have slope that are the negative reciprocal of each other: that is, the product of their slopes is -1 or m1*m2 = -1.

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Slope-Intercept Form

Linear equations of the formy = mx + b

have slope-intercept form.

The coefficient of the x term is the slope and the b value is the y-coordinate of the y-intercept which is (0, b).

We can easily read the slope and discover the coordinate for the y-intercept by inspection of the equation.

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Example 7

Find the slope and y-intercept for the following linear equations: 3 5

2 3

34

40.3 2.5

y x

y x

y x

y x

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Transposing

Often we are given the equation of a line in the standard form

Ax + By = C

Then we are asked to change it to the slope-intercept form

y = mx + b

This is easily accomplished by solving the standard form for the variable y.

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Example 3

Write 3x – 4y = 4 in slope-intercept form.

3 4 4

3 4 4

31

43

14

x y

x y

x y

y x

What is the slope for this line?What are the coordinates of they-intercept?

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Parallel or Perpendicular

Other types of problems will ask us to compare the slopes of two lines in order to decide if the lines are parallel (same slope) or perpendicular (the product of slopes is -1).

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Example 8

TThus, the line represented by these equations are parallel.

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Finding the equation

We explore in the next example the first type of “finding the linear equation” type of problems. Here we are given the slope and the y-intercept. Using the slope-intercept form, y = mx +b, renders these type of problems quickly.

We will explore other types in our next lecture.

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Example 9

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Finding equations

In the previous study we have developed strategies for finding the equation of a line. Given the slope, m, and the coordinates of the y-

intercept, (0, b) we can use the slope-intercept form, y = mx + b, to generate the equation of a line.

Here we explore some other situations. You are given: The slope and a point of the line Two points.

For these situations we use the point-slope form.

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Point-Slope Form

The point-slope form of the equation of a line is

y – y1 = m(x – x1)

where m is the slope of the line and

(x1, y1) is a point on the line.

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Example 10:

Find the equation of the line passing through (-1, 5) with slope -2.

Since the slope of the line is given, we know that m = -2. Thus, we have

y – y1 = -2(x – x1)

Now using the coordinates of the point (-1, 5), we have the following y – 5 = -2(x – (-1))which becomes y – 5 = -2(x + 1)

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Example 10 (continued)

If we want to express the equation in slope-intercept form, we have

y – 5 = - 2x - 2 y = - 2x + 3But if we want to express the equation in

standard form, we have y = - 2x + 3 2x + y = 3

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Two Points

Most often when we are looking for the equation of a line we are only given two points. Thus, the technique becomesFinding the slopeUsing the slope-point form

Our next example illustrates this situation.

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Example 11:

Find an equation of the line through (2, 5) and (-3, 4).

First we must find the slope.

2 1

2 1

'

'

4 5 1 1

3 2 5 5

differenceof y sm

differenceof x s

y ym

x x

m

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Example 11 (continued)

Now we use the point-slope form to generate the equation. We will use the point (2, 5). Thus

15 ( 2)

51 2

55 51 23

5 5

y x

y x

y x

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Example 11 (continued)

In the previous slide the equation is in slope-intercept form. What does the equation look like in standard form?

1 235 * 5( )

5 55 23

5 23

y x

y x

x y

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Special Lines

The equation of a vertical line can be written in the form x = c. You should recall that this line is parallel to y-axis

The equation of a horizontal line can be written y = c. You should recall that this line is parallel to the x-axis.

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Example 12

Find an equation of the vertical line through ( -3, 5).

The equation of a vertical line can be written in the form x = c, so an equation for a vertical line passing through (-3, 5) is x = -3.

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Example 12 (continued)

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Example 13:

Find an equation of the line parallel to the line y = 5 and passing through the point (-2, -3).

Since the graph of y = 5 is a horizontal line, any line parallel to it is also horizontal.

The equation of a horizontal line can be written in the form y = c.

An equation for the horizontal line passing through (-2, -3) is y = -3.

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Example 3 (continued)

Table of Values

1. Enter the linear function y = 3x – 5

2. Now press the graph key.

3. To see a table of values, press the 2nd key followed by the graph key table view.

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Table Functions

You can alter the setup values in the table by pressing the 2nd key followed by the Window key table set function

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Intersecting Lines

Consider the two equations:y = 3x – 5y = -2x + 6

The graph is shown at the right. You should observe that they intersect.

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Find the Intersection

Press 2nd Trace. Choose option

5:intersection You want to use the first

equation. You want to use the

second equation. Yes, you want the

calculator to “guess” or calculate the intersection

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Forms of Linear Equations

Form Description

Ax + By = C Standard Form.

A and B are not both 0

y = mx + b Slope-intercept Form.

Slope is m; y-intercept is (0, b)

y – y1 = m(x - x1) Point-slope Form

Slope is m; (x1, y1) is a point of the line

y = c Horizontal Line

Slope is 0; y-intercept is (0, c)

x = c Vertical Line

Slope is undefined; x-intercepet is (c, 0)