section 2.3 linear functions and slopes

Section 2.3Linear Functions and Slopes

The Slope of a Line

change in y 5 1 6 6

change in x 2 3 5 5m or

Find the slope of the line that passes

through (-2,5) and (3,-1)

Example

Find the slope of the line passing through the pair of points. (5,-2) and (-1,7)

The Point-Slope Form

of the Equation of a Line

1 1

Write the point-slope form of the equation of the

line with slope of 3 that passes through (-1,2).

Substitute into the point-slope form; y-y ( )

2 3( 1)

2 3( 1)

m x x

y x

y x

Solving in both forms

• A.Write the equation in point slope form of the line with slope 4 that passes through the point (4,-3). B.Then solve the equation for y

y1x1

y-(-3) = 4(x-4) Substituting the values into the euation

y+3 = 4(x-4)This is Point Slope Form. Apply the distributive property for the parentheses. This will give us the slope intercept form. (The equation is solved for y.)

-3 -3

y= 4(x-4)-3-3y= 4x-16-3

• y-y1 = m(x-x1)

(slope intercept form)

Y=4x-19

Example

Write the point slope form of the equation of the line with slope of -4 that passes through (2,5). Then solve for y.

If you are given two points

and you need to write an

equation in point-slope

form, then you can use

either point for (x1,y1).

2 1

2 1

Write the point-slope form of the equation of the

line that passes through (-1,2) and (-4,5). Then

solve for y.

5 2 3First: Find the slope. 1

4 1 3

Second: Substitue into the point-slope fo

y y

x x

1 1

rm.

y-y ( )

5 1( 4)

Third: Solve for y.

5 1( 4)

5 1 4

y=-1x+1

m x x

y x

y x

y x

Example

Write the point slope form of the equation of the line that passes through (2,5) and (-1,0). Then solve for y.

The Slope-Intercept Form

of the Equation of a Line

Point Slope FormFor a nonvertical line with slope m that passes through (x1,y1) the equation is

y-y1 = m(x-x1)

Example: slope = -3

point on the line(-1,-2)

Y-(-2)= -3(x-(-1))

Y+2= -3(x+1)

Slope Intercept FormFor a nonvertical line with slope m and y-intercept b the equation is y=mx+b

Example: slope =2

y-intercept of 6

Y=2x +6

Two forms for Equations of Lines

Graph the linear equation y= 2/3x+4

x

y

First: Plot the y-intercept of 4Rise by 2 unitsRun ( go to the right) by 3 units.Plot the second point (3, 6)Connect the two points with a straight edge or ruler.

(0,4)

(3,6)

Example

Graph the linear equation y= -3x+5

x

y

Example

x

y

1Graph the linear equation y= 3

2x

Equations of Horizontal

and Vertical Lines

Example Graph x=4. Graph y=-2

x

y

The General Form

of the Equation of a Line

Find the slope and the y intercept of the line.

4 5 20 0

4 5 20

-5y=-4x-20

-5y 4 20

-5 5 54

y= 45

x y

x y

x

x

Y intercept

slope

Example

Find the slope and the y intercept of the line whose equation is 2x+5y-10=0.

Using Intercepts to Graph

Ax + By + C = 0

Find x and y intercepts to graph a line 6x-2y=12

X intercept so let y=0 Y intercept so let x=000

6x-2(0)=12

6x=12

X=2(2,0)

Y=-6(0,-6)

-2y=12

x

y

6(0)-2y=12

X intercept - Let y=0

4x-3 0 6 0

4x-6=0

4x=6

6 3 x=

4 23

,02

Y-intercept - Let x=0

4 0-3y-6=0

-3y-6=0

-3y=6

6 y= 2

-3 0, 2

Example

Find the x and y intercepts then graph using those points.

X-4y-8=0

x

y

Summary

Applications

The graph gives the median age of the

US population in the indicated year. The

data is displayed as a scatterplot with two

points on the line indicated. Find the

equation of the line, in order to

make predictions of the US

population in the future.

Now we will use the equation to predict

the median age of the US population in 2010.

That means we will substitute in 40 for the x.

The reason we use 40 is the initial date was 1970.

If we add 40 to 1970 we will get 2010.

y=0.265x+27.35

y=0.265(40)+27.35

y=37.95

This means that the median age of the US population

will be 37.95 in 2010.

Example

Diameter 8 10 12 16

Price 6.40 8.00 9.60 12.80

The local pizza shop has a special sale on pizzas. Write

the slope-intercept equation of the line that describes the

price as a function of the diameter of the pizza.

If this company decides to make an 18 inch pizza, how

much should they charge?

Graphing Calculator-Linear Regression

D $

8 6.40

10 8.00

12 9.60

16 12.8

Take the data from the previous pizza problem.

Put the data into List1 & List2 in the graphing

calculator.

To do that Press STAT,

then 1 for Edit. Type in

the numbers.

Press STAT, move the

cursor to the right to CALC,

then press 4 for LinReg.

The next screen gives you the values of

a and b for the equation.

The equation is y=.8x.More on the next slide.

To see the scatterplot of the data, we need to

change the Window. Press WINDOW, and

type in what you see at left.

Press the GRAPH key. You will see the scatterplot

at left. The equation y=.8x will go through these

points. Press Y= and type in the equation. Press

GRAPH to see the line and scatterplot.

Graphing Calculator-Linear Regression continued

Press 2nd Y= to get STAT PLOT. First make

certain that all plots are off by pressing 4. Then

return to STAT PLOT and press Plot1. On the

word ON press ENTER.

Cursor down and press the appropriate

1

2

keys so

you get what you see in the picture at left. L is

obtained by pressing 2nd then 1. L - 2nd then 2.

(a)

(b)

(c)

(d)

Find the equation of the line in slope-intercept form for a line that passes through (0,-4) and has a slope of -2.

2 4

4 2

2 4

2 4

y x

y x

y x

y x

(a)

(b)

(c)

(d)

Find the equation of the line in slope-intercept form of the line that passes through (-3,-2) and (0,-2).

x 3

2

0

2 3

y

y

y x

(a)

(b)

(c)

(d)

What is the slope of the line 3x - 7y – 4 = 0.

7

37

44

73

7