chapter 11: graphing lines

Chapter 11: Chapter 11: Graphing LinesGraphing Lines

Regular MathRegular Math

Section 11.1: Graphing Section 11.1: Graphing Linear EquationsLinear Equations

A A linear equation linear equation is an equation whose is an equation whose solutions fall on a line on the coordinate solutions fall on a line on the coordinate grid.grid.

A linear equation’s graph will always be a A linear equation’s graph will always be a straight line. straight line.

5 Steps to Graph any 5 Steps to Graph any EquationEquation

1.1. Choose a value for x.Choose a value for x.

2.2. Substitute the x-value into the equation, and find the Substitute the x-value into the equation, and find the corresponding y-values.corresponding y-values.

3.3. Form an ordered pair with the x-value and y-value.Form an ordered pair with the x-value and y-value.

4.4. Graph the ordered pair.Graph the ordered pair.

5.5. Repeat the process until you have at least 3 points.Repeat the process until you have at least 3 points. Remember – One point must be a negative point.Remember – One point must be a negative point.

Graphing EquationsGraphing Equations

Graph each equation and tell whether it is Graph each equation and tell whether it is linear.linear.

y = 2x – 3y = 2x – 3

y = x squaredy = x squared

y = 2/3 xy = 2/3 x

y = -3y = -3

Try these on your own…Try these on your own…

Graph each equation and tell whether is it Graph each equation and tell whether is it linear.linear. y = 3x -1y = 3x -1

LinearLinear

y = x cubedy = x cubed Not LinearNot Linear

y = -3/4 xy = -3/4 x LinearLinear

y = 2y = 2 LinearLinear

Sports ApplicationSports Application

In bowling, the equation In bowling, the equation h = 160 – 0.8s h = 160 – 0.8s represents the handicap represents the handicap (h) calculated for a (h) calculated for a bowler with average bowler with average score (s). How much will score (s). How much will the handicap be for each the handicap be for each bowler listed in the bowler listed in the table? Draw a graph that table? Draw a graph that represents the represents the relationship between the relationship between the average score and the average score and the handicap.handicap.

BowlerBowler Average Average ScoreScore

SandiSandi 145145

DominicDominic 125125

LeoLeo 160160

SheilaSheila 140140

TawanaTawana 175175

Try this one on your Try this one on your own…own…

A lift on a ski slope rises A lift on a ski slope rises according to the equation a = according to the equation a = 130t + 6250, where a is the 130t + 6250, where a is the altitude in feet and t is the altitude in feet and t is the minutes that a skier has been minutes that a skier has been on the life. Five friends are on on the life. Five friends are on the lift. What is the altitude of the lift. What is the altitude of each person if they have each person if they have been on the ski lift for the been on the ski lift for the times listed in the table? Draw times listed in the table? Draw a graph that represents the a graph that represents the relationships between the relationships between the time on the lift and the time on the lift and the altitude.altitude.

SkierSkier Time of Time of LiftLift

AnnaAnna 4 minutes4 minutes

TracyTracy 3 minutes3 minutes

KwaniKwani 2 minutes2 minutes

TonyTony 1.5 1.5 minutesminutes

GeorgeGeorge 1 minute1 minute

SkierSkier Time on LiftTime on Lift AltitudeAltitude

AnnaAnna 4 minutes4 minutes 6770 ft6770 ft

TracyTracy 3 minutes3 minutes 6640 ft6640 ft

KwaniKwani 2 minutes2 minutes 6510 ft6510 ft

TonyTony 1.5 minutes1.5 minutes 6445 ft6445 ft

GeorgeGeorge 1 minute1 minute 6380 ft6380 ft

Section 11.2: Slope of a Section 11.2: Slope of a LineLine

Finding Slope, Given To Finding Slope, Given To PointsPoints

Find the slope of the line that passes through Find the slope of the line that passes through (2,5) and (8,1).(2,5) and (8,1).

Try this one on your own…Try this one on your own… Find the slope of the line that passes through (-2, -Find the slope of the line that passes through (-2, -

3) and (4, 6).3) and (4, 6).

Finding Slope from a Finding Slope from a GraphGraph

Use the graph of the Use the graph of the line to determine its line to determine its slope. slope.


Use the graph of the Use the graph of the line to determine its line to determine its slope.slope.

Parallel and Parallel and Perpendicular SlopesPerpendicular Slopes

Parallel LinesParallel Lines have the same slope. have the same slope.

Perpendicular LinesPerpendicular Lines have complete have complete opposite slopes.opposite slopes.

Identifying Parallel and Identifying Parallel and Perpendicular Lines by Perpendicular Lines by SlopeSlope

Identifying Parallel Identifying Parallel and Perpendicular and Perpendicular Lines by SlopeLines by Slope

Line 1: (1,9) and (-1,5)Line 1: (1,9) and (-1,5)

Line 2: (-3, -5) and (4,9)Line 2: (-3, -5) and (4,9)

Line 1: (-10, 0) and Line 1: (-10, 0) and (20,6)(20,6)

Line 2: (-1, 4) and (2, -Line 2: (-1, 4) and (2, -11)11)

Graphing a Line Using a Graphing a Line Using a Point and the SlopePoint and the Slope

Graph the line Graph the line passing through (1,1) passing through (1,1) with slope -1/3.with slope -1/3.

Graph the line Graph the line passing through (3,1) passing through (3,1) with slope 2.with slope 2.

Section 11.3: Using Section 11.3: Using Slopes and InterceptsSlopes and Intercepts

The The x-interceptx-intercept of a line is the value of x of a line is the value of x where the line crosses the x-axis. (y = 0)where the line crosses the x-axis. (y = 0)

The The y-intercepty-intercept of a line is the value of y of a line is the value of y where the line crosses the y-axis. (x = 0)where the line crosses the y-axis. (x = 0)

Finding x-intercepts and y-Finding x-intercepts and y-intercepts to Graph Linear intercepts to Graph Linear EquationsEquations

Find the x-intercept and y-intercept of the Find the x-intercept and y-intercept of the line 2x + 3y = 6. Use the intercepts to line 2x + 3y = 6. Use the intercepts to graph the equations.graph the equations.

Step One: Solve for y.Step One: Solve for y. Step Two: Find the x – intercept and the y – Step Two: Find the x – intercept and the y –

intercept.intercept. Step Three: Graph.Step Three: Graph.


Find the x-intercept and y-intercept of the Find the x-intercept and y-intercept of the line 4x – 3y = 12. Use the intercepts to line 4x – 3y = 12. Use the intercepts to graph the equation.graph the equation.

Slope – Intercept FormSlope – Intercept Form

Slope – Intercept Form : y = mx + bSlope – Intercept Form : y = mx + b

m = slopem = slope b = y-interceptb = y-intercept

Notice that y is all by itself on one side and Notice that y is all by itself on one side and everything else is on the other.everything else is on the other.

Using Slope-Intercept Form Using Slope-Intercept Form to Find Slope and y-to Find Slope and y-interceptintercept

Write each equation in slope-intercept Write each equation in slope-intercept form, and then find the slope and y-form, and then find the slope and y-intercept.intercept. y = xy = x

7x = 3y7x = 3y

2x + 5y = 82x + 5y = 8


Write each equation in slope-intercept Write each equation in slope-intercept form, and then find the slope and the y-form, and then find the slope and the y-intercept.intercept.

2x + y = 32x + y = 3

5y = 3x5y = 3x

Entertainment Entertainment ApplicationApplication

An arcade deducts 3.5 points from your An arcade deducts 3.5 points from your 50-point game card for each Skittle-ball 50-point game card for each Skittle-ball game you play. The linear equation game you play. The linear equation y = -3.5x + 50 represents the number of y = -3.5x + 50 represents the number of points (y) on your card after (x) games. points (y) on your card after (x) games. Graph the equation using the slope and Graph the equation using the slope and y-intercept.y-intercept.


A video club charges $8 to join, and A video club charges $8 to join, and $1.25 for each DVD that is rented. The $1.25 for each DVD that is rented. The linear equation y = 1.25x + 8 represents linear equation y = 1.25x + 8 represents the amount of money (y) spent after the amount of money (y) spent after renting (x) DVDs. Graph the equation renting (x) DVDs. Graph the equation using the slope and y – intercept.using the slope and y – intercept.

Writing Slope-Intercept Writing Slope-Intercept FormForm

Write the equation of the line that passes Write the equation of the line that passes through (-3,1) and (2, -1) in slope-through (-3,1) and (2, -1) in slope-intercept form.intercept form.

Try this one on your own…Try this one on your own… Write the equation of the line that passes Write the equation of the line that passes

through (3, -4) and (-1,4) in slope-intercept through (3, -4) and (-1,4) in slope-intercept form.form.

Section 11.4: Point-Slope Section 11.4: Point-Slope FormForm

The point-slope form of an equation of a The point-slope form of an equation of a line with slope (m) passing through line with slope (m) passing through (x1,y1) is y – y1 = m (x – x1).(x1,y1) is y – y1 = m (x – x1).

Use Point-Slope Form to Use Point-Slope Form to Identify Information About a Identify Information About a LineLine

Use the point-slope form of each equation Use the point-slope form of each equation to identify a point the line passes through to identify a point the line passes through and the slope of the line.and the slope of the line.

y – 9 = -2/3 (x - 21)y – 9 = -2/3 (x - 21) m = -2/3m = -2/3 Point = (21, 9)Point = (21, 9)

y – 3 = 4 (x + 7)y – 3 = 4 (x + 7) m = 4m = 4 Point = (-7, 3)Point = (-7, 3)


Use the point-slope form of each Use the point-slope form of each equation to identify a point the line equation to identify a point the line passes through and the slope of the line.passes through and the slope of the line. Y – 7 = 3 (x – 4)Y – 7 = 3 (x – 4)

m = 3m = 3 Point = (4,7)Point = (4,7)

Y – 1 = 1/3 ( x + 6)Y – 1 = 1/3 ( x + 6) m = 1/3m = 1/3 Point = (-6, 1)Point = (-6, 1)

Writing the Point-Slope Writing the Point-Slope Form of an EquationForm of an Equation

Write the point-slope form of the equation Write the point-slope form of the equation with the given slope that passes through with the given slope that passes through the indicated point.the indicated point.

the line with slope -2 passing through (4,1)the line with slope -2 passing through (4,1)

the line with slope 7 passing through (-1,3)the line with slope 7 passing through (-1,3)


Write the point-slope form of the equation Write the point-slope form of the equation with the given slope that passes through with the given slope that passes through the indicated point.the indicated point.

the line with slope 4 passing through (5, -2)the line with slope 4 passing through (5, -2) y + 2 = 4 (x – 5)y + 2 = 4 (x – 5)

the line with slope -5 passing through (-3, 7)the line with slope -5 passing through (-3, 7) y – 7 = -5 (x + 3)y – 7 = -5 (x + 3)

Medical ApplicationMedical Application

Suppose that laser eye surgery is Suppose that laser eye surgery is modeled on a coordinate grid. The laser modeled on a coordinate grid. The laser is positioned at the y-intercept so that the is positioned at the y-intercept so that the light shifts down 1 mm for each 40 mm it light shifts down 1 mm for each 40 mm it shifts to the right. The light reaches the shifts to the right. The light reaches the center of the cornea of the eye at (125,0). center of the cornea of the eye at (125,0). Write the equation of the light beam in Write the equation of the light beam in point-slope form, and find the height of point-slope form, and find the height of the laser.the laser.


A roller coaster starts by ascending 20 feet A roller coaster starts by ascending 20 feet for every 30 feet in moves forward. The for every 30 feet in moves forward. The coaster starts at a point 18 feet above the coaster starts at a point 18 feet above the ground. Write the equation of the line that ground. Write the equation of the line that the roller coaster travels along in point-slope the roller coaster travels along in point-slope form, and use it to determine the height of form, and use it to determine the height of the coaster after traveling 150 feet forward. the coaster after traveling 150 feet forward. Assume that the roller coaster travels in a Assume that the roller coaster travels in a straight ling for the first 150 feet.straight ling for the first 150 feet.

Section 11.5: Direct Variation

For direct variation, two variable quantities are related proportionally by a constant positive ratio. The ratio is called constant of proportionality.

Equation: y = kx k = constant

Determining Whether a Data Set Varies Directly

Determine whether the data set shows direct variation.

Shoe Sizes…US Size 7 8 9 10 11

European Size

39 41 43 44 45

Determine whether the data set shows direct variation. Distance Sound Travels at 20 degrees Celcius (m)

Time (s) 0 1 2 3 4

Distance (m)

0 350 700 1050 1400

Try these on your own…

Determine whether the data set shows direct variation. Adam’s Growth Chart

Distance Traveled by TrainTime (Min) 10 20 30 40

Distance (mi)

25 50 75 100

Age (mo) 3 6 9 12

Length (in.) 22 24 25 27

Finding Equations of Direct Variation

Find each equation of direct variation, given that y varies directly with x.

y is 52 when x is 4

x is 10 when y is 15

y is 15 when x is 2


Find each equation of direct variation, given that y varies directly with x.

y is 54 when x is 6

x is 12 when y is 15

y is 8 when x is 5

Story Problem…

Mrs. Perez has $4000 in a CD and $4000 in a money market account. The amount of interest she has earned since the beginning of the year is organized in the following table. Determine whether there is a direct variation between either data set and time. If so, find the equation of direct variation.

Time (mo) Interest in CD ($)

Interest in Money Market ($)

0 0 0

1 17 19

2 34 37

3 51 55

4 68 73

Section 11.6: Graphing Inequalities in Two Variables

When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality.

A boundary line is the set of points where the two sides of a two-variable linear inequality are equal.

Graphing Inequalities

Graph each inequality.

1243

1

1

xy

xy

xy


Graph each inequality.

652

12

1

xy

xy

xy

Science Application…

Solar powered rovers landing on Mars in 2004 will have a range of up to 330 feet per Martian day. Graph the relationship between the distance a rover can travel and the number of Martian days. Can a rover travel 3000 feet in 8 days?

Try this one on your own…

A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200 page screenplay in 30 days?

Section 11.7: Lines of Best Fit

To estimate the equation of a line of best fit: Find the mean of the x-coordinates and y-

coordinates. Create a new point. Draw a line through the new point that

appears to fit the data the best. Estimate the coordinates of another point on

the line. Find the equation of the line.

Finding a Line of Best Fit

X Y

2 4

4 8

5 7

1 3

3 4

8 8

6 5

7 9


X Y

4 4

7 5

3 2

8 6

8 7

6 4

Sports Application

Find a line of best fit for the women’s 3000-meter speed skating. Use the equation of the line to predict the winning time in 2006.

Let 1960 represent year 0.

Year Winning Time (minutes)

1964 5.25

1968 4.94

1972 4.87

1976 4.75

1980 4.54

1984 4.41

1988 4.20

1992 4.33

1994 4.29

1998 4.12

2002 3.96


Find a line of best fit for the Main Street Elementary annual softball toss. Use the equation of the line to predict the winning distance in 2006.

Let x = 0 represent the year 1990.

Year Distance (ft)

1990 98

1992 101

1994 103

1997 106

2002 107

chapter 11: graphing lines

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