Chapter 11: Graphing Lines
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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
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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
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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.
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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.
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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
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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.
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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)
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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)
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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.
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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
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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
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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
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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?
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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
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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
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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
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