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DRILL. Given each table write an equation to find “y” in terms of x. 2) Find the value of x:. Chapter 5 Analyzing Linear Equations. 5-1 Patterns and Slope Connection. Linear Patterns. - PowerPoint PPT Presentation
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DRILLDRILL1) Given each table write an equation
to find “y” in terms of x.
2) Find the value of x:
X 1 2 3 4
Y 7 11 15 19
x
10
6
4
Chapter 5Chapter 5Analyzing Linear EquationsAnalyzing Linear Equations
5-1 5-1
Patterns and Slope ConnectionPatterns and Slope Connection
Linear PatternsLinear Patterns
• In order for a pattern to be linear the common difference in “y” divided by the common difference in “x” must be the same for all given values.
DRILLDRILL
• Is this pattern linear? Why/Why Not
• Solve for x:
X 1 3 6 10
Y 5 13 25 41
208
12 x
x
18
5
3
SlopeSlope
• Slope is defined in numerous ways some of which are:
1) 2) Change in “y”
Change in “x”
3)
run
rise
x
ym
Slope FormulaSlope Formula
The formula for finding slope is:
Where the coordinates of two points are (x1, y1) and (x2, y2)
12
12
xx
yym
Types of SlopesTypes of Slopes
+ _
0
Undefined
DRILLDRILL
• Find the Slope Given the following points:
a) (2, 5) and (4, 13)
b) (-3, 3) and (7, -2)
c) (-4, 0) and (4, 24)
d) (5, 7) and (5, 13)
7.4 SlopeObjectives: •To find the slope of a line given two points on the line•To describe slope for horizontal and vertical lines
Slope
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
rise
runSlope =riserun
Example 1Graph the line containing points (2,1) and (7,6) and find the slope.
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
rise
runSlope =riserun
55
1
Practice
1) (-2,3) (3,5)
Graph the line containing these points and find their slopes.
2) (0,-3) (-3,2)
Slope
Slope =riserun
change in y coordinateschange in x coordinates
2 1
2 1
y ym
x x
1 2
1 2
y ym
x x
or
Example 2Find the slope of the line containing points (1,6) and (5,4).
2 1
2 1
y ym
x x
4 6m
5 1
2m
4
1m
2
Practice
1) (2,2) (8,9)
Find the slope of the lines containing these points.
2) (-2,3) (2,1)
3) (5,-11) (-9,4)
Slope of a Horizontal Line
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
What about the slope of a horizontal line?
What is the rise?
What is the run?
0
6
Slope =riserun
06
0
The slope of ANY horizontal line is 0.
Slope of a Vertical Line
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
What about the slope of a vertical line?
What is the rise?
What is the run?
5
0
Slope =riserun
50
The slope of ANY vertical line is undefined.
Example 1
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
Find the slope of the line y = -4.
Slope =riserun
05
0
Example 2
-4 -2
2
42 6 8 10
4
6
-4
-6
-8
-2
-10
10
8
Find the slope of the line x = 7.
Slope =riserun
30
The line has no slope.
Practice
1) (9,7) (3,7)
Find the slopes, if they exist, of the lines containing these points.
2) (4,-6) (4,0)
3) (2,4) (-1,5)
Warm-UpGraph. Then find the slope.
5 minutes
1) y = 3x + 22) y = -2x +5
7.5.1 Equations and SlopeObjectives: •To find the slope and y-intercept of a line from an equation
Find the slope of the line y = 2x - 4.
Slope-Intercept Equation
y = 2x - 4
x y
-4-20
012
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
rise: -4
run: -2rise
sloperun
42
2
The y-intercept is -4.
y = mx + b
Slope-Intercept Equation
slope y-intercept
y = 4x + 8
slope y-intercept= 4
= 8
y = 2x - 3
slope y-intercept= 2
= -3
Find the slope and y-intercept of the line y = -4x + 4.
Example 1
The slope is -4.The y-intercept is 4.
Find the slope and y-intercept of the line y = 5x - 7.
Example 2
The slope is 5.The y-intercept is -7.
Practice
1) y = x + 3
2) y = -4x – 7
3) y = 3x - 9
Find the slope and y-intercept of each line.
Find the slope and y-intercept of the line 3x + 4y = 12.
Example 3
3x + 4y = 12
solve for y-3x -3x
4y = 12 – 3x
4 4
3y 3 x
4
3y x 3
4
slope y-intercept= 334
Practice
1) y = -x - 3
Find the slope and y-intercept of each line.
2) 8x + 2y = 10
3) 3y – 6x = 12
Homework
p.326 #1-10,19-27 odds
Warm-UpFind the slope and y-intercept.
4 minutes
1) y = 3x + 4
2) y = -2x - 8
3) 2x + 7y = 9
4) 4x = 9y + 7
7.5.2 Equations and SlopeObjectives: •To graph lines using the slope-intercept equation
Graph y = 3x + 2.
Example 1
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
riseslope
run 3
1
m 3
What is the y-intercept?
What is the slope of this line?
b = 2
Graph
Example 2
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
riseslope
run 4
7
4m
7
What is the y-intercept?
What is the slope of this line?
b = -3
4y x 3
7
Practice
1) y = 3x - 5
Graph each line.
2) y = -2x + 4
3) 3
y x 35
Graph 3x + 4y = 12.Example 3
3x + 4y = 12
solve for y-3x -3x
4y = -3x + 12
4 43
y x 34
Graph 3x + 4y = 12.Example 3
3y x 3
4
slopey-intercept
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
Practice
1) 7x + 2y = 4
Graph each line.
2) 5y – 10 = 4x
Homework
p.326 #29,31,37,39
*Use graph paper for the graphs
Warm-Up1. Find the slope of the line containing the points (-2,5) and (4,6).
6 minutes
2. Find the slope of the line y = x – 9.
3. Find the slope of the line 3y – 4x = 9.
7.6.1 Finding an Equation of a LineObjectives: •To write an equation of a line using the slope-intercept equation
The Slope-Intercept Equation
y = mx + b
slopey-intercept
Create an equation of a line with a slope of -3 and a y-intercept of 4.y = -3x +
4y = 4 – 3x3x = 4 - y-4 = -y – 3x
Example 1Write an equation for the line with slope 3 that contains the point (-2,4)
y = mx + b4 =3(-2) +
b4 = -6 + b+6
+610 = b
y = 3x + 10
substitute
solve for b
simplify
Practice
1) (5,10); m = 4
Write an equation for the given line that contains the given point and has the given slope.
2) (-3,8); m = 2
Example 2Write an equation for the line containing the points (1,5) and (2,8).
change in y-coordinatesm
change in x-coordinates
2 1
2 1
y ym
x x
8 5m
2 1
3m 3
1
Example 2Write an equation for the line containing the points (1,5) and (2,8).
m 3
y = mx + b5 =3(1)+
b5 = 3 + b-3 -32 = b
y = 3x + 2
substitute
simplify
Practice
1) (-4,1) (-1,4)
Write an equation for the line that contains the given points.
2) (-3,5) (-1,-3)
Homework
p.331 #3,5,7,15,19
Warm-Up1. Write an equation for the line with slope -2 and containing the point (-3,0).
6 minutes
2. Write an equation for the line containing the points (0,0) and (4,2).
7.6.2 Finding an Equation of a LineObjectives: •To write an equation of a line using the point-slope equation
The Point-Slope Equation
y – y1 = m(x – x1)
Create an equation of a line with a slope of -3 that contains the point (7,2).
y – 2 = -3(x – 7)
y – 2 = -3x + 21+2
+2y = -3x + 23
Example 1Write an equation for the line with slope 7 that contains the point (3,4).
y – y1 = m(x – x1)y – 4 = 7(x – 3)
y – 4 = 7x - 21+4
+4y = 7x - 17
Practice
1) (-3,0), m = -3
Write an equation for the line with the given point and slope.
2) (4,3), m = ¾
Example 2Write an equation for the line containing (5,7) and (2,1).
y – y1 = m(x – x1)
First, find the slope: 2 1
2 1
y ym
x x
1 7m
2 5
63
2
y – 7 = 2(x – 5)
y – 7 = 2x - 10+7
+7y = 2x - 3
Practice
1) (12,16) (1,5)
Write an equation for a line containing the following points.
2) (-3,5) (-1,-3)
Example 3Write an equation for the line shown below.
-4 -2
2
42
4
-4
-2
First, find any two points on the line.
(-3,-3) and
(1,-1)
2m
4
12
y – y1 = m(x – x1) y + 3 = ½(x +
3)y + 3 = ½x + 1½ -3 -3
y = ½x – 1½
Homework
p.331 #9,13,21,23,25
Quiz Tomorrow
Warm-Up1. Graph the line y = 3x + 4.
5 minutes
2. Graph the line y = 3x - 2
3. What is the slope of the lines in the equations above?
7.8.1 Parallel and Perpendicular LinesObjectives: •To determine whether the graphs of two equations are parallel
Parallel Lines
Parallel lines are lines in the same plane that never intersect.
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
Parallel lines have the same slope.
Example 1Determine whether these lines are parallel.y = 4x -6
and y = 4x + 2
The slope of both lines is 4.
So, the lines are parallel.
Example 2Determine whether these lines are parallel.y – 2 = 5x + 4
and -15x + 3y = 9+2 +2
y = 5x + 6
+15x +15x 3y = 9 +
15x3 3
y = 3 + 5x
y = 5x + 3
The lines have the same slope.So they are parallel.
Example 3Determine whether these lines are parallel.y = -4x + 2 and -5 = -2y + 8x
+2y + 2y2y - 5 = 8x
+5 +52y = 8x + 5
2 2
5y 4x
2
Since these lines have different slopes, they are not parallel.
Practice
2) 3x – y = -5 and 5y – 15x = 10
Determine whether the graphs are parallel lines.
3) 4y = -12x + 16 and y = 3x + 4
1) y = -5x – 8 and y = 5x + 2
Example 4Write the slope-intercept form of the equation of the line passing through the point (1, –6) and parallel to the line y = -5x + 3.
slope of new line =
-5
y – y1 = m(x – x1)
y – (-6) = -5(x – 1)y + 6 = -5x +
5 y = -5x - 1
PracticeWrite the slope-intercept form of the equation of the line passing through the point (0,2) and parallel to the line 3y – x = 0.
Homework
p.340 #3-11 odds
Warm-UpDetermine whether the graphs of the equations are parallel lines.
4 minutes
1) 3x – 4 = y and y – 3x = 8
2) y = -4x + 2 and -5 = -2y + 8x
7.8.2 Parallel and Perpendicular LinesObjectives: •To determine whether the graphs of two equations are perpendicular
Perpendicular Lines
Perpendicular lines are lines that intersect to form a 900 angle.
-8 -6 -4 -2
2
42 6 8
4
6
-4
-6
-8
-2
8
The product of the slopes of perpendicular lines is -1.
4m 2
2
2 1m
4 2
Example 1Determine whether these lines are perpendicular.
and y = -3x - 21
y x 73
1m
3 m = -3
13
3 1
Since the product of the slopes is -1, the lines are perpendicular.
Example 2Determine whether these lines are perpendicular.
and y = -5x - 2
m 5 m = -5
5 5 25
Since the product of the slopes is not -1, the lines are not perpendicular.
y = 5x + 7
PracticeDetermine whether these lines are perpendicular.
1) 2y – x = 2 and y = -2x + 4
2) 4y = 3x + 12 and -3x + 4y – 2 = 0
Example 3Write an equation for the line containing (-3,-5) and perpendicular to the line y = 2x + 1.First, we need the slope of the line y = 2x + 1.
m = 2Second, we need to find out the slope of the line that is perpendicular to y = 2x + 1. 1
m2
Lastly, we use the point-slope formula to find our equation.
1 1(y y ) m(x x ) 1
(y 5) (x 3)2
1y 5 (x 3)
2
PracticeWrite an equation for the line containing the given point and perpendicular to the given line.1) (0,0); y = 2x + 4
2) (-1,-3); x + 2y = 8
Homework
p.340 #13,15,21,27,29,31,33
DRILLDRILL
1) What is the equation of a line containing the two points (2, 4) and (5, 16)?
2) What is the slope and y-intercept of the line y = -3x – 8 ?
3) Write the equation of a line given the slope is ½ and the y-intercept is -5.
PGCC CHM 103 Sinex
The Best-Fit Line
Linear Regression
Age (months)
Height (inches)
18 76.1
19 77
20 78.1
21
22 78.8
23 79.7
24 79.9
25 81.1
26 81.2
27 82.8
28
29 83.5
Work with your group to make a prediction for the height at:
• 21 months
• 28 months
• 20 years
Line of Best Fit• Definition - A Line of Best is a straight line on a
Scatterplot that comes closest to all of the dots on the graph.
• A Line of Best Fit does not touch all of the dots.• A Line of Best Fit is useful because it allows us
to:– Understand the type and strength of the relationship
between two sets of data– Predict missing Y values for given X values, or
missing X values for given Y values
Equation For Line of Best Fit
y = 0.6618x + 64.399
X (months) Formula Y (inches)
21 0.6618(21) + 64.399
28 0.6618(28) + 64.399
240 0.6618(240) + 64.399
78.3
82.9
223.3
Predicting Data with Scatterplots
• Interpretation - Making a prediction for an unknown Y value based on a given X value within a range of known data
• Extrapolation - Making a prediction for an unknown Y value based on a given X value outside of a range of known data
• More accurate: Interpretation• Less accurate: Extrapolation
How do you determine the best-fit line through data points?
x-variable
y-variable Fortunately technology, such as the graphing calculatorand Excel, can do a betterjob than your eye and a ruler!
The Equation of a Straight Line
y = mx + b
where m is the slope or y/x and b is the y-intercept
In some physical settings, b = 0 so the equation simplifies to:
y = mx
x-variable
y-variable
deviation = residual= ydata point – yequation
Linear regression minimizes the sum of the squared deviations
y = mx + b
Linear Regression
• Minimizes the sum of the square of the deviations for all the points and the best-fit line
• Judge the goodness of fit with r2
• r2 x100 tells you the percent of the variation of the y-variable that is explained by the variation of the x-variable (a perfect fit has r2 = 1)
x-variable
y-variable
Goodness of Fit: Using r2
r2 is high
r2 is low
How about the value of r2?
y = 2.0555x - 0.1682
R2 = 0.9909
0
5
10
15
20
25
0 2 4 6 8 10
x-variable
y-v
aria
ble
Strong direct relationship
99.1% of the y-variation is due tothe variation of the x-variable
y = -2.2182x + 25
R2 = 0.8239
05
1015
202530
0 2 4 6 8 10
x-variable
y-v
aria
ble
Noisy indirect relationship
Only 82% of the y-variation is due tothe variation of the x-variable - whatis the other 18% caused by?
R2 = 0.0285
0
5
10
15
20
0 2 4 6 8 10
x-variable
y-v
ari
able
When there is no trend!
No relationship!
In Excel
• When the chart is active, go to chart, and select Add Trendline, choose the type and on option select display equation and display r2
• For calibration curves, select the set intercept = 0 option Does this make physical sense?
Calibration Curvey = 0.8461x + 0.0287
R2 = 0.9954y = 0.8888x
R2 = 0.99110
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1concentration
abso
rban
ce
Using the set intercept = 0 option lowers the r2 value by a small amount
and changes the slope slightly
Does the set intercept = 0 option make a difference?
The equation becomes
A = mc
or
A = 0.89c
99.1% of the variation of the absorbance is due to the variation of the concentration.
DrillDrill1) Write the equation of a line that
passes through the two points (-2, 4) and (4, 7).
2) Find the value of “y” when “x” is 12, if y = 14 when x = 2.
3) What is the equation of a line that passes through the point (-9, 4) and has a slope of 1/3?
EECCRR
1) Find the equation for the line of best fit.
2) What is the slope of the line and what does it mean?
3) Predict the distance from home after 8 hours.
Hours Driven
Distance From House
1 42
2 58
3 78
4 90
5 109
