Chapter 4 Analyzing Linear Equations. 4.1 Rate of Change & Slope Rate of Change- a ratio that...
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Transcript of Chapter 4 Analyzing Linear Equations. 4.1 Rate of Change & Slope Rate of Change- a ratio that...
![Page 1: Chapter 4 Analyzing Linear Equations. 4.1 Rate of Change & Slope Rate of Change- a ratio that describes, on average, how much one quantity changes compared.](https://reader036.fdocuments.in/reader036/viewer/2022072015/56649ebf5503460f94bca493/html5/thumbnails/1.jpg)
Chapter 4
Analyzing Linear Equations
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4.1 Rate of Change & Slope
• Rate of Change- a ratio that describes, on average, how much one quantity changes compared to another
Rate of change=
Change in y
Change in x
Ex:
Time
Walking(sec)
Distance
Walked(ft)
1 4
2 8
3 12
4 16
x y
+1
+1
+1
+4
+4
+4
Rate of change =
4
1
= 4 ft/sec
Negative slope Positive slope
Zero slope Undefined slope
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• Slope- the ratio of the change in y-coordinates over the change in x-coordinates for a line
5
4
5
4
41
62
m
Slope=rise
run
Slope:
m =y2 – y1
x2 – x1
Ex: Ex:Find the slope (4, 6) (-1, 2)x1 y1 x2 y2
11
10
65
55
m
Ex: Find the value of r so that the line through (10, r) and (3, 4) has a slope of -2/7 x1 y1 x2 y2
103
4
7
2
r
)4(7)7(2 rr72814
-28 -28-14= -7r/-7 /-7
r = 2Plug in the m and the coordinates for the slope
Simplify and solve for r
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4.2 Slope and Direct Variation
• Direct Variation- a proportional relationshipy = kx
*this represents a constant rate of change and k is the constant of variation
Ex:a. Name the constant of variation
b. Find the slope
c. Compare slope and the constant of variation. What do you notice?
k = -1/2
They are the same thing
2
1
4
2
04
02
m
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• Graph a direct Variation:
1. Write the slope as a ratio
2. Start at the origin (0, 0)
3. Move up and across according to the slope
4. Draw a line to connect the ordered pairs
1
4m
x
y
a. y = 4x b. y = -1/3x
Up 4 and right 13
1m
Down 1 and right 3
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4.3 Slope-Intercept Form
y = mx + b
slope y-intercept
Ex: Write an equation in slope-intercept form given slope and the y-intercept
Slope = 3 y-intercept = -5
y = 3x - 5
a. b. y-intercept= (0, 6) m = 2/5
y = 2/5x + 6
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Ex:
Ex:
Write an equation in slope-intercept form given a graph.
Graph an equation given in slope-intercept form
Graph from an equation:
1. write the slope as a ratio and y-intercept as an ordered pair
2. Graph the y-intercept
3. Use slope to find other points and connect with a line3
5
30
32
m
y-intercept= (0, 2)
So b=2
y = 5/3x + 2
x
y
a. y = -2/3x + 1
b. 5x – 3y = 6
m = -2/3 y-int= (0, 1)
5x – 3y = 6
-5x -5x
-3y = -5x + 6
/-3 /-3
y = 5/3x - 2 m = 5/3 y-int= (0, -2)
![Page 8: Chapter 4 Analyzing Linear Equations. 4.1 Rate of Change & Slope Rate of Change- a ratio that describes, on average, how much one quantity changes compared.](https://reader036.fdocuments.in/reader036/viewer/2022072015/56649ebf5503460f94bca493/html5/thumbnails/8.jpg)
6.7 Graphing Inequalities with Two Variables
• The equation makes the line to define the boundary• The shaded region is the half-plane
1. Get the equation into slope-intercept form2. List the intercept as an ordered-pair and the slope as a
ratio3. Graph the intercept and use the slope to find at least 2
more points4. Draw the line (dotted or solid)5. Test an ordered-pair not on the line
1. If it is true shade that side of the line2. If it is false shade the other side of the line
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< or > or
Dotted Line Solid Line
Ex1: y 2x - 3
m =
b = -3 = (0, -3)1
2
Use a solid line because it is
Test: (0, 0)
0 2(0) – 3
0 0 – 3
0 -3 false (shade other side)
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• Ex2: y – 2x < 4
y – 2x < 4
+ 2x +2x
y < 2x + 4
m =
b = 4 = (0, 4)1
2
Test: (0, 0)
0 < 2(0) + 4
0 < 0 + 4
0 < 4 true (shade this side)
Use a dotted line because it is <
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• Ex3: 3y - 2 > -x + 7
3
1
3y – 2 > -x + 7
+2 +2
3y > -x + 9
/3 /3 /3
y > - x + 3
m = -
b = 3 = (0, 3)3
1
Test: (0, 0)
0 > - (0) + 3
0 > 0 + 3
0 > 3 false (shade other side)
3
1
Use a dotted line because it is >
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4.4 Writing Equations in Slope-Intercept Form
y = mx + b
slope y-intercept
Ex. Write an equation given slope and one point
Write the equation of the line that passes through (1, 5) with slope 2
y = mx + b
y = 2x + b
5 = 2(1) + b
5 = 2 + b
-2 -2 3 = b
y = 2x + 3
Replace m with the slope
Replace the x and y with the ordered pair coordinates
Solve for b (the y-intercept)
Replace the numbers for slope and the y-intercept
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Ex. Write an equation given two points
Write an equation for the line that passes through (-3, -1) and (6, -4)
y = mx + b
y = -1/3x + b
3
1
9
3
36
14
m
-1 = -1/3(-3) + b
-1 = 1 + b-1 -1
-2 = b
y = -1/3x - 2
Find slope
Replace m with the slope
Replace the x and y with one of the ordered pair coordinates
Solve for b (the y-intercept)
Replace the numbers for slope and the y-intercept
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4.5 Writing Equations in Point-Slope Form
Point-Slope Form: y – y1 = m(x – x1)
Ex:
Ex:
Write an equation given slope and one point
y – y1 = m(x – x1)Write the equation of a line that passes through (6, -2) with slope 5
y – -2= 5(x – 6)y + 2 = 5(x – 6)
Replace m with the slope and y1 and x1 with the ordered pair
Simplify
Write an equation of a horizontal line
y – y1 = m(x – x1)y – 2 = 0(x – 3)y – 2 =0
Replace m with the slope and y1 and x1 with the ordered pair
Simplify
y = 2
Write the equation for the line that passes through (3, 2) and is horizontal
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Ex: Ex:
Ex:
Write an equation in Standard Form
Write an equation in Slope-Intercept Form
Write an equation in Point-Slope Form given two points
y + 5= -5/4(x -2)
4[y + 5= -5/4(x -2)]4y + 20= -5(x -2)
4y + 20= -5x + 10+5x +5x
5x + 4y + 20= 10
- 20 -205x + 4y = -10
Multiply by 4Distribute
Add 5x(move it to the other side)
Subtract 20(move it to the other side)
Simplify
y -2= ½(x+ 5)2[y -2= ½(x+ 5)]2y -4= x+ 5 +4= +42y = x+ 9/2 /2y = 1/2x+ 9/2
Multiply by 2
Add 4(move it to the other side)
Divide by 2
Simplify
Write the equation for the line that passes through (2, 1) (6, 4)
4
3
26
14
m
y - 1= 3/4(x -2)
y - 4= 3/4(x -6)or
Find slope
Replace m with slope and y1 and x1 with one of the ordered pairs
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4.6 Statistics: Scatter Plots and Lines of Fit
• Scatter Plot- a graph in which two sets of data are plotted as ordered pairs in a coordinate plane – Used to investigate a relationship between
two quantities
Positive Correlation
Negative Correlation
No Correlation
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• If the data points do not lie in a line, but are close to making a line you can draw a Line of Fit– This line describes the trend of the data (once you have this line
you can use ordered pairs from it to write an equation)
Ex:
years 1 3 5 8 12 12 13 15
Vertical Drop
151 156 225 230 306 300 255 400
a. Make a scatter plot b. Draw a line of fit. What correlation do you find?
c. Write an equation in slope-intercept form for the line
Positive correlation
1 3 5 7 9 11 13 15
100
200
300
400
500
(8, 230) (12, 306) Find slope
m= 19
Plug in an ordered pair and slope to find the y-intercept
y = 19x +78
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4.7 Geometry : Parallel and Perpendicular Lines
• Parallel lines- do not intersect and have the same slope
• Perpendicular lines- make right angles and have opposite slopes
a
b Line a has slope 6/5
Line b has slope 6/5
c
d
Line c has slope 3/2
Line d has slope -2/3