5.1 Equation of Lines Using Slope-Intercept Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach Objectives : • Learn how to use slope intercept form to write an equation of a line • Learn how to model a real-life situation with a linear equation Slope Intercept Form: y = mx + b Where m = slope b = y-intercept Write an equation when given the slope and y-intercept: m = –2 m = 5 b = 4 b = 1 2 − y = –2x + 4 y = 5x – 1 2
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5.1 Equation of Lines Using Slope-Intercept
Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach
Objectives:
• Learn how to use slope intercept form to write an equation of a line
• Learn how to model a real-life situation with a linear equation
Slope Intercept Form: y = mx + b Where m = slope b = y-intercept Write an equation when given the slope and y-intercept:
m = –2 m = 5
b = 4 b = 12−
y = –2x + 4 y = 5x – 12
5.1 Equation of Lines Using Slope-Intercept
Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach
Write an equation from the graph
y = mx + b
y = 32 x +(–3)
y = mx + b
y = –x –1
y = mx + b
y = 3 Real-life: the Phone Company charges a flat fee of $0.75 for the first minute of long
distance plus $0.10 per minute after that. Write an equation to use to figure out each call.
1. Verbal Model:
Total cost = first minute cost + rateminute · number of minutes
2. Labels:
y = total cost
c = cost for first minute ($0.75)
x = # of additional minutes
r = rate per additional minute ($0.10)
5.1 Equation of Lines Using Slope-Intercept
Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach
3. Algebraic Model:
Fill in what you know and write in slope-intercept form (y = mx + b)
y = .75 + .10x 4. Solve/Give Answer:
Find values for extra minutes: 0, 5, 10 and graph
x y
0 .75
5 1.25
10 1.75
1 2 3 4 5 6 7 8 9 10 11 x
1
2
3
y
minutes
Cost ($)
5.2 Equations of Lines Given the Slope and a Point
Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach
Objectives:
• Learn how to use slope and any point to write an equation of the line
• Learn how to model a real-life situation with a linear equation
Slope intercept form: y = mx + b Need to know m and b Given the slope, m = –2 and the point (6, –3), find equation of the line:
y = mx + b
–3 = –2(6) + b substitute what you know into the slope-intercept form of the equation, and then solve for b
–3 = –12 + b +12 +12 9 = b
y = –2x + 9 graph check:
1 2 3 4 5 6 7 8 910–1–2–3–4–5–6–7–8–9–10 x
123456789
10
–1–2–3–4–5–6–7–8–9
–10
y
5.2 Equations of Lines Given the Slope and a Point
Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach
Write the equation for a line that passes through point (6, 7) and has a slope 23
y = mx + b graph:
7 = 6(32 ) + b
7 = 4 + b –4 –4 3 = b
y = 23 x + 3
Real-life: Find an equation for vacation trips y (in millions) in terms of the
year, t. Let t = 0 correspond to 1980. From 1980 – 1990, vacation trips increased by about 15 million per year. In 1985, Americans went on 340 million vacation trips.
Because change is constant, you can model this as a linear equation y = mt + b Constant is 15 trips per year, so you know slope. In 1985, the year would be t = 0 + 5 and the number of vacation trips (y) would be 340 million.
y = 15t + b
340 = 15(5) + b
340 = 75 + b –75 –75 265 = b
1 2 3 4 5 6 7 8 9–1–2–3–4–5–6–7–8–9 x
123456789
–1–2–3–4–5–6–7–8–9
y
5.2 Equations of Lines Given the Slope and a Point
Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach
y = 15t + 265
In 1998, the value of t = 1998 - 1980 = 18
So the number of vacations taken in 1998 would be:
Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach
Objectives:
• Learn how to write an equation of a line given 2 points on the line
• Learn how to model a real-life problem with a linear equation
Equation of a line: y = mx + b Need slope and y-intercept Given 2 points, can you find the slope and y-intercept?
Points (3, 5) and (–7, 2)
1. Find the slope: m = 2 1
2 1
riserun
y yx x−
=−
m = 2 57 3−
− − = 3
10
y = mx + b Substitute the value of m back into the slope-intercept form of the linear equation
y = 310 x + b
2. Substitute the x and y values from one point to find the y-intercept (the value of b):
y = mx + b (3, 5)
5 = 310 (3) + b
5 = 910 + b
– 910 – 9
10
5.3 Equations of Lines Given 2 Points
Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach
4 110 = b
3. Substitute both m and b back into the slope-intercept form of the linear equation and solve:
y = 3
10 x + 4 110 This is the equation of the line passing through points
(3, 5) and (–7, 2)
(3, 5)
(–7, 2)
1 2 3 4 5 6 7 8 910–1–2–3–4–5–6–7–8–9–10 x
123456789
10
–1–2–3–4–5–6–7–8–9
–10
y
Example: Find linear equation with points (–3, 2) and (5,–2), and then graph
Let’s say (x1, y1) = (–3, 2) and (x2, y2) = (5,–2) (Though it really doesn’t matter)
1. Find the slope, m:
m = 2 25 ( 3)− −− −
= 48− = 1
2−
y = mx + b Substitute m into the slope-intercept form of the linear equation
y = 12− x + b
5.3 Equations of Lines Given 2 Points
Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach
2. Substitute the x and y values from one of the two points given in the question to find the y-intercept (the value of b):
y = mx + b (–3, 2)
2 = 12− (–3) + b
2 = 32 + b
– 32 – 3
2
12 = b
3. Substitute both m and b back into the slope-intercept form of the linear equation and solve:
y = 1
2− x + 12
(–3, 2)
(5, –2)
1 2 3 4 5 6 7 8 910–1–2–3–4–5–6–7–8–9–10 x
123456789
10
–1–2–3–4–5–6–7–8–9
–10
y
5.3 Equations of Lines Given 2 Points
Mr. Noyes, Akimel A-al Middle School 4 Heath Algebra 1 - An Integrated Approach
Can we find the equation of a line if we know both the x, and y-intercepts?
Write an equation that has points whose y-intercept is –4 and x-intercept is –6
What are your 2 points? (0, –4) (–6, 0)
m = 2 1
2 1
y yx x−−
= 0 ( 4)6 0− −− −
= 46−
= – 23
y = mx + b 0 = – 2
3 (–6) + b
0 = +4 + b –4 –4
–4 = b
y = mx + b
y = – 23 x – 4
(0, –4)
(–6, 0)
1 2 3 4 5 6 7 8–1–2–3–4–5–6–7–8 x
12345678
–1–2–3–4–5–6–7–8
y
5.3 Equations of Lines Given 2 Points
Mr. Noyes, Akimel A-al Middle School 5 Heath Algebra 1 - An Integrated Approach
Real–Life: (see textbook, page 253, Question #29) The Tunnel (aka, “The Chunnel”) from Calais, France to Dover, England Write a linear equation of the line formed from A to B
Point A: (0, 60) Point B: (15, – 70)
m = 2 1
2 1
y yx x−−
= 70 6015,000 0− −
−
= 130 1315000 1500− = −
y = mx + b
60 = 131500− (0) + b
60 = b
y = mx + b
y = – 131500 x + 60
5.4 Exploring Data: Fitting a Line to Data
Mr. Noyes, Akimel A-al Middle School 1 Heath Algebra 1 - An Integrated Approach
Objectives:
• Learn how to find a linear equation that approximates a set of data points
• Learn how to use scatter plots to determine positive, negative or no correlation
The winning Olympic times for the 100-meter run from 1928 to 1988 are plotted in the graph below. Approximate the best-fitting line for these times. Let y represent the winning time and x the year (x = 0, corresponding to 1928).
20 40 60 x
10
11
12
13
y
Looking at the graph, do you see the trend in the info given?
Can we write a linear equation to match?
Often in life the data collected (no matter how carefully done) to analyze whether a relationship exists between two variables will not appear as a nice and neat straight line. However, while all the data may not fall on one line, they may still exhibit a trend that can best be described as linear. When this happens we draw a single line that best represents (or approximates) the set of data points. This line is called the line of best fit.
5.4 Exploring Data: Fitting a Line to Data
Mr. Noyes, Akimel A-al Middle School 2 Heath Algebra 1 - An Integrated Approach
Draw a line of best fit and pick any 2 points along the line (they don’t have to be actual points plotted on the scatter plot).
Let’s say points (0, 12) and (50, 11) Use the method described in Chapter 5.3
m= 2 1
2 1
riserun
y yx x−
=−
=11 1250 0−−
m = 150−
y = mx + b
11 = 150− (50) + b
11 = –1 + b +1 +1
12 = b
y = 150− x + 12
Correlation: A quantitative assessment of whether a relationship exists between
two variables. Describes the slope of the line of best fit.
x
y
x
y
x
y
Positive correlation Negative correlation No correlation When a line cannot be drawn to represent the set of data points, we say there is no correlation.
5.4 Exploring Data: Fitting a Line to Data
Mr. Noyes, Akimel A-al Middle School 3 Heath Algebra 1 - An Integrated Approach
Real-life: Make a graph, find the equation of the line of best fit (if possible), and make a prediction.
Find approximate wing area of a 400 g bird
Bird Weight (g) Wing Area (cm2)
Sparrow 25 87 Martin 47 186 Blackbird 78 245 Starling 93 190 Dove 143 357 Crow 607 1344 Gull 840 2006 Blue Heron 2090 4436 Let x represent weight and y wing area