What’s slope
got to do with it?
Today we are going to see how slope
and y-intercept are actually used in many situations in the “real world”
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A linear equation that describes a real world
situation is called aLinear model
Our learning goals are to be able to:
Identify and explain the meaning of the slope as a rate of change in the context of a given model
Identify and explain the meaning of the y-intercept in the context of a given model
What do you remember about….
•Slope
•Slope intercept form
• y-intercept
Remember what we already know about slope:
•Slope is the ratio of vertical rise to horizontal run.
• Slope is often described as “rise over run”.
•Slope is also called the rate of change.
• Given two points with coordinates (x1, y1) and (x2,y2) the formula for the slope of the line containing the points is
my2 y1x2 x1
€
slope =change in y
change in x
What do you remember about……
y = mx + b
What do you remember about……y = mx + b
This is the equation of a line. It tells us:
m is the slope of the line• It is the ratio between the change in y and the change in x. • It tells the “rate of change” of y and x • It shows how x and y are changing together.• It describes how steep the line is and how it is slanted.
b is the y-intercept• It tells us where the line crosses the y-axis• It shows us the value of y when x equals zero.
Consider this linear equation……
y = 1,000x + 1,500
This equation can be used in a real world example about saving money.
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x = the number of years since you opened a saving account y = the total amount of money in the account ($). 1. How much money will you have after 2 years?
1. How much money will you have after 5 years?
1. How much did the amount in your account change over the 3 years? How much was that per year?
4. Identify the slope and tell what it represents in this example.
5. What is the y – intercept and what does it represent?
y = 1,000x + 1,500
1.How much money will you have after 2 years?
1.How much money will you have after 5 years?
1.How much did the amount in your account change over those 3 years? How much was that per year?
2.Identify the slope and tell what it represents in this example.
3.What is the y – intercept and what does it represent?
Identify and explain what the slope representsSlope = m = 1,000 and it represents that the money in the account
increases $ 1,000 per year ( m= $1,000 / 1 year )
b) What does the y – intercept represent? y- intercept =1,500 and it is the amount of money in dollars
that was in the account when it was opened. (year 0 )
Given the linear equation y = -800 x + 8,000 where
x is the age of the car in years and y is the value of the car in dollars.
•What is the value of the car after 1 year?•What is the value of the car after 6 years?•How did the value of the car change between those 5 years?•Identify the value of the slope and explain what it represents in this example•What is the y – intercept and what does it represent in this example?
In this “real world” example, let’s see what slope means:
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Given y = - 800 x + 8,000 where x is the age of the car in years and
y is the cost of the car in dollars.
1.What is the value of the car after 1 year?2.What is the value of the car after 6 years?3.How did the value of the car change between those 5 years?
1.Identify the value of the slope; explain what it represents in this example.
Slope: m = -800 This is the “change in y” (value of the car) compared to the “change in x” (the age of the car). Connect the numbers and the words:
The value of the car decreases by $800 every year•What is the y – intercept and what does it represent in this example?
The y-intercept is $ 8,000. It represents the cost of the car when x (years) = 0.
The car was worth $8,000 when it was first bought.
a)Write the linear equation for this situation using C=the total cost and h=hours worked.
b) Identify the value of the slope; explain what slope represents in this example
c) What is the y-intercept and what does it represent in this example?
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An electrician charges $30 for a home visit plus $60 for each hour of service.
An electrician charges $30 for a home visit plus $60 per hour of service.
a) Find the equation of the line for this situation using C=the total cost, and h = hours worked
C = 60h + 30
b) Identify and explain what the slope represents:
Slope = 60 which represents the cost for each hour of service, or slope= $60/1 hour .
The slope is positive, that means that for each hour of service the cost will increase by $60.
c) What does the y-intercept represent?The y intercept = 30. It means that every service call has an automatic cost of $30, before any time is worked (the number of
hours = zero)
The value of a used car decreases every year. (The older the car, the less it is worth.) A car that is 4 years old has a value of $5,200;
when the same car is 7 years old, its value is $2,500.
1)Write two ordered pairs to represent the information in this example, using the form (x, v), with x = age of the car and y = the value of the car.
2) Find the slope of the line containing these two points.
3) Find the equation of this line.
4) What is the y-intercept? What does it represent?
5) What does the slope represent in this example?
Let x= the age of the car, and v = the value of the carFrom this, we see that both x (the car’s age) and v (the car’s value) are changing. And we can also tell that the value of
the car depends on the age of the car.
1) Write two ordered pairs to represent the information in this example, using the form (x, v). Answers: (4, 5200), (7, 2500)
2) Find the slope of the line containing these two points. m = -900
3) Find the equation of this line. v = -900 x + 8800
4) What is the y-intercept? What does it represent? The y-intercept is 8,800; the value of the car was $8,800 when the car was new.
5) What does the slope represent in this example? Every year the car’s value drops by $900.
John just graduated from high school and started a new job.
He received some cash as gifts from his relatives.
Instead of spending this gift money, he decided to save it for the future.
He also decided to start saving a set amount
from his paycheck each week.
After 12 weeks, he had a total of $720.
After 30 weeks he had a total of $1,440.
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1. Choose and define two variables for this example.2. How are these variables changing? 3. Write two ordered pairs to represent the
information in this example, using the form (w, d). 4. Find the slope of the line containing these two
points. 5. Find the equation of this line. 6. What is the y-intercept? What does it represent? 7. What does the slope represent in this example?
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Try this one on your own
1) Determine and identify the two variables for this example.The variables are number of weeks and amount of money
saved. Let w = number of weeks, and let d = number of dollars saved.
2) How are these variables changing? As the number of weeks increases, the amount of money
saved by John also increases.
3) Write two ordered pairs to represent the information in this example, using the form (w, d).
(12, 720) and (30, 1,440).
4) Find the slope of the line containing these two points. m = 40.
5 )Find the equation of this line. y = 40x + 240.
6 )What is the y-intercept? What does it represent? The y intercept is 240.It means that when the number of weeks was 0, John had $240. So we can say that John received $240 in graduation gifts.
7 )What does the slope represent in this example? Use both the number value and the verbal meaning of the variables to the slope. There was an increase of $40 per week. This is the meaning of the slope: it telsl what 2 quantities are changing and by how much (amount saved and weeks).
A taxi cab ride costs a $10 flat fee
plus $3 per mile traveled. A linear equation
can represent the total cost of a taxi ride.
a) Write an equation that shows how the miles traveled and the total cost of the taxi ride are related.b) Identify and explain what the slope means
c) What does the y-intercept mean?
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a) Write an equation: C = 3d + 10 This uses d for distance in miles and
C for total cost of the taxi ride.
b) Slope = 3. The slope represents the rate of change of the miles and the cost of the service. Slope (m) = $ 3 / 1 mile. This means that for every 1 mile, the cost will increase by $3. As the distance increases, the
cost of the service will increase. (Positive slope)
c) The y intercept is 10. That means that even if the distance is ZERO miles, you
must to pay a fee of $10 for the taxi cab ride.
Write an equation that shows how miles traveled and the total cost of the taxi ride are related.
•3 things you learned about the meaning of slope
•2 examples of how slope is used in the real world
•1 thing you learned about the meaning of y-intercept
EXIT TICKET
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