Direct variation lesson.notebook
April 30, 2013
Direct Variation
Opener
Marshall bought delicious beef biscuits at the bulk food store.
The cost was $1.10 for 100 g.
Here is a table of costs for different masses.
Mass of beef
biscuits (g)
Costs ($)
200 400 600 800
2.20 4.40 6.60 8.80 Graph the data is the chart on the next slide.
Find the rate of change.
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Direct variation lesson.notebook
April 30, 2013
The graph is a straight line that passes though ______.
This illustrates ___________________________.
In ______________________________, we say that the cost varies directly as the mass.
When two quantities vary directly they are ________________________________.
The cost is __________________________________
to the mass.
Finding the rate of change:
The rise is:
The run is : The rate of change is ___________.
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Direct variation lesson.notebook
April 30, 2013
Write a Rule:
Cost if beef biscuits in $ = (0.011) x (mass of snacks)
the rate of change in the mass
dollars per gram in grams
We can use the rule to determine how much Marshall has to pay for 350 g of beef biscuits.
Substitute 350 for the mass of the beef biscuits.
Cost of beef biscuits in $ = 0.011 x 350
= 3.85
The cost of 350 g of beef biscuits is $3.85.
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Direct variation lesson.notebook
April 30, 2013
Now use the rule to determine how much Marshall has to pay for 450 g of beef biscuits.
What about if Marshall wanted to buy 800 g of beef biscuits?
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Direct variation lesson.notebook
April 30, 2013
We can use this rule to help us develop an equation for the line.
If $ = (0.011) x (mass of biscuits) (where $ represents the y­axis, mass of biscuits represents the x­axis and the 0.011 represents the rate of change or slope of the line)
Our equation for the line would be:
y = 0.011x
The equation for any line with direct variation is:
y = mx
(where m represents the slope of the line)
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Direct variation lesson.notebook
April 30, 2013
The equation for any line with direct variation is:
y = mx
(where m represents the slope of the line)
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Direct variation lesson.notebook
April 30, 2013
So, direct variation means that the two quantities are proportional and vary directly. When looking at a graph, you can tell it is direct variation because the straight line passes through the origin!
Which of these graphs illustrates direct variation?
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Direct variation lesson.notebook
April 30, 2013
Partial Variation
Example
The cost of pizza with tomato sauce is $9.00.
It costs $0.75 for each additional topping.
This table shows the cost of a pizza with up to 8 additional toppings.
Toppings
Costs ($)
0
2
4
6 8 9.00 10.50 12.00 13.50 15.00 The cost, in dollars, for a 4­topping pizza is 9+ (4 x 0.75) = 12.00
Graph the data is the chart on the next slide.
Find the rate of change.
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Direct variation lesson.notebook
April 30, 2013
All points make a straight line, but the graph does not passes though _________.
This illustrates ___________________________.
In ___________________________, we say that the cost of a pizza is the sum of a fixed cost and a variable cost.
The point where the line crosses the vertical axis is the ___________________________.
Finding the rate of change:
The rise is:
The run is :
The rate of change is ___________.
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Direct variation lesson.notebook
April 30, 2013
Write a Rule:
Cost of pizza in $ = 9.00 + (0.75 X number of toppings)
the fixed cost in dollars this is the variable cost in dollars because it depends on the number of toppings.
We can use the rule to determine how much someone would have to pay if they ordered a pizza with 12 toppings.
Substitute 12 for the number of toppings.
Cost of pizza in $ = 9.00 + (0.75 X number of toppings) = 9.00 + (0.75 X 12)
= 9.00 + 9
= 18.00
The cost of ordering a pizza with 12 toppings is $18.00.
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Direct variation lesson.notebook
April 30, 2013
Now use the rule to determine how someone would pay if they ordered a pizza with 15 toppings.
Cost of pizza in $ = 9.00 + (0.75 X number of toppings) = 9.00 + (0.75 X 15)
= 9.00 + 11.25
= 20.25
The cost of a 15 topping pizza would be $20.25.
What about if someone ordered a pizza but did not want any toppings? Is the pizza free? Explain your thinking.
If the pizza did not have any toppings, it would NOT be free.
The $9.00 is a fixed cost, which means that the pizza costs $9.00 initially before they even start adding toppings on.
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Direct variation lesson.notebook
April 30, 2013
We can use this rule to help us develop an equation for the line.
If the cost of the pizza is $ = 9.00 + (0.75 X number of toppings) (where $ represents the y­axis, number of toppings represents the x­
axis and the 0.75 represents the rate of change or slope of the line)
Our equation for the line would be:
y = 0.75x + 9
The equation for any line with direct variation is:
y = mx + b
(where m represents the slope of the line and b represents the y­intercept)
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Direct variation lesson.notebook
April 30, 2013
So, partial variation is when one quantity equals a fixed value plus a constant multiple of another quantity. When looking at a graph, you can tell it is partial variation because the straight line that does not passes through the origin!
Which of these graphs illustrates partial variation? Find the equation for each line using the y= mx + b formula.
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Direct variation lesson.notebook
April 30, 2013
Remember, the more work you finish in class, the less homework you will have!
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Direct variation lesson.notebook
April 30, 2013
Remember, the more work you finish in class, the less homework you will have!
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