Unit 4: Direct Proportion Update your Table of Contents:

IT / NOT IT Look at the IT / NOT IT examples on the next four slides and decide what you think makes “IT” fit each of the categories…. IT/NOT IT

Direct Proportion also known as Direct Variation

Consider this: 1 2 5 Complete the chart to the right. On Friday chicken biscuits are sold before school for $2.50 each. Complete the chart to the right. What number do you think should go in the “?” column? How do you know? # of biscuits times? =Total cost 1 2 5 $2.50, because every biscuit costs $2.50.

Direct Proportion Defined: The relationship between 2 quantities where one quantity changes based on what happens to the other quantity.

Direct Proportion Look at the BISCUIT table. Does the cost per biscuit change? What affects the total cost? The number of biscuits determines the total cost. The number of biscuits is “x” and the total cost is “y”. What equation could we write to represent the total cost of the biscuits? y = 2.50 x no – it is always $2.50 the # of biscuits

Direct Proportion: Equations The cost of the biscuits is the same no matter how many you buy. Each time the number of biscuits changes, it changes by the same amount….$2.50 per biscuit. It is the constant of proportionality. (k) y = 2.50 x All equations in a direct proportion are in the form y = k x Constant of Proportionality (AKA: unit rate)

Direct Proportion : Equations IS a DIRECT VARIATION NOT! y = 3x y = 3x +1 y = x y = x-1 y = ½ x or y=x/2 y = x + 3 y = 1000000x y = 3/x What is the constant of proportionality for each of these equations?

Direct Proportion : Graphs Look at the graph at the bottom of your notes Take 5 minutes to work with your shoulder partner to graph the biscuit table What do you notice about your graph?

Direct Variation and its graph Observations: the graph will always be a straight line The graph will always go through…

the ORIGIN!!!!! (0,0) Special point known as the “ORIGIN”

Tell if the following graph is a Direct Variation or not. Yes No No No

Tell if the following graph is a Direct Variation or not. Yes No No Yes

Direct Proportion: Tables Spend 5 minutes completing the Movies Rock! Table. How did you know what to put for y values?

You can find the constant of proportionality k by using k = y/x direct proportion : the table People (x) Total Cost (y) 3 $33 5 $55 9 $99 Notice… As x increases in value, y increases by the same factor …. This “same factor” is the constant of proportionality. You can find the constant of proportionality k by using k = y/x

Is it a Direct Proportion? Note, x increases: 6 , 7 , 8 and y increases: 12, 14, 16 Is it a direct proportion?   Find the constant of proportionality for each row using k =y/x.  k = 12/6 or 2 The constant of proportionality in the table above is 2. If all the values are the same it is a Direct Proportion The equation would be y=2x

Is this a direct variation Is this a direct variation? If yes, give the constant of variation (k) and the equation. Yes! k = 10/20 or ½ k= 9/18 or ½ k= 7/14 or ½ k= 4/8 or ½ Equation? y = 1/2 x

The k values are different! Is this a direct variation? If yes, give the constant of variation (k) and the equation. k No! The k values are different!

Direct Proportion: Word Problems A car uses 8 gallons of gasoline to travel 280 miles. If the gas used and miles driven are proportional, how far will the car go on 10 gallons of gas? 25 gallons of gas? How many gallons are needed to drive 420 miles? Spend 5 minutes with your shoulder partner completing the table from the word problem. What is the constant of proportionality? What equation would represent the scenario? What would the graph look like?

x gallons used y miles driven 8 280 10 350 25 875 12 420 k = y/x 280/8= 35 10 x 35 = 350 25 x 35 = 875 35x= 420 x=12 constant of proportionality k = 35 equation: y = 35x a straight line through the origin