Connecting Variation, Slope, and First Differences
A relation can be expressed in four ways (the text refers to this as the rule of four): • using words (the description of the situation)
• using numbers (a table of values)
• using a diagram or graph
• using an equation
Recall that the equation of a line has the form y = mx + b, where m represents the slope and b represents the vertical intercept.
Example 1: Consider two points Ax1, y1 and Bx2, y2.
Calculate the slope of line segment AB.
mAB=change in ychange in x
mAB=∆ y∆ x,
∆ y is read as “delta y”
=y2-y1x2-x1
mAB=y2-y1x2-x1
Example 2: Given A(2, 13) and B(4, 5), calculate the slope of the line segment AB.
Example 3: Determine the slope of the following using 2 different methods
M=y2-y1x2-x1
=0–102–4
=0+102+4
=106
=53
Example 4: a) For the following relation described in words, express the relation in 3 other ways (table, graph, and equation) Words: A house painter charges $400 plus $200 per room to paint the interior of a house.
Equation:
C=200r+400
Let r represent the number of rooms and let C represent the cost.
Total:
Graph:
Plot the points and join them with a straight line
b) Describe what the slope means.
The slope one room is the amount charged to paint (cost per room).
c) How can you identify from each of the 4 forms if a relationship demonstrates direct or partial variation?
Example 5: a) Calculate the slope using 2 methods.
Method 1: Pick two points
(2,11), (4,16)
m=y2-y1x2-x1
=16-114-2
=52
Method 2: m=riserun
=52
b) Determine the vertical intercept.
The vertical intercept is the point where the line crosses the y-axis.
The vertical intercept is 6 (as a point (0,6)).
So, b=6.
c) Write an equation for the relation
y=mx+b (Sub. m=52 and b=6)
y=52x+6
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