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Direct Variation
Section 4.6
Page 253
• When x and y vary directly, there is a nonzero number a such that the following is true:
y = ax
a is the constant of variation
For example: y = 3x or y = -1/4x
non-example : y = 4x + 5
• y = ax shows direct variation between x & y
• What else is true about this type of equation?
• YES = it goes through the origin!
EXAMPLE 1 Identify direct variation equations
Tell whether the equation represents direct variation. If so, identify the constant of variation.
2x – 3y = 0a. – x + y = 4b.
EXAMPLE 1 Identify direct variation equations
ANSWER
Because the equation 2x – 3y = 0 can be rewritten in the form y = ax, it represents direct variation. The constant of variation is. 2
3
SOLUTION
To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax.
Write original equation.
Subtract 2x from each side.– 3y = – 2x
y =23
x Simplify.
2x – 3y = 0
EXAMPLE 1 Identify direct variation equations
– x + y = 4 Write original equation.
Add x to each side.y = x + 4
ANSWER
Because the equation – x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation.
b.
EXAMPLE 2 Graph direct variation equations
Graph the direct variation equation.
SOLUTION
a. y = x2
3y = – 3xb.
a. Plot a point at the origin. The slope is equal to the constant of variation, or Find and plot a second point, then draw a line through the points.
2 3
EXAMPLE 2 Graph direct variation equations
Plot a point at the origin. The slope is equal to the constant of variation, or – 3. Find and plot a second point, then draw a line through the points.
b.
What can you tell about these two lines?
Let’s look at the graph
Y =3x (red)
Y = -2x (blue)
What is the slope of each line above?
Graph these equations
Y = - 3x
Y = 0.4 x
EXAMPLE 3 Write and use a direct variation equation
The graph of a direct variation equation is shown.
SOLUTION
y = ax Write direct variation equation.Substitute.2 = a (– 1)
Write the direct variation equation.a.Find the value of y when x = 30.b.
Because y varies directly with x, the equation has the form y = ax. Use the fact that y = 2 when x = – 1 to find a.
a.
Solve for a.– 2 = a
• Assignment:
• P. 256 (#1-15, 19-20)• You will need graph paper
