2. • 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 variationFor example: y = 3x or y = -1/4xnon-example : y = 4x + 5
3. • y = ax shows direct variation between x & y• What else is true about this type of equation?• YES = it goes through the origin!
4. EXAMPLE 1 Identify direct variation equations Tell whether the equation represents direct variation. If so, identify the constant of variation. a. 2x – 3y = 0 b. – x + y = 4
5. EXAMPLE 1 Identify direct variation equations SOLUTION To tell whether an equation represents direct variation, try to rewrite the equation in the form y = ax. 2x – 3y = 0 Write original equation. – 3y = – 2x Subtract 2x from each side. 2x Simplify. y= 3 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
6. EXAMPLE 1 Identify direct variation equations b. –x+y=4 Write original equation. y = x+4 Add x to each side. ANSWER Because the equation – x + y = 4 cannot be rewritten in the form y = ax, it does not represent direct variation.
7. EXAMPLE 2 Graph direct variation equations Graph the direct variation equation. 2 a. y = x b. y = – 3x 3 SOLUTION a. Plot a point at the origin. The slope is equal to the constant of variation, or 2 Find and plot a second 3 point, then draw a line through the points.
8. EXAMPLE 2 Graph direct variation equations b. 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.
9. What can you tell about these two lines?Let’s look at thegraphY =3x (red)Y = -2x (blue)What is the slope ofeach line above?
10. Graph theseequationsY = - 3xY = 0.4 x
11. EXAMPLE 3 Write and use a direct variation equation The graph of a direct variation equation is shown. a. Write the direct variation equation. b. Find the value of y when x = 30. SOLUTION a. 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. y = ax Write direct variation equation. 2 = a (– 1) Substitute. –2=a Solve for a.
12. • Assignment:• P. 256 (#1-15, 19-20)• You will need graph paper