Company Name 7.1- Inverse Variation

Direct vs. Inverse Variation Two variables x and y show Direct Variation when y=ax for some nonzero constant a. Ex: y = 5x, x = y/3, y = (1/2)x . Two variables show Inverse Variation when they are related as follows: The constant a is the constant of variation, and y is said to vary inversely with x.

Examples Do x and y show direct variation, inverse variation, or neither?

Inverse Variation Relationship When we talk about an inverse variation, we are talking about a relationship where x increases, y decreases or x decreases, y increases by a CONSTANT FACTOR

Example Table shows length x(inches) and width y(inches) of a rectangle. Area of every rectangle formed is 36 square inches. What is the equation to model the relationship between the length and width? x y 1 36 2 18 3 12 4 9 6 Follows form of , so it’s an inverse equation

Example(cont.d) Visual Representation

Examples of Inverse Variation: What is the constant of variation of the table above? Since , we can say k = xy. Therefore: (-2)(-18)=k or k = 36 (72)(0.5)=k or k = 36 (4)(9)=k or k =36 xy = 36 or y = Note: k is constant

Using Inverse Variation to find Unknowns Given that y varies inversely with x and y = -30 when x= -3, find y when x = 5. 2 step process 1. Find the constant variation. k = xy or k = -3(-30) k = 90 2. Use k = xy. Find the unknown (y). 90 = xy so 90= 5y y= 18 Therefore: when x = 5, y=18

Classifying Data (b) (a) Find the products xy and ratios y/x. x and y show neither direct nor inverse variation. x and y show inverse variation.

Using Inverse Variation to Solve Word Problems The time y that it takes a plane to reach a certain destination varies inversely as the average speed x of the plane. If it took a plane 3 hours to reach its destination when it traveled at an average speed of 150 mi/hr, what was the average speed of the plane if it took 4 hours to reach the same destination?

Using Inverse Variation to Solve Word Problems Write the equation that relates the variables then solve. The time y that it takes a plane to reach a certain destination varies inversely as the average speed x of the plane. If it took a plane 3 hours to reach its destination when it traveled at an average speed of 150 mi/hr, what was the average speed of the plane if it took 4 hours to reach the same destination? t(time) varies inversely as s(speed) so Time is the y variable and Speed is the x variable K = xy K = 150(3) K = 450 The equation is 450 = xy Substituting the new values: 450 = x(4) x = 112.5 The average speed of the plane to reach the destination in 4 hours was 112.5 mi/hr.

Graphs of Inverse Variation Equations