Over Chapter 10 Over Chapter 10 Chapter 11 Rational Functions and Equations Essential Question: How can simplifying

mathematical expressions be useful? Section 11-1 Inverse Variations Learning Goal: To identify, graph, and use inverse variations. inverse variation

product rule Identify Inverse and Direct Variations A. Determine whether the table represents an inverse or a direct variation. Explain. Notice that xy is not constant. So, the table does not

represent an indirect variation. Identify Inverse and Direct Variations Answer: The table of values represents the direct variation .

Identify Inverse and Direct Variations B. Determine whether the table represents an inverse or a direct variation. Explain. In an inverse variation, xy equals a constant k. Find xy for each ordered pair in the table.

1 12 = 12 2 6 = 12 3 4 = 12 Answer: The product is constant, so the table represents an inverse variation. Identify Inverse and Direct Variations

C. Determine whether 2xy = 20 represents an inverse or a direct variation. Explain. 2xy = 20 xy = 10 Write the equation. Divide each side by 2.

Answer: Since xy is constant, the equation represents an inverse variation. Identify Inverse and Direct Variations D. Determine whether x = 0.5y represents an inverse or a direct variation. Explain. The equation can be written as y = 2x.

Answer: Since the equation can be written in the form y = kx, it is a direct variation. A. Determine whether the table represents an inverse or a direct variation. A. direct variation B. inverse variation

B. Determine whether the table represents an inverse or a direct variation. A. direct variation B. inverse variation C. Determine whether 2x = 4y represents an inverse or a direct variation.

A. direct variation B. inverse variation D. Determine whether or a direct variation. A. direct variation B. inverse variation

represents an inverse Write an Inverse Variation Assume that y varies inversely as x. If y = 5 when x = 3, write an inverse variation equation that relates x and y. xy = k

Inverse variation equation 3(5) = k x = 3 and y = 5 15 = k Simplify. The constant of variation is 15. Answer: So, an equation that relates x and y is xy = 15 or

Assume that y varies inversely as x. If y = 3 when x = 8, determine a correct inverse variation equation that relates x and y. A. 3y = 8x B. xy = 24 C. D.

Solve for x or y Assume that y varies inversely as x. If y = 5 when x = 12, find x when y = 15. Let x1 = 12, y1 = 5, and y2 = 15. Solve for x2. x1y1 = x2y2 12 5 = x2 15

60 = x2 15 Product rule for inverse variations x1 = 12, y1 = 5, and y2 = 15 Simplify. Divide each side by 15. 4 = x2

Answer: 4 Simplify. If y varies inversely as x and y = 6 when x = 40, find x when y = 30. A. 5 B. 20

C. 8 D. 6 Use Inverse Variations PHYSICAL SCIENCE When two people are balanced on a seesaw, their distances from the center of the seesaw are inversely proportional to their weights. How far

should a 105-pound person sit from the center of the seesaw to balance a 63-pound person sitting 3.5 feet from the center? Let w1 = 63, d1 = 3.5, and w2 = 105. Solve for d2. w1d1 = w2d2 63 3.5 = 105d2 Product rule for inverse variations

Substitution Divide each side by 105. 2.1 = d2 Simplify. Use Inverse Variations

Answer: To balance the seesaw, the 105-pound person should sit 2.1 feet from the center. PHYSICAL SCIENCE When two objects are balanced on a lever, their distances from the fulcrum are inversely

proportional to their weights. How far should a 2-kilogram weight be from the fulcrum if a 6-kilogram weight is 3.2 meters from the fulcrum? A. 2 m B. 3 m

C. 4 m D. 9.6 m Graph an Inverse Variation Graph an inverse variation in which y = 1 when

x = 4. Solve for k. Write an inverse variation equation. xy = k (4)(1) = k 4=k Inverse variation equation

x = 4, y = 1 The constant of variation is 4. The inverse variation equation is xy = 4 or Graph an Inverse Variation Choose values for x and y whose product is 4.

Answer: Graph an inverse variation in which y = 8 when x = 3. A. B.

C. D.