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8-1: Model Joint and Inverse Variation. Objectives: Identify inverse and direct variation equations. Solve for the constant in variation equations. Common Core Standards: A-REI-2 Assessments: Define all vocab for this section Do worksheet 8-1.
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8-1: Model Joint and Inverse Variation Objectives: Identify inverse and direct variation equations. Solve for the constant in variation equations. Common Core Standards: A-REI-2 Assessments: Define all vocab for this section Do worksheet 8-1
We have studied many types of linear functions. One special type of linear function is called direct variation. A direct variation is a relationship between two variables x and y that can be written in the form y = kx, where k ≠ 0. In this relationship, k(sometimes the variable a is used instead of k, but any variable can be used)is the constant of variation. For the equation y = kx, y varies directly as x.
A direct variation equation is a linear equation in the form y = mx + b, where b = 0 and the constant of variation k is the slope. Because b = 0, the graph of a direct variation always passes through the origin.
k k This type of variation is an inverse variation. An inverse variation is a relationship between two variables x and y that can be written in the form y = , where k ≠ 0. For the equation y = , y varies inversely as x. x x Another type of variation describes a situation in which one quantity increases and the other decreases. For example, the table shows that the time needed to drive 600 miles decreases as speed increases.
y = y c. = x 4 7 x EXAMPLE 1 Classify direct and inverse variation Tell whether xand yshow direct variation, inverse variation, or neither. Type of Variation Given Equation Rewritten Equation a.xy = 7 b.y = x + 3 y = 4x
EXAMPLE 2 Write an inverse variation equation The variables xand yvary inversely, and y = 7 when x=4. Write an equation that relates xand y. Then find ywhen x = –2 .
The number of songs that can be stored on an MP3 player varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes (MB). Write an inverse variation model EXAMPLE 3 • Write a model that gives the number nof songs that will fit on the MP3 player as a function of the average song size s(in megabytes).
The variables xand yvary inversely. Use the given values to write an equation relating xand y. Then find ywhen x=2. GUIDED PRACTICE • x = 4,y = 3
A joint variation is a relationship among three variables that can be written in the form y = kxz, where k is the constant of variation. For the equation y = kxz, y varies jointly as x and z. It is not always y that varies jointly. The variable that varies jointly = k(Variable)(Variable). Note that the variables for Direct, Inverse, and Joint variation can change. The parts must all be there, a constant and variables.
Write a joint variation equation EXAMPLE 5 The variable zvaries jointly with xand y. Also, z= –75 when x = 3 and y = –5. Write an equation that relates x, y, and z. Then find zwhen x = 2 and y = 6.
y = y = z = atr x = s ay a a x2 x x Compare different types of variation EXAMPLE 6 Write an equation for the given relationship. Relationship Equation a.yvaries inversely with x. b.zvaries jointly with x, y, and r. z = axyr c.y varies inversely with the square of x. d.zvaries directly with yand inversely with x. e.xvaries jointly with tand rand inversely with s.
The variable zvaries jointly with xand y. Use the given values to write an equation relating x, y, and z. Then find zwhen x = –2 and y = 5. GUIDED PRACTICE x = 1,y = 2,z = 7
