Rational Functions * Inverse/Direct

Direct Variation What you’ll learn … To write and interpret direct variation equations

This is a graph of direct variation This is a graph of direct variation.  If the value of x is increased, then y increases as well. Both variables change in the same manner.  If x decreases, so does the value of y.  We say that y varies directly as the value of x. 

(see any similarities to y = mx + b?) Definition: Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx + b?) Another way of writing this is k =

Example 1 Identifying Direct Variation from a Table For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y 1 4 2 7 5 16 x y 2 8 3 12 5 20 k = _______ k = _______ Equation _________________ Equation _________________

Example 2 Identifying Direct Variation from a Table For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. x y -1 -2 3 4 6 7 x y -6 -2 3 1 12 4 k = _______ k = _______ Equation _________________ Equation _________________

Example 3 Using a Proportion Suppose y varies directly with x, and x = 27 when y = -51. Find x when y = -17.

Example 4 Using a Proportion Suppose y varies directly with x, and x = 3 when y = 4. Find y when x = 6.

Example 5 Using a Proportion Suppose y varies directly with x, and x = -3 when y = 10. Find x when y = 2.

Inverse Variation What you’ll learn … To use inverse variation To use combined variation

In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.   Inverse variation:  when one variable increases, the other variable decreases. 

Inverse Variation An inverse variation between 2 variables, y and x, When two quantities vary inversely, one quantity increases as the other decreases, and vice versa. Generalizing, we obtain the following statement. An inverse variation between 2 variables, y and x, is a relationship that is expressed as:             where the variable k is called the constant of proportionality. As with the direct variation problems, the k value needs to be found using the first set of data.

Example 1 Identifying Direct and Inverse Variation Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations. x 0.5 2 6 y 1.5 18 x 0.2 0.6 1.2 y 12 4 2 x 1 2 3 y 0.5

Example 2 Identifying Direct and Inverse Variation Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations. x 0.8 0.6 0.4 y 0.9 1.2 1.8 x 2 4 6 y 3.2 1.6 1.1 x 1.2 1.4 1.6 y 18 21 24

Example 3 Real World Connection Zoology. Heart rates and life spans of most mammals are inversely related. Us the data to write a function that models this inverse variation. Use your function to estimate the average life span of a cat with a heart rate of 126 beats / min. Mammal Heart Rate (beats per min) Life Span (min) Mouse 634 1,576,800 Rabbit 158 6,307,200 Lion 76 13,140,000 Horse 63 15,768,000

A combined variation combines direct and inverse variation in more complicated relationships. Equation Form y varies directly with the square of x y = kx2 y varies inversely with the cube of x y = z varies jointly with x and y. z = kxy z varies jointly with x and y and inversely with w. z = z varies directly with x and inversely with the product of w and y. k x3 kxy w kx wy

Example 1 Finding a Formula The volume of a regular tetrahedron varies directly as the cube of the length of an edge. The volume of a regular tetrahedron with edge length 3 is . Find the formula for the volume of a regular tetrahedron. e e 9 √ 2 4

Example 2 Finding a Formula The volume of a square pyramid with congruent edges varies directly as the cube of the length of an edge. The volume of a square pyramid with edge length 4 is . Find the formula for the volume of a square pyramid with congruent edges. e e e 32 √ 2 3