1. Rational Functions * Inverse/Direct

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

3. 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. 

4. (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 =

5. 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 _________________

6. 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 _________________

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

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

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

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

11. 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. 

12. 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.

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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