1. Modeling with Linear Functions
2.2
SECTION
Modeling with Linear Functions
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2. Learning Objectives
1 Determine if two quantities are directly proportional
2 Construct linear models of real-world data sets and use them to predict results
3 Find the inverse of a linear function and interpret its meaning in a real-world Context

3. Recognizing When to Use a Linear Model

4. Recognizing When to Use a Linear Model
Several key phrases alert us to the fact that a linear model may be used to model a data set.
Some of the simplest linear models to construct are those that model direct proportionalities, where one quantity is a constant multiple of another quantity.

5. Recognizing When to Use a Linear Model
Solving the equation y = kx for k yields
Thus another way to define direct proportionality is to say that two quantities are directly proportional if the output divided by the input is a constant.

6. Direct Variation, Example 1
Example: w is directly proportional to z. If w = −6 when z = 2,find w when z = −7.
Solution:

7. Direct Variation, Example 1

8. Direct Variation, Example 2
Garth’s front lawn is a rectangle measuring 120 feet by 40 feet. If a 25lb. bag of “Weed and Feed” will treat 2,000 square feet, how many bags must Garth buy to treat his lawn? (Assume that he must buy whole bags.)
Solution:

9. Direct Variation, Example 2

10. Inverses of Linear Functions

11. Inverses of Linear Functions
We have known that the domain of a function f was the range of its inverse function f –1 and the range of the function f was the domain of its inverse function f –1.
This notion is represented as follows.
The phrase “y is the function of x” corresponds with the phrase “x is the inverse function of y.”

12. Inverses of Linear Functions
Symbolically, y = f (x) is related to x = f –1(y).
Recognizing this relationship between a function and its inverse is critical for a deep understanding of inverse functions, especially in a real-world context.
Many students struggle with the concept of inverse functions.
To help you get a better grasp on this concept, we will work a straightforward example before summarizing the process of finding an inverse of a linear function.

13. Example 5 – Finding the Inverse of a Linear Function
Find the inverse of the function
Solution: In the function we are given, x is the independent variable and y is the dependent variable. We solve this equation for x.

14. Example 5 – Solution
cont’d
In this new equation, y is the independent variable and x is the dependent variable. We can write the original function as
Similarly, we can write the inverse function as
The function is the inverse of

15. Inverses of Linear Functions
We now summarize the process of finding the inverse of a linear function. Observe that since dividing by 0 is undefined, this process only works for linear functions with nonzero slopes.

16. Inverses of Linear Functions
We have known that horizontal lines can be written in the form y = b.
The inverse of a horizontal line is a vertical line x = a; however, a vertical line is not a function. Therefore, only linear functions with nonzero slopes have an inverse function.

17. Inverses of Linear Functions Example
Based on data from June 2006, the forecast for maximum 5-day snow runoff volumes for the American River at Folsom, CA, can be modeled by
v = f(t) = −2.606t thousand acre-feet,
where t is the number of days since the end of May That is, the model forecasts the snow runoff for the 5-day period beginning on the selected day of June Find the inverse function.

18. Inverses of Linear Functions Example
v = f(t) = −2.606t
Solution:

19. Exit Ticket:

20. Inverses of Linear Functions Example