1. Linear Functions and Equations
Chapter 2
Linear Functions and Equations

2. More Modeling with Functions
2.4
Model data with a linear function
Evaluate and graph piecewise-defined functions
Evaluate and graph the greatest integer function
Use direct variation to solve problems

3. Modeling with Linear Functions
To model a quantity that is changing at a constant rate with f(x) = ax + b, the following formula may be used.
f(x) = (constant rate of change)x + initial amount
The constant rate of change corresponds to the slope of the graph of f, and the initial amount corresponds to the y-intercept. 3

4. Example: Finding a symbolic representation
A 100-gallon water tank, initially full of water, is being drained at a rate of 5 gallons per minute.
(a) Write a formula for a linear function f that models the number of gallons of water in the tank after x minutes.
(b) How much water is in the tank after 4 minutes?
(c) Graph f. Identify the x- and y-intercepts and interpret each.
(d) Discuss the domain of f. 4

5. Example: Finding a symbolic representation
Solution
(a) Water in tank decreasing at 5 gal/min, so constant rate of change is –5. The initial amount of water is 100 gal.
(b) After 4 min the tank contains: 5

6. Example: Finding a symbolic representation
(c) Slope: –5 y-intercept: the number of gallons of water initially in the tank x-intercept: 20 - time in min to empty the tank 6

7. Example: Finding a symbolic representation
(d) From the graph: the domain of f must be restricted to ≤ x ≤ 20.
It makes sense:
can’t have negative time,
f(21) = –5(21) = –5 tank can’t hold –5 gallons of water 7

8. Piecewise-Defined Functions
A function f that models data and is defined on pieces of its domain is called a piece-wise function.
If each piece is linear, the functions is a piece-wise linear function. 8

9. Fujita Scale - Intensity of Tornadoes
An F1 tornado has winds speeds between 40 and 72 miles per hour. Here is the piece-wise function for the F-scale. 9

10. Example: Evaluating and graphing a piecewise-defined function
Use f(x) to complete the following:
(a) What is the domain of f ?
(b) Evaluate f(–3), f(2), f(4), and f(5).
(c) Sketch a graph of f.
(d) Is f a continuous function on its domain? 10

11. Example: Evaluating and graphing a piecewise-defined function
Solution
(a) f is defined for –4 ≤ x < 2 or 2 ≤ x ≤ 4 domain is D = {x |–4 ≤ x ≤ 4}, or [–4, 4]
(b) f(–3) = –3 – 1 = –4 f(2) = –2 • 2 = –4 f(4) = –2 • 4 = –8 f(5) is undefined (5 is not in the domain) 11

12. Example: Evaluating and graphing a piecewise-defined function
(c) Sketch a graph of f.
From –4 up to 2, graph y = x – 1, open circle at 2
From 2 to 4, graph y = –2x
(d) f is not continuous because there is a break at x = 2 12

13. Greatest Integer Function
The greatest integer function is defined as follows.
is the greatest integer less than or equal to x. 13

14. Direct Variation
Let x and y denote two quantities. Then y is directly proportional to x, or y varies directly with x, if there exists a nonzero number k such that
y = kx
k is called the constant of proportionality or the constant of variation. 14

15. Hooke’s Law
Hooke’s law states that the distance that an elastic spring stretches beyond its natural length is directly proportional to the amount of weight hung on the spring, as illustrated in the Figure. 15

16. Hooke’s Law
This law is valid whether the spring is stretched or compressed.
The constant of proportionality is called the spring constant. Thus if a weight or force F is applied and the spring stretches a distance x beyond its natural length, then the equation F = kx models this situation, where k is the spring constant. 16

17. Example: Working with Hooke’s law
A 12-pound weight is hung on a spring, and it stretches 2 inches.
(a) Find the spring constant.
(b) Determine how far the spring will stretch when a 19-pound weight is hung on it.
Solution
Let F = kx: F = 12 pounds, x = 2 inches
F = k(2) so k = 6
The spring constant equals 6 17

18. Example: Working with Hooke’s law
(b) F = 19 18

19. Solving a Variation Problem
When solving a variation problem, the following steps can be used.
STEP 1: Write the general equation for the type of variation problem that you are solving.
STEP 2: Substitute given values in this equation so the constant of variation k is the only unknown value in the equation. Solve for k. 19

20. Solving a Variation Problem
STEP 3: Substitute the value of k in the general equation in Step 1.
STEP 4: Use this equation to find the requested quantity. 20

21. Example: Solving a direct variation problem
Let T vary directly with x, and suppose that T = 33 when x = 5. Find T when x = 31.
Solution
STEP 1: Direct variation is T = kx
STEP 2: Substitute 33 for T, 5 for x , solve for k 21

22. Example: Solving a direct variation problem
STEP 3:
STEP 4: When x = 31, we have: 22