![Greatest-Integer Function
• The greatest-integer function, usually written
is defined by saying that [x] denotes the largest integer
that is less than or equal to x.
• Be careful with negatives! Notice that –3 is less than –2.6!
( ) ,
f x x
=
[8] 8
[7.45] 7
[] 3
[-1] -1
[-2.6] -3](https://image.slidesharecdn.com/3-230707013640-a2c1567d/85/3-2-Graphs-of-Functions-27-320.jpg)
![Greatest-Integer Function
• Here’s another way to look at it:
• Here’s how the graph looks:
x –2 x –1 –1 x 0 0 x 1 1 x 2
[x] –2 –1 0 1
The graph should
demonstrate why
this is an example of
a step function.](https://image.slidesharecdn.com/3-230707013640-a2c1567d/85/3-2-Graphs-of-Functions-28-320.jpg)
![Step Functions
• Example: To mail a letter to Canada in 2016, the U.S. Post
Office charged $1.20 for the first ounce, an additional
$0.47 for an additional two ounces, and then an additional
$0.49 up to a maximum weight of 3.5 ounces. If L(x)
represents the cost of sending a letter weighing x ounces,
graph L(x) for x in the interval (0, 3.5].](https://image.slidesharecdn.com/3-230707013640-a2c1567d/85/3-2-Graphs-of-Functions-29-320.jpg)
![Step Functions
• Example: To mail a letter to Canada in 2016, the U.S. Post
Office charged $1.20 for the first ounce, an additional
$0.47 for an additional two ounces, and then an additional
$0.49 up to a maximum weight of 3.5 ounces. If L(x)
represents the cost of sending a letter weighing x ounces,
graph L(x) for x in the interval (0, 3.5].
• Let’s figure out our steps:
0 1: $1.20
1 3: $1.20 $0.47 $1.67
3 3.5: $1.67 $0.49 $2.16
x
x
x
+ =
+ =](https://image.slidesharecdn.com/3-230707013640-a2c1567d/85/3-2-Graphs-of-Functions-30-320.jpg)
3.2 Graphs of Functions
- 1. MATH 1324 – Business College Algebra 3.2 Graphs of Functions Chapter 3 Functions and Graphs
- 2. Concepts and Objectives The objectives for this section are • Connect functions to their graphs • Graph piecewise-defined functions • Graph absolute value functions • Graph greatest-integer functions • Interpret graphs • Use the vertical line test to determine a function
- 3. Graphs of Functions • The graph of a function f is defined to be the graph of the equation y = f(x). • It consists of all points (x, f(x)) – that is, every point whose first coordinate is an input number from the domain of f and whose second coordinate is the corresponding output number from the function.
- 4. Graphs of Functions (cont.) • For the most part, we use f (x) and y interchangeably to denote a function of x, but there are some subtle differences. • y is the output variable, while f (x) is the rule that produces the output variable. • An equation with two variables, x and y, may not be a function at all. Example: is a circle, but not a function 2 2 4 x y + =
- 5. Graphs of Functions (cont.) • Example: The graph of the function g(x) = .5x – 3 is the graph of the equation y = .5x – 3. This equation is in slope-intercept form, which means we can plot the y-intercept (0, –3), and then count the slope (up 1, over 2) to find the second point (2, –2). We could also have found the x-intercept at (6, 0).
- 6. Graphs of Piecewise Functions • Example: Graph the following function: The graph of f consists of: the part of the line y = x +1 where x 2 and the part of the line y = –2x + 7 where x > 2. ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 7. Graphs of Piecewise Functions • Example: Graph the following: • For the first equation, plot the y- intercept and count the slope to find the second point. ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 8. Graphs of Piecewise Functions • Example: Graph the following: • For the first equation, plot the y- intercept and count the slope to find the second point. • Now, draw a line through the two points, extending it until you reach x = 2. End your line with a closed circle. ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 9. Graphs of Piecewise Functions • Example: Graph the following: • If we evaluate the second equation at x = 2, we get (2, 3), which is the same point on our first equation (this doesn’t always happen). • Use the slope (–2) to find the second point. ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 10. Graphs of Piecewise Functions • Example: Graph the following: • Draw the line through the two points to complete the graph. ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 11. Graphs of Piecewise Functions • Example: Graph the following: • We can also use technology to graph the function. I prefer using Desmos. You control the interval being graphed by using braces: ( ) 1 if 2 2 7 if 2 x x f x x x + = − +
- 12. Graphs of Piecewise Functions • Example: Graph the following: • Using Desmos, we list the two parts. ( ) 2 if 3 8 if 3 x x f x x x − = − +
- 13. Graphs of Piecewise Functions • Example: Graph the following: • Unlike the last problem, the two lines do not intersect. This means we have to indicate which line actually includes x = 3. To do this, we use open and closed circles. • For the first equation, 3 – 2 = 1, so (3, 1) is closed. • For the second, –(3) + 8 = 5, so (3, 5) is open. ( ) 2 if 3 8 if 3 x x f x x x − = − +
- 14. Graphs of Piecewise Functions • Example: Graph the following: ( ) 2 if 3 8 if 3 x x f x x x − = − + Use the setup wheel to change the color and type of circle.
- 15. Graphs of Piecewise Functions • Example: Graph the following: ( ) 2 if 3 8 if 3 x x f x x x − = − +
- 16. Graphs of Piecewise Functions • Example: Graph the following: ( ) 2 if 3 8 if 3 x x f x x x − = − +
- 17. Graphs of Piecewise Functions • Example: Graph the following: • Click “Done” to finish ( ) 2 if 3 8 if 3 x x f x x x − = − +
- 18. Absolute-Value Function • The absolute-value function is an example of a piecewise function. It’s defined as and looks like this: ( ) if 0 if 0 x x f x x x x = = −
- 19. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. a) How much would a customer who used 10 CCF of water pay?
- 20. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. a) How much would a customer who used 10 CCF of water pay? 25 + 2(10) = $45
- 21. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. b) How much would a customer who used 20 CCF of water pay?
- 22. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. b) How much would a customer who used 20 CCF of water pay? 25 + 2(15) + 4(5) = $75
- 23. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. c) Find a rule for the function f(x) that gives the amount of the water bill when x CCF of water are used.
- 24. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. c) Find a rule for the function f(x) that gives the amount of the water bill when x CCF of water are used. For the higher rate, notice that 25 + 2(15) = 55.
- 25. Writing Piecewise Functions • Example: The Birmingham Water Works charges its residential customers with a ⅝-inch water line a basic service charge of $25 per month. The first 15 hundred cubic feet (CCF) of water cost $2 per CCF. To encourage water conservation, excessive water usages is charged at a higher rate of $4 per CCF above 15. c) Find a rule for the function f(x) that gives the amount of the water bill when x CCF of water are used. ( ) ( ) 25 2 if 0 15 55 4 15 if 15 x x f x x x + = + −
- 26. Writing Piecewise Functions d) Graph the function.
- 27. Greatest-Integer Function • The greatest-integer function, usually written is defined by saying that [x] denotes the largest integer that is less than or equal to x. • Be careful with negatives! Notice that –3 is less than –2.6! ( ) , f x x = [8] 8 [7.45] 7 [] 3 [-1] -1 [-2.6] -3
- 28. Greatest-Integer Function • Here’s another way to look at it: • Here’s how the graph looks: x –2 x –1 –1 x 0 0 x 1 1 x 2 [x] –2 –1 0 1 The graph should demonstrate why this is an example of a step function.
- 29. Step Functions • Example: To mail a letter to Canada in 2016, the U.S. Post Office charged $1.20 for the first ounce, an additional $0.47 for an additional two ounces, and then an additional $0.49 up to a maximum weight of 3.5 ounces. If L(x) represents the cost of sending a letter weighing x ounces, graph L(x) for x in the interval (0, 3.5].
- 30. Step Functions • Example: To mail a letter to Canada in 2016, the U.S. Post Office charged $1.20 for the first ounce, an additional $0.47 for an additional two ounces, and then an additional $0.49 up to a maximum weight of 3.5 ounces. If L(x) represents the cost of sending a letter weighing x ounces, graph L(x) for x in the interval (0, 3.5]. • Let’s figure out our steps: 0 1: $1.20 1 3: $1.20 $0.47 $1.67 3 3.5: $1.67 $0.49 $2.16 x x x + = + =
- 31. Step Functions • This would give us a function that looked like: • And a graph of ( ) 1.2 if 0 1 1.67 if 1 3 2.16 if 3 3.5 x L x x x =
- 32. Reading and Interpreting Graphs • Graphs are often used in business and the social sciences to present data (recall the NASDAQ graph from last class). It is just as important to know how to read such graphs as it is to construct them.
- 33. Reading and Interpreting Graphs • Example: The graph below shows the median sales prices (in thousands of dollars) of new privately owned, one- family houses by region of the United States (Northeast, Midwest, South, and West) for the years 2008 to 2015. a) How do prices in the West compare with those in the Midwest?
- 34. Reading and Interpreting Graphs a) How do prices in the West compare with those in the Midwest? The houses in the West started out about $100,000 higher than those in the Midwest. In both regions, prices declined initially, but they rebounded. By 2012, the gap between the two regions had shrunk to less than $50,000, but then widened again to about $75,000 by 2015.
- 35. Reading and Interpreting Graphs b) Which region typically had the highest prices?
- 36. Reading and Interpreting Graphs a) Which region typically had the highest prices? The Northeast had the highest median sales price for all years shown.
- 37. Reading and Interpreting Graphs • Example: The average price f(x) of a barrel of crude oil (in dollars) in year x is shown for the years 2005 to 2015, with projected prices shown through the year 2025.
- 38. Reading and Interpreting Graphs a) Estimate the price of crude in 2015 and 2025.
- 39. Reading and Interpreting Graphs a) Estimate the price of crude in 2015 and 2025. 2015: $50/barrel 2025: $80/barrel
- 40. Reading and Interpreting Graphs b) During what period did oil prices drop below $40 per barrel?
- 41. Reading and Interpreting Graphs b) During what period did oil prices drop below $40 per barrel? 2016
- 42. Vertical Line Test • No vertical line intersects the graph of a function y = f(x) at more than one point. • Notice the difference: Not a function Function
- 43. By Next Class • 3.2 Graphs of Functions in MyMathLab • Quiz 3.2 in Canvas • Notes on Section 3.3