1. Bell Ringer
Get out your notebook and prepare to take notes on Chapter 8
From Chapter 1, how do we plot points on a graph?
i.e. −2, 4

2. Linear Functions and Graphing
Chapter 8
Linear Functions and Graphing

3. 8.1 – Relations and Functions (Page 404)
Essential Question:
What is the difference between a function and a relation?

4. 8.1 cont. Relation: A set of ordered pairs
Note: { } are the symbol for "set"
Examples:
{ (0,1) , (55,22), (3,-50) }
{ (0, 1) , (5, 2), (-3, 9) }
{ (-1,7) , (1, 7), (33, 7), (32, 7) }
Any group of numbers is a relation as long as the numbers come in pairs

5. 8.1 cont. DOMAIN: 𝟐, 𝟔, 𝟗, 𝟐 RANGE: 𝟓, 𝟖, 𝟓, 𝟏𝟏 Example 1:
What is the domain and range of the following relation: 2,5 , 6,8 , 9,5 , 2,11 ?
DOMAIN: 𝟐, 𝟔, 𝟗, 𝟐
RANGE: 𝟓, 𝟖, 𝟓, 𝟏𝟏

6. 8.1 cont.
Domain
First coordinates of the relation
Range
Second coordinates of the relation
Tip: Alphabetically x comes before y, and “domain” comes before “range”
DOMAIN
𝒙,𝒚
RANGE

7. 8.1 cont. Function: Special type of relation
Set of ordered pairs containing a domain and range (like a relation)
Each member of the domain 𝑥 is paired with EXACTLY one member of the range 𝑦
SOME relations are functions
Tip: Think of a function as a number generator

8. 8.1 cont. Mapping diagrams: Shows whether a relation is a function
Steps:
List domain values and range values in order
Draw arrows from domain values to corresponding range values
2 range values for domain value 1
NOT a function
1 range value for each domain value
IS a function
1 range value for each domain value
IS a function

9. No, there are two range values for the domain value 2
8.1 cont.
Example 2:
Is the following relation a function: −2,3 , 2,2 , 2,−2 ?
Explain. 3 −2 2 2 −2
No, there are two range values for the domain value 2

10. This relation is a function!
8.1 cont.
Functions can model everyday situations when one quantity depends on another
One quantity is a function of the other
Example 3:
Is the time needed to cook a turkey a function of the weight of the turkey? Explain.
The time the turkey cooks (range value) is determined by the weight of the turkey (domain value).
This relation is a function!

11. 8.1 cont. Vertical-Line Test
Visual way of telling whether a relation is a function
If you can find a vertical line that passes through two points on the graph, then the relation is NOT a function

12. Therefore, the relation is NOT a function.
8.1 cont.
Example 4:
Graph the relation shown in the following table:
Using the vertical line test, you would pass through both 𝟐,𝟎 and 𝟐,𝟑 .
Therefore, the relation is NOT a function.

13. 8.1 - Closure
What is the difference between a function and a relation?
Any set of ordered pairs is a relation
A function is a relation with the restriction that no two of its ordered pairs have the same first coordinate

14. 8.1 - Homework
Page , 2-28 even

15. No, there are two range values for the domain value −𝟐.
Bell Ringer
Get out your notebook and prepare to take notes on Section 8.2
Draw a mapping diagram for the following relation:
−1, 1 , −2, 1 , −2, 2 , 0, 2
Is the relation a function? Explain. −𝟏 −𝟐 𝟎 𝟏 𝟐
No, there are two range values for the domain value −𝟐.

16. 8.2 – Equations With Two Variables (Page 409)
Essential Questions:
What is the solution of an equation with two variables?
How can you graph an equation that has two variables?

17. 8.2 cont. 𝒚=𝟑𝒙+𝟒 Equations With Two Variables:
Solution: ordered pair that makes the equation true
Can have multiple ordered pair solutions [i.e. 𝑥, 𝑦 = ?, ? ]
𝒚=𝟑𝒙+𝟒

18. 8.2 cont. SOLUTION = −𝟏, 𝟏 𝒚=𝟑𝒙+𝟒 𝒚=𝟑 −𝟏 +𝟒 𝒚=−𝟑+𝟒 𝒚=𝟏 Example 1:
Find the solution of 𝑦=3𝑥+4 for 𝑥=−1.
𝒚=𝟑𝒙+𝟒
𝒚=𝟑 −𝟏 +𝟒
𝒚=−𝟑+𝟒
𝒚=𝟏
SOLUTION = −𝟏, 𝟏

19. 8.2 cont. Linear Equations: Any equation whose graph is a line
ALL equations in this lesson are linear equations
Solutions can be shown in a table or graph

20. 8.2 cont.
Example 2:
For the following equation, make a table of values to show solutions. Then, graph your results.
𝒚=𝟐𝒙+𝟏 𝒙
𝟐𝒙+𝟏 𝒚
2 0 +1=0+1=1 1
2 1 +1=2+1=3 3 2
2 2 +1=4+1=5 5
2 3 +1=6+1=7 7

21. 8.2 cont. 𝒚=𝟐𝒙+𝟏 Vertical Line Test NOT A FUNCTION!!
Every x-value has exactly 1 y-value
Therefore, this relation IS a function
Linear equations are functions unless its graph is a vertical line
NOT A FUNCTION!!
𝒚=𝟐𝒙+𝟏

22. 8.2 cont. 𝒚=−𝟑: YES 𝒙=𝟒: NO Example 3:
Graph the following equations: 𝑦=−3 and 𝑥=4. Is each equation a function? 𝒙 𝒚 −3 1 2 3 𝒙 𝒚 4 1 2 3
𝒚=−𝟑: YES
𝒙=𝟒
𝒚=−𝟑
𝒙=𝟒: NO

23. 8.2 cont. 𝟐𝒙+𝒚=𝟑 −𝟐𝒙 −𝟐𝒙 𝒚=𝟑−𝟐𝒙 Example 4:
Solve 2𝑥+𝑦=3 for 𝑦. Then, graph the equation.
𝟐𝒙+𝒚=𝟑
−𝟐𝒙
−𝟐𝒙
𝒚=𝟑−𝟐𝒙 𝒙
𝟑−𝟐𝒙 𝒚
3−2 0 =3−0=3 3 1
3−2 1 =3−2=1 2
3−2 2 =3−4=−1 −1
3−2 3 =3−6=−3 −3

24. 8.2 - Closure What is the solution of an equation with two variables?
Any ordered pair that makes the equation a true statement
How can you graph an equation that has two variables?
Make a table of values to show ordered-pair solutions of the equation
Graph the ordered pairs, then draw a line through the points

25. 8.2 - Homework
Page 412, 2-36 even

26. Bell Ringer
Get out your 8.2 homework assignment
Get out your notebook and prepare to take notes on Section 8.3
Is the ordered pair −3, −2 a solution of 4𝑥−3𝑦=6? Why or why not?
𝟒 −𝟑 −𝟑 −𝟐 =𝟔
−𝟏𝟐+𝟔=𝟔
−𝟔≠𝟔

27. 8.3 – Slope and y-intercept (Page 415)
Essential Question:
What is an easier way to graph linear equations?

28. 8.3 cont. Slope: Ratio that describes the tilt of a line
To calculate slope, use the following ratio:

29. 8.3 cont.
NEGATIVE SLOPE
POSITIVE SLOPE
UNDEFINED SLOPE
ZERO SLOPE

30. 8.3 cont.
Example 1:
Find the slope of a line that includes the points −6, 2 and −2, 4 . 𝟒
−𝟐, 𝟒 𝟐
−𝟔, 𝟐
Slope = 𝟐 𝟒 = 𝟏 𝟐

31. 8.3 cont.
Horizontal and Vertical Lines:

32. 8.3 cont. = 𝟏−𝟔 𝟖−𝟐 =− 𝟓 𝟔 𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏 Example 2:
Find the slope of the line through the following pair of points:
2, 6 and 8, 1
How can we find the slope without graphing??
𝒙 𝟏 , 𝒚 𝟏
𝒙 𝟐 , 𝒚 𝟐
= 𝟏−𝟔 𝟖−𝟐 =− 𝟓 𝟔
𝒎= 𝒚 𝟐 − 𝒚 𝟏 𝒙 𝟐 − 𝒙 𝟏

33. 8.3 cont. y-intercept: Slope-intercept form:
Point where the line crosses the y-axis
y-intercept is the “Constant” in the equation
Slope-intercept form:
𝑦=𝑚𝑥+𝑏
Where m is the slope of the line, and b is the y-intercept
Slope-intercept form is helpful in graphing equations

34. 8.3 cont.
Example 3:
Find the slope and y-intercept of 𝑦=− 1 3 𝑥+2; then, graph the equation.

35. 8.3 - Closure USE SLOPE-INTERCEPT FORM!!
What is an easier way to graph linear equations?
USE SLOPE-INTERCEPT FORM!!

36. Page , 2-18 even, even
8.3 - Homework

37. Bell Ringer 𝒚=𝟓+ 𝟐 𝟑 𝒙 Get out your 8.3 homework assignment
Get out your notebook and prepare to take notes on Section 8.4
Solve 3𝑦−2𝑥=15 for y. Then identify the slope and the y-intercept of the equation.
y-intercept
𝒚=𝟓+ 𝟐 𝟑 𝒙
slope

38. 8.4 – Writing Rules for Linear Functions (Page 422)
Essential Question:
How can we use tables and graph to write a function rule?

39. 8.4 cont. Function Notation: Function Rule: Use 𝑓 𝑥 instead of 𝑦
𝑓 𝑥 is read “f of x”
Function Rule:
Equation that describes a function

40. 8.4 cont.
Example 1:
A long-distance phone company charges its customers a monthly fee of $4.95 plus 9 cents for each minute of a long-distance call.
Write a function rule that relates the total monthly bill to the number of minutes a customer spent on long-distance calls.
Find the total monthly bill if the customer made 90 minutes of long-distance calls.
Let 𝒎= minutes spent on long distance calls
Let 𝒕 𝒎 = total monthly bill
𝒕 𝒎 =𝟒.𝟗𝟓+.𝟎𝟗𝒎
Evaluate the function for 𝒎=𝟗𝟎
𝒕 𝟗𝟎 =𝟒.𝟗𝟓+.𝟎𝟗 𝟗𝟎
𝒕 𝟗𝟎 =𝟏𝟑.𝟎𝟓
$𝟏𝟑.𝟎𝟓

41. 8.4 cont. Writing Function Rules From Tables or Graphs
Look for a pattern!
May need to add, subtract, multiply, divide, or use a power
OR a combination of these operations

42. 8.4 cont. Example 2: 𝒇 𝒙 =𝟐𝒙 𝒇 𝒙 =−𝟐𝒙 𝒚=𝟐𝒙+𝟏
Write a rule for each of the following linear function tables:
𝒇 𝒙 =𝟐𝒙
𝒇 𝒙 =−𝟐𝒙
𝒚=𝟐𝒙+𝟏

43. 8.4 cont.
Example 3:
Write a rule for the linear function graphed below:

44. 8.4 - Closure
How can we use tables and graphs to write a function rule?
Look for a pattern using a combination of addition, subtraction, multiplication, division, and powers
Use slope and y-intercept to write a linear function

45. 8.4 - Homework
Page , 2-22 even

46. Bell Ringer
Plot your given point on the coordinate plane:

47. 8.5 – Scatter Plots (Page 427) Essential Question:
How can we make scatter plots and use them to find a trend?

48. 8.5 cont.
Scatter Plots:
Shows a relationship between two sets of data

49. 8.5 cont.
Example 1:
Make a scatter plot for the data in the table below:
Value (in thousands)
Age (in years)

50. 8.5 cont.
Example 2:
Make a scatter plot for the data below:

51. 8.5 cont.
Trends:

52. 8.5 cont. Trend Line: Shows relationship between data sets
Allows us to make predictions about data values
Possible to have no trend line

53. Example 3:
Use the following scatter plot to predict the height of a tree that has a circumference of 175 in:
88 ft

54. 8.5 - Closure
How can we make scatter plots and use them to find a trend?
Plot ordered pairs
Draw a trend line (positive, negative, or no trend)
Predict values

55. 8.5 - Homework
P ; 2-30 even