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
Linear Functions and
Graphing
8.1 – Relations and Functions
(Page 404)
 Essential Question:
 What is the difference between a function and a relation?
8.1 cont.
 Relation:
 A set of ordered pairs
 Note: { } are the symbol for "set"
 Examples:
1.
2.
3.
{ (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
8.1 cont.
 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
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
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
8.1 cont.
 Mapping diagrams:
 Shows whether a relation is a function
 Steps:
1.
2.
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 1 range value for each domain value
IS a function
IS a function
8.1 cont.
 Example 2:
 Is the following relation a function:
Explain.
−2
2
−2,3 , 2,2 , 2, −2
3
2
−2
No, there are two range values for the domain value 2
?
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!
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
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.
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
8.1 - Homework
Page 407-408, 2-28 even
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
−𝟏
𝟏
Is the relation a function? Explain.
No, there are two range values
for the domain value −𝟐.
−𝟐
𝟐
𝟎
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?
8.2 cont.
 Equations With Two Variables:
𝒚 = 𝟑𝒙 + 𝟒
 Solution: ordered pair that makes the equation true
 Can have multiple ordered pair solutions [i.e. 𝑥, 𝑦 = ? , ? ]
8.2 cont.
 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
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
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
0
2 0 +1=0+1= 1
1
1
2 1 +1=2+1= 3
3
2
2 2 +1=4+1= 5
5
3
2 3 +1=6+1= 7
7
8.2 cont.
 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
𝒚 = 𝟐𝒙 + 𝟏
8.2 cont.
 Example 3:
 Graph the following equations: 𝑦 = −3 and 𝑥 = 4. Is each
equation a function?
𝒙 𝒚
𝒙 𝒚
𝒙=𝟒
0
−3
4
0
1
−3
4
1
2
−3
4
2
3
−3
4
3
𝒚 = −𝟑: YES
𝒚 = −𝟑
𝒙 = 𝟒: NO
8.2 cont.
 Example 4:
 Solve 2𝑥 + 𝑦 = 3 for 𝑦. Then, graph the equation.
𝟐𝒙 + 𝒚 = 𝟑
−𝟐𝒙
−𝟐𝒙
𝒚 = 𝟑 − 𝟐𝒙
𝒙
𝟑 − 𝟐𝒙
𝒚
0
3−2 0 =3−0=3
3
1
3−2 1 =3−2=1
1
2
3 − 2 2 = 3 − 4 = −1
−1
3
3 − 2 3 = 3 − 6 = −3
−3
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
8.2 - Homework
Page 412, 2-36 even
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
8.3 – Slope and y-intercept
(Page 415)
 Essential Question:
 What is an easier way to graph linear equations?
8.3 cont.
 Slope:
 Ratio that describes the tilt of a line
 To calculate slope, use the following ratio:
8.3 cont.
NEGATIVE SLOPE
POSITIVE
SLOPE
UNDEFINED
SLOPE
ZERO SLOPE
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
𝟐
𝟒
Slope = =
𝟏
𝟐
8.3 cont.
 Horizontal and Vertical Lines:
8.3 cont.
 Example 2:
 Find the slope of the line through the following pair of points:
2, 6 and 8, 1
𝒙𝟏 , 𝒚𝟏
𝒙𝟐 , 𝒚𝟐
 How can we find the slope without graphing??
𝟓
𝒚𝟐 − 𝒚𝟏 𝟏 − 𝟔
=
=−
𝒎=
𝟔
𝒙𝟐 − 𝒙 𝟏 𝟖 − 𝟐
8.3 cont.
 y-intercept:
 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
8.3 cont.
 Example 3:
1
3
 Find the slope and y-intercept of 𝑦 = − 𝑥 + 2; then, graph the equation.
8.3 - Closure
 What is an easier way to graph linear equations?
USE SLOPE-INTERCEPT FORM!!
8.3 - Homework
Page 418-419, 2-18
even, 24-38 even
Bell Ringer
 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
𝟐
𝒚=𝟓+ 𝒙
𝟑
slope
8.4 – Writing Rules for Linear
Functions (Page 422)
 Essential Question:
 How can we use tables and graph to write a function rule?
8.4 cont.
 Function Notation:
 Use 𝑓 𝑥 instead of 𝑦
 𝑓 𝑥 is read “f of x”
 Function Rule:
 Equation that describes a function
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.
a.
Write a function rule that relates the total monthly bill to the number of
minutes a customer spent on long-distance calls.
b.
Find the total monthly bill if the customer made 90 minutes of longdistance 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
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
8.4 cont.
 Example 2:
 Write a rule for each of the following linear function tables:
𝒇 𝒙 = 𝟐𝒙
𝒇 𝒙 = −𝟐𝒙
𝒚 = 𝟐𝒙 + 𝟏
8.4 cont.
 Example 3:
 Write a rule for the linear function graphed below:
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
8.4 - Homework
Page 424-425, 2-22
even
Bell Ringer
 Plot your given point on the coordinate plane:
8.5 – Scatter Plots (Page 427)
 Essential Question:
 How can we make scatter plots and use them to find a trend?
8.5 cont.
 Scatter Plots:
 Shows a relationship between two sets of data
8.5 cont.
 Example 1:
Value (in thousands)
 Make a scatter plot for the data in the table below:
Age (in years)
8.5 cont.
 Example 2:
 Make a scatter plot for the data below:
8.5 cont.
 Trends:
8.5 cont.
 Trend Line:
 Shows relationship between data sets
 Allows us to make predictions about data values
 Possible to have no trend line
 Example 3:
 Use the following scatter plot to predict the height of a tree that has a
circumference of 175 in:
88 ft
8.5 - Closure
 How can we make scatter plots and use them to find a trend?
Plot ordered pairs
2. Draw a trend line (positive, negative, or no trend)
3. Predict values
1.
8.5 - Homework
P 430-432; 2-30 even