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```M3: Section 12.8, 8.1-8.3 Notes
Page 1 of 20
Chapter 8 Notes
Linear Functions
Sections
12.8, 8.1-8.3
Name_____________________Pd.___
M3: Section 12.8, 8.1-8.3 Notes
Sections 12.8, 8.1-8.3
List of Vocabulary Words:
Section 12.8:
 Sequence
 Term
 Arithmetic sequence
 Common difference
 Geometric sequence
 Common ratio
Section 8.1:
 relation
 domain
 range
 input
 output
 function
 vertical line test
Section 8.2:
 equation in two variables
 solution of an equation in two variables
 graph
 linear equation
 function form
Section 8.3:
 x-intercept
 y-intercept
Page 2 of 20
M3: Section 12.8, 8.1-8.3 Notes
Page 3 of 20
Section 12.8: Sequences
Learning Goal: We will extend and graph sequences.
Vocabulary:
 Sequence –
 Term –
 Arithmetic sequence – a sequence in which the difference
between consecutive terms is constant
 Common difference – the difference between consecutive terms
of an arithmetic sequence
EXAMPLE 1: Extending Arithmetic Sequences
Find the common difference for the arithmetic sequence. Then find
the next three terms.
a. 32, 39, 46, 53, 60,…
b. 88, 83, 78, 73, 68,…
M3: Section 12.8, 8.1-8.3 Notes
c. 12, 17, 22, 27, 32,…
Page 4 of 20
d. 5, 3, 1, -1, -3, …
 Geometric sequence – a sequence in which the ratio of any term
to the previous term is constant
 Common ratio – the ratio of any term of a geometric sequence to
the previous term of the sequence
EXAMPLE 2: Extending Geometric Sequences
Find the common ratio for the geometric sequence. Then find the next
three terms.
a. 24, 48, 96, 192, …
b. 625, 125, 25, 5, …
c. 2, 6, 18, 54, …
d. 160, 80, 40, 20, …
M3: Section 12.8, 8.1-8.3 Notes
Page 5 of 20
EXAMPLE 3: Using Sequences
You are saving for \$1100 laptop. In August, you save \$380 from a
summer job. Starting in September, you save \$120 per month from a
part time job. After how many months of saving will you have enough
money for the laptop?
M3: Section 12.8, 8.1-8.3 Notes
EXAMPLE 4: Graphing an Arithmetic Sequence
Graph the arithmetic sequence 25, 21, 17, 13, 9, …
Graph the arithmetic sequence 5, 10, 15, 20, 25, …
EXAMPLE 5: Graphing a Geometric Sequence
Graph the sequence 20, 30, 45, 67.5, 101.25, ….
Page 6 of 20
M3: Section 12.8, 8.1-8.3 Notes
Page 7 of 20
Section 8.1: Relations and Functions
Learning Goal: We will use graphs to represent relations and functions.
Vocabulary:
 Relation – a pairing of numbers in one set (the domain) with the
numbers in another set (range)
 Domain –
 Range –
 Input –
 Output –
EXAMPLE 1: Identifying the Domain and Range
Identify the domain and range of the relation represented by the
table below, which shows one alligator’s length at different ages.
M3: Section 12.8, 8.1-8.3 Notes
Page 8 of 20
Identify the domain and range of the relation.
**In addition to using ordered pairs or a table to represent a relation,
you can also use a ______________ or a _____________________.
EXAMPLE 2: Representing a Relation
Represent the relation (2, 0), (1, -1), (2, 2), (0, 0), (-1, 1) as indicated.
a. A graph.
b. A mapping diagram
Represent the relation (-1, 1), (2, 0), (3, 1), (3, 2), (4, 5), as indicated.
a. A graph.
b. A mapping diagram
M3: Section 12.8, 8.1-8.3 Notes
Page 9 of 20
 Function – a relation with the property that for each input there
is exactly one output
EXAMPLE 3: Identifying Functions
Tell whether the relation is a function using a mapping diagram. Explain
a. (1, 60), (2, 100), (3, 160),
b. (2, 0), (1, 1), (2, 2), (0, 0),
(4, 210), (5, 270)
(-1, 1)
Tell whether the relation is a function. Explain your reasoning.
a. (0, 3), (1, 2), (2, -1), (4, 4),
b. (-2, -1), (0, 2), (2, 3), (-2, -4)
(5, 4)
 Vertical line test – for a relation represented by a graph, if any
vertical line passes through more than one point of the graph,
then the relation is not a function. If no vertical line passes
through more than one point of the graph, then the relation is a
function.
M3: Section 12.8, 8.1-8.3 Notes
Page 10 of 20
EXAMPLE 4: Using the Vertical Line Test
Tell whether the relation represented by the graph is a function.
a.
b.
M3: Section 12.8, 8.1-8.3 Notes
Page 11 of 20
Section 8.2: Linear Equations in Two Variables
Learning Goal: We will find solutions of equations in two variables.
Vocabulary:
 Equation in two variables – an equation that contains two
different variables
 Solution – an ordered pair (x, y) that produces a true statement
when the coordinates of the ordered pair are substituted for the
variables in the equation
EXAMPLE 1: Checking Solutions
Tell whether the ordered pair is a solution of
a. (4, 2)
b. (2, 1)
x  3 y  1.
M3: Section 12.8, 8.1-8.3 Notes
Page 12 of 20
EXAMPLE 2: Finding Solutions
The number of pages p that Donald has left to write for his 30-page
research paper depends on how many days d he writes 5 pages per day.
This situation can be modeled by the equation p  30  5d .
a. Make a table of solutions for the equation.
b. How many days does he need to finish the paper?
M3: Section 12.8, 8.1-8.3 Notes
Page 13 of 20
 Graph – the set of points in a coordinate plane that represent all
the solutions of the equation
 Linear equation – an equation whose graph is a line
EXAMPLE 3: Graphing a Linear Equation
Graph
y
1
x 1.
2
Graph the equation.
a. y   x  3
b.
y  3x  4
M3: Section 12.8, 8.1-8.3 Notes
**The graph of the equation y
Page 14 of 20
 b is the _________________ line
through (0, b).
**The graph of the equation x  a is the _________________ line
through (a, 0).
EXAMPLE 4: Graphing Horizontal and Vertical Lines
Graph the equation.
a. x  1
b. y  4
Graph y  1 and x  4 . Tell whether each equation is in function form.
M3: Section 12.8, 8.1-8.3 Notes
Page 15 of 20
 Function form – an equation that is solved for y
EXAMPLE 5: Writing an Equation in Function Form
a. Write 5 x  y  8 in function
b. Write 3 x  2 y  6 in function
form.
form.
c. Write
x  3 y  3 in function form. Then graph the equation using
a t-chart.
M3: Section 12.8, 8.1-8.3 Notes
Page 16 of 20
1
a. Write x  y  3 in function form. Then graph the equation using
2
a t-chart.
b. Write
2 x  3 y  3 in function form. Then graph the equation
using a t-chart.
M3: Section 12.8, 8.1-8.3 Notes
Page 17 of 20
Section 8.3: Using Intercepts
Learning Goal: We will use x- and y-intercepts to graph linear equations.
**You can graph a linear equation quickly by recognizing that only ____
points are needed to draw a line.
Vocabulary:
 x-intercept – the x-coordinate of a point where a graph crosses
the x-axis
 y-intercept – the y-coordinate of a point where a graph crosses
the y-axis
M3: Section 12.8, 8.1-8.3 Notes
Page 18 of 20
EXAMPLE 1: Finding Intercepts of a Graph
Find the intercepts of the graph of
2y  x  2 .
Find the intercepts of the equation’s graph.
1. x  2 y  2
2. 4 x  3 y  12
3. y  2 x  8
EXAMPLE 2: Using Intercepts to Graph a Linear Equation
Graph the equation
2y  x  2 .
(Use the intercepts from Example 1)
M3: Section 12.8, 8.1-8.3 Notes
Page 19 of 20