Transcript Section 5 – 1 Relating Graphs to Events

Section 5 – 1 Relating Graphs to Events
• In this section you will mostly be reading and
interpreting graphs
• Things to look for/remember:
A) Notice the units on each of the axes
B) Read the graph from left to right
C) Points are named by their coordinates:
(x-coordinate, y-coordinate)
D) The x-coordinate tells you how to move
horizontally (left or right)
• Things to look for/remember (cont’d):
E) The y-coordinate tells you how to move
vertically (up or down)
F) The point where the x-axis meets the yaxis is called the origin
• Open your book to page 236 and we
will look at the examples together
Section 5 – 2 Relations and Functions
• A relation is a set of ordered pairs
• The domain is all of the x-coordinates in a
set of ordered pairs (all possible x-values)
• The range is all of the y-coordinates in a set
of ordered pairs (all possible y-values)
• When you are given a set of ordered pairs,
name the domain and range by the set of
numbers in set brackets { }
• A function is a relation that passes the
vertical line test
If you were to draw a graph, you could draw a
vertical line anywhere on that graph and it would
never hit two or more points at the same time
(no two points have the same x-value) in order
to be a function
See example 2 on page 242
• If any two y-values have the same x-value,
the relation is NOT a function
• A function rule is just an equation
• The domain in an equation is all of the input
values (all of the possible x-values)
• The range in an equation is all of the output
values (all of the possible y-values)
• f(x) is function notation (say “f of x”)
• Treat y = 2x + 5 the same as f(x) = 2x + 5
• Ex1. Find the domain and range of {(-2,4),
(3,5), (6,0), (-1,7)}. Is it a function?
• Ex2. Find the range of the function rule
f(x) = 3x – 6 when the domain is {-2, 0, 4}.
• Read ex. 3 on pg. 242 (mapping diagram)
Section 5-3 Function Rules, Tables,
and Graphs
• x is the independent variable (the one you
are inputting values for in the equation)
• y is the dependent variable (the values are
getting out from the equation)
• y is the dependent variable because it
depends on what x value you use
• To graph a relation, you can make a table of
x and y values and plot those values
• If none of the variables has an exponent
other than 1, your graph will be a line
• Linear graphs: y = mx + b
b is the y-intercept (it tells you where to put
a point on the y-axis)
m is the slope (from that y-intercept it tells
you how to move to make another point)
• Slope = rise/run = vertical change/horizontal change
• If a variable has an exponent other than
1 or absolute value symbols, you will
have to make a chart and plot enough
points to see the shape of the graph
• Graph each of the following:
• Ex1. y = -3x + 2
Ex2. y = ⅔x – 6
• Ex3. y = |x| + 2
Ex4. y = x² – 2
Section 5-4 Writing a Function Rule
• To write a function rule from a chart you must
determine what operation(s) you can perform on the
x-coordinate to get the corresponding y-coordinate
each and every time
• Ex1. Write a function rule
x
1
2
3
5
f(x)
3
6
9
15
• When writing a function rule from a word
problem, you will have to analyze the given
information and figure out how to put it all
together
• Good place to start: determine what is the
independent variable and what is the
dependent variable
• Ex2. Charles charges $15 per hour for
babysitting and a flat fee of $3 for bringing
his own movie with him. Write a rule to
describe his profit as a function of how many
n = # of hours
P(n) = total profit
hours he works.
P(n) = 15n + 3
Section 5 – 5 Direct Variation
• Direct variation describes the relationship
between two things
• For the variation to be direct, as one goes up
so must the other
• i.e. as you work more hours, you earn more
money (these two things are direct variation)
• Direct variation is written in the form y = kx
where k ≠ 0 (k is called the constant of
variation)
• With direct variation (y = kx), y is said to vary
directly with x (and vice versa)
• To determine whether or not an equation is
direct variation, solve for y and see if it ends
up in the form y = kx
If it does, then it is direct variation
If it does not (i.e. y = ⅜x – 5), then it is not
• Ex1. Write an equation of the direct variation
that includes the point (-2, 6)
y = kx
6 = k(-2)
k = -3
y = -3x
• Open your book to page 263 and read
example 4
• Ex2. Do the “Check Understanding” on
page 264
• Proportions are often used in direct
variation questions (see example 5 on page
264)
Section 5 – 6 Describing Number
Patterns
• Inductive reasoning is when you make
conclusions based on patterns you have
observed
• A conclusion you reach is conjecture,
because you are not sure whether or not it is
true yet (it is just an educated guess)
• You can use inductive reasoning to figure out
number patterns (called sequences)
• Each number in a sequence is called a term
• Arithmetic sequences are those in which
each term is determined by adding or
subtracting the same number to the
previous term (i.e. 3, 5, 7, 9, 11, …)
• They are said to have a common difference
• Arithmetic sequence A(n) = a + (n – 1)d
A(n) = nth term of the sequence
a = first term (many books write a1)
n = term number
d = common difference
• Ex1. Find the 4th, 7th, and 10th terms of the
sequence A(n) = 3 + (n – 1)(-5)
• Ex2. Find the next three terms in the
sequence and the constant difference:
3, -5, -13, -21,… . Then find the formula for
the sequence.