Bell Work
• Solve each equation for x.
1. 22 = 10 + .5(x – 1)
2. 30 = 16 + (1/4)(x – 1)
Lesson 4.5 Page 275 - Well,
Maybe It Is a Function!
Sequences and Functions
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
• The graphs of the arithmetic sequences from Lesson 4.4,
Thank Goodness Descartes Didn’t Drink Some Warm Milk!
are shown.
1. Identify the function family that represents the graphs of
the arithmetic sequences shown. Do you think all
arithmetic sequences belong to this function family?
Explain your reasoning
– The graphs belong to the linear function family.
– The graph of any arithmetic sequence is linear, because both
arithmetic sequences and linear functions have a constant rate
of change.
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
• Consider the explicit formula for the
arithmetic sequence shown in the first graph.
an= -4 + 2(n – 1)
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
3. Based on the formulas, identify the function
family of these arithmetic sequences. Explain
your reasoning.
– The arithmetic sequences belong to the linear
function family because they are all in the form
f(x)=mx+b.
4. What is the relationship between the common
difference of an arithmetic sequence and the
slope of a linear function?
– The common difference of an arithmetic sequence is
the slope of a linear function.
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
5. Hank says that the y-intercept of a linear
function is the same as the first term of an
arithmetic sequence. Is he correct? Why or why
not?
– Hank is not correct. Every term of a sequence begins
with term number 1. When the sequence is graphed,
the x-coordinate of the first term is 1. The y-intercept
of a linear function is on the y-axis (where x=0). So,
the first term can never be plotted on the y-axis.
6. Represent the y-intercept of an arithmetic
sequence algebraically.
– a1 – d
Problem #1 – If It Looks Like a Function
and Quacks Like a Function…
7. Complete the table by writing each part of
the linear function that corresponds to each
part of the arithmetic sequence
Dependent variable f(x)
Slope m
Independent variable x
y-intercept b
Problem #2 - And if it Swims like a
Function and Smells like a Function…
• The graphs of the geometric sequences from Lesson 4.4,
Thank Goodness Descartes Didn’t Drink Some Warm Milk!
are shown.
1. Do all of the graphs of the geometric sequences belong to
the same function family? Why or why not?
– The graphs of the geometric sequences do not belong to the
same function family. Five of the graphs belong to the
exponential function family. Two of the graphs do not appear to
belong to any of the function families we’ve seen earlier in this
chapter.
Problem #2 - And if it Swims like a
Function and Smells like a Function…
• Consider the explicit formula for the
geometric sequence shown in the first graph.
gn=45*2n-1
Problem #2 - And if it Swims like a
Function and Smells like a Function…
Problem #2 - And if it Swims like a
Function and Smells like a Function…
3.
Based on the formulas, do the geometric sequences belong to the
same function family? Explain your reasoning.
– No. Only the geometric sequences with positive common ratios belong
to the exponential function family because the definition of an
exponential function states that the base of the exponential function
must be greater than 0 but not equal to 1.
Problem #2 - And if it Swims like a
Function and Smells like a Function…
4.
What is the relationship between the
common ratio of a geometric sequence
and the base of the power in an
exponential function?
–
5.
What is the relationship between the
first term of a geometric sequence and
the coefficient of the power in an
exponential function?
–
6.
The common ratio of a geometric sequence is
the base of an exponential function.
The first term of a geometric sequence is the
coefficient of the power of an exponential
function.
Complete the table by writing each part
of the exponential function that
corresponds to each part of the
geometric sequence.
Dependent variable
Coefficient of a
power a
Base of power b
Independent variable
(exponent) x
Talk the Talk
1. Complete each statement with always,
sometimes, or never. Explain your reasoning
for each statement.
a. An arithmetic sequence can ____________be
always
represented as a linear function whose domain is
the set of natural numbers.
sometimes
b. A geometric sequence can ___________be
represented as an exponential function whose
domain is the set of natural numbers
Talk the Talk
2.
Determine whether the statement is true or
false. If it is false, explain why it is false.
Remember, the coordinate plane is split into
four quadrants, as shown (at right)
a.
An arithmetic sequence will always begin in
Quadrant 1.
a.
b.
An arithmetic sequence will never begin in
Quadrant 3.
a.
c.
True
A geometric sequence will sometimes begin
in Quadrant 2.
a.
d.
False – It can begin in quadrant 4
False – all geometric sequences are in Quadrants
1 and 4
A geometric sequence will always begin in
Quadrant 4.
a.
False it can begin in Quadrant 1