Bell Work Solve each equation for x = (x – 1) 2.30 = 16 + (1/4)(x – 1)

Lesson 4.5 Page 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. a n = (n – 1)

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. – a 1 – 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. g n =45*2 n-1

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? – The common ratio of a geometric sequence is the base of 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? – The first term of a geometric sequence is the coefficient of the power of an exponential function. 6.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 represented as a linear function whose domain is the set of natural numbers. b.A geometric sequence can ___________be represented as an exponential function whose domain is the set of natural numbers always sometimes

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.False – It can begin in quadrant 4 b.An arithmetic sequence will never begin in Quadrant 3. a.True c.A geometric sequence will sometimes begin in Quadrant 2. a.False – all geometric sequences are in Quadrants 1 and 4 d.A geometric sequence will always begin in Quadrant 4. a.False it can begin in Quadrant 1