ࡱ>  57 ./01234F bjbjVV 3K<<u7NN4h1Mwt &"&&&MMMvvvvvvv$xc{j>vM HMMM>v&&C!wXXXMT&&vXMvXXqKJXt&`vtN( rvw0Mwr{XR{4XtXt8{ttMMXMMMMM>v>v`XdMMMMwMMMM{MMMMMMMMMN W:  Functions Video 1, Popper 04 Increasing/decreasing graphs Basic Graphs Worksheets Linear Quadratic Cubic Cube Root Square Root Absolute Value Exponential Logarithmic Video 2, Popper 05 Rational Functions Video 3, Popper 06 Polynomials End behavior x-intercepts and behavior at them Working backwards Sequences Notation Domain and Range Bounded Sequences Appendices Interval Notation Factoring The Guaranteed Method Complete the Square Method Factoring by Grouping In this module, we will be looking more deeply at functions in particular, the graphs of functions. Starting off with 3 important properties which describe the behavior of the graph, well define increasing, decreasing, and neither and see how certain graphs exhibit these properties. Then well review of various families of graphs lots of types of functions, all except rational functions and polynomials Well look at only rational functions in the second video and only polynomials in the third video. These types of functions are so important that they each get a separate video! Increasing and decreasing graphs We often talk of a turn-around point, a place where a graph changes directionso it is important to be able to talk about what is happening at a turnaround point. Most graphs can be described as increasing, decreasing or neitherand knowing where the graph is exhibiting these properties helps us understand the behavior that the graph represents. Lets look at a graph that has all three movements to get an intuitive sense of the descriptions:  Domain Range Intercepts Increasing Decreasing Neither An application of the concept: If you graph a revenue function for a retail item, the most common shape is a cupped down parabola. Now, firms are always wanting to increase revenue. Lets look at how increasing/decreasing/neither would work to alert a manager of what to do to keep the revenues flowing in. We have months on the x-axis and dollars on the y- axis here. First off, why is revenue cupped down? And what do the parts of the graph below the x-axis mean?  Where is the graph incr, decr, and n? Secondly: When is the BEST time to have a sale? Identifying exactly WHICH x is the x of the turn-around point is a focus of this class. We will learn how to do this in the Derivatives section. We can identify increasing and decreasing on a graph and note that neither occurs at a turn-around point (or if the graph is a flat horizontal line). We always do this in a certain way. First we look at the xs and we scan the axis from left to right. Note that the smaller xs are on the left. In fact, if we number a first x and a second x moving left to right, we have:  EMBED Equation.3 . Then we check the motion of the y values on the y axis as the xs get larger and larger. So identifying the relevant property is a sort of scanning process. If the y values are moving UP the y axis as they get used, then the function is increasing. If the y values are moving DOWN the y axis as they get used, then the function is decreasing. This has nothing to do with the sign of the y value a graph can be in Quadrant 3 with negative numbers for x and y and still be increasing (see the first example below). For example: Here is the graph of y = 0.5x ( 6 Domain Range Intercepts Increasing Decreasing Turn-around points: This graph of a line is increasing everywhere on its domaineven in Quadrant 3 and Quadrant 4 where the ys are negative numbers. Another example: Here is the graph of  EMBED Equation.3 . What are the domain, range, and intercepts? Where is the graph increasing, decreasing, or neither? Domain Range Intercepts Increasing Decreasing Turnaround point: Note that you always specify where a graph is increasing or decreasing on intervals of the x axis, even though youre talking about the behavior of the y values! Here is the graph of  EMBED Equation.3 , an exponential graph. The domain is all real numbers and the range is  EMBED Equation.DSMT4 . Note the leftward half-asymptote, too! Are there any turn-around points? Lets trace out how we know its increasing on its domain:  Heres a polynomial. The domain and range are each all real numbers. What is the formula for this polynomial and how would you know? Where is the graph increasing? Decreasing? Neither?  Basic Graphs Worksheets Here well look at various families of graphs, always starting with the basic or parent graph and then looking at a couple of shifted graphs. Well review the function facts like domain, range, and intercepts along with discussing increasing/decreasing/neither for each family. For an excellent review of graph shifting, check out the videos in the UH Math 1310 textbook for College Algebra:  HYPERLINK "http://online.math.uh.edu/Math1310/" http://online.math.uh.edu/Math1310/ Look for Chapter 3, Section 4 on transforming graphs. While youre there you might check around for any other topics you want a review onthe videos are just excellent! Basic Linear Function:   EMBED Equation.DSMT4  Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: (0, 0) y intercept: (0, 0) Increasing everywhere. This function is one-to-one. f(x) =  EMBED Equation.DSMT4   Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: ((5/3, 0) y intercept: (0, 5) Increasing on its domain! Telling whats happening depends on the slope, m, of the line. Lets see how  EMBED Equation.DSMT4  m = 0 Intercepts: Neither!  EMBED Equation.DSMT4  m > 0 Intercepts: Increasing on its domain!  EMBED Equation.DSMT4  m < 0 Intercepts: Decreasing on its domain! Graph below for demo! Lets trace out the x and y movements on the axis to help see the decreasing description Popper 04, Question 1 Basic Quadratic function:  EMBED Equation.DSMT4   Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: (0, 0) y intercept: (0, 0) Decreasing:  EMBED Equation.DSMT4  Increasing:  EMBED Equation.DSMT4  Neither at (0, 0), the turn-around point This function is not one-to-one. Example of a shifted graph:  EMBED Equation.DSMT4   Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercepts: ((5, 0) and (1, 0) y intercept: (0, (5) vertex: ("2, "9), the turn around point! decreasing  EMBED Equation.3  increasing  EMBED Equation.3  neither at the vertex! Let s work out a whole problem here, reviewing as we go. Here s the graph of a nice cupped-up parabola  Where are the intercepts on the graph? What are the point coordinates for the intercepts? x-intercepts? y-intercept? What do the x-intercepts tell us? What is the formula? Lets multiply that formula out and then find the vertex using algebra and see if it looks right where the graph has it. Now, where is the graph increasing, decreasing, or doing neither? Report your answer in interval notation. Popper 04, Question 2  EMBED Equation.DSMT4  Basic Cubic function:  EMBED Equation.3   Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (0, 0) y intercept: (0, 0) Increasing everywhere. This function is one-to-one. Example of a shifted graph:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (1, 0) y intercept: (0, (1/2) no turn around points, increasing on its domain Lets look at a cubic with a HUGE change:  EMBED Equation.3  How are we going to handle this? We need to graph it. Lets get the intercepts:  EMBED Equation.DSMT4  is the y-intercept: To get the x-intercept(s), solve  EMBED Equation.3  for x. And lets get a few points f(1) = f(2) = f("1) = f("2) = [Let s try to remember graph shifting from College Algebra  a negative sign in front of the formula means a reflection about the x-axis.] Let s graph what we ve got so far So, now let s analyze it by working with the process that helps us identify whether the graph is increasing or decreasing: Taking the xs left to right. Are there any turnaround points? No? well then the graph will be doing the same thing on its whole domain. So f(x) means a reflection about the x-axis! Watch for those. We saw one earlier with EMBED Equation.DSMT4 lets review that graph on page 7! Popper 04, Question 3 Basic Cube Root function:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (0, 0) y intercept: (0, 0) Increasing everywhere. This function is one-to-one. Example of a shifted graph: f(x + 1) ( 2 shifting instructions: left 1, down 2 new formula:  EMBED Equation.3   Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (7, 0) y intercept: (0, (1) increasing everywhere track the key point (0, 0) to ((1, (2) Example: Graph  EMBED Equation.3 . Where is the graph increasing or decreasing? The domain is all real numbers and so is the range. The negative sign means to reflect about the x-axis. Heres the parent graph:  EMBED Equation.3 . Lets do the reflection: Popper 04, Question 4 Basic Square Root function:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (0, 0) y intercept: (0, 0) Increasing on its domain. This function is one-to-one. *Domain restriction! Example of a shifted graph: shifting instructions: reflect about the x axis, left 3, reflect about the y axis new formula:  EMBED Equation.3   Domain*:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (3, 0) y intercept:  EMBED Equation.3  increasing on its domain note the signs on the ys Examples: Graph  EMBED Equation.3 . Where are the intercepts? What is the domain and range? Where is the graph increasing or decreasing? Graph  EMBED Equation.3 . Where are the intercepts? What is the domain and range? Where is the graph increasing or decreasing? Popper 04, Question 5 Basic Absolute Value function:  EMBED Equation.3   Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (0, 0) y intercept: (0, 0) Decreasing:  EMBED Equation.DSMT4  Increasing:  EMBED Equation.DSMT4  This function is not one-to-one. Example of a shifted graph: (f(x ( 5) +2 shifting instructions: reflect about the x axis, right 5, up 2 new formula:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercepts: (7, 0) and (3, 0) y intercept: (0, (3) increasing  EMBED Equation.3  decreasing  EMBED Equation.3  Examples: Where is the following function increasing and decreasing?.  EMBED Equation.3  Increasing/decreasing/neither One to one? Graph  EMBED Equation.DSMT4  Discuss increasing/decreasing/neither Popper 04, Question 6 Basic exponential function:  EMBED Equation.DSMT4  Domain:  EMBED Equation.3  Range:  EMBED Equation.DSMT4  x intercept: none y intercept: (0, 1) horizontal asymptote: y = 0 Increasing everywhere. (the illustration is with b = 3 see the point (1, 3)) Example of a shifted graph: shifting instructions: reflect about the x axis, left 3, up 9, new formula:  EMBED Equation.DSMT4   Domain:  EMBED Equation.3  Range:  EMBED Equation.DSMT4  x intercept: ((1, 0) y intercept: (0, (18) horizontal asymptote: y = 9 decreasing everywhere. Examples Tell me everything about:  EMBED Equation.3  Popper 04, Question 7 Basic logarithmic function:  EMBED Equation.3   Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: (0, 1) y intercept: none Vertical asymptote: x = 0 Increasing on its domain. This function is 1:1. (illustration is with b = 10)  Example of a shifted graph: shifting instructions: left 2, reflect y new formula:  EMBED Equation.DSMT4  Domain:  EMBED Equation.DSMT4  Domain Restriction! Range:  EMBED Equation.DSMT4  x intercept: (1, 0) y intercept:  EMBED Equation.DSMT4  vertical asymptote: x = 2 decreasing on its domain Basic Logarithmic Review: Example: Tell me everything about  EMBED Equation.3  Popper 04, Question 8 Popper 04, Question 9 Popper 04, Question 10 End Video 1 Functions Video 2 Rational Functions In this piece we will look at various types of rational functions. Now a rational function is a polynomial divided by another polynomial. Division by a number and not an expression in x is NOT a rational function. Lets look at two functions:  EMBED Equation.DSMT4  Similarly, well look at lines divided by parabolas and cubics divided by parabolas and so on. In the etexts: Math 1310, College Algebra (4.4) Math 1330, Pre-calculus (2.3) Ill start with the parent graph for the line over line graphs just to give you a feel for what we need to consider. Then Ill do some presentations on calculating the asymptotes and intercepts for these. Then well move into more general rational functions. Basic Rational function:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: none y intercept: none vertical asymptote: x = 0 horizontal asymptote: y = 0 Decreasing on its domain This function is one-to-one. Example of a shifted graph: shifting instructions: left 2, down 3 new formula:  EMBED Equation.3  Well track on (0, 0) which is the intersection of the asymptotes and NOT a graph point for shifting.   EMBED Equation.3   vertical asymptote: x = (2 horizontal asymptote y = (3 Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: ("5/3, 0)  EMBED Equation.3  y intercept: (0, "5/2)  decreasing on it s domain Popper 05, Question 1 More discussion on Asymptotes: The disallowed values in the domain create vertical asymptotes& .x = 0 is a vertical asymptote for this graphthe unattainable values in the range create horizontal asymptotesy = 0 is a horizontal asymptote. Asymptotes are lines that shape the graph. Graphs cannot cross a vertical asymptote these create branches ( but sometimes cross horizontal asymptotes the graph resembles a horizontal line for xs far down either end of the x axis when theres a horizontal asymptote. Lets look at f(x) =  EMBED Equation.3   Domain: Asymptotes: VA HA What are the x and y intercepts? VA:  Behavior at the VA: unbounded y values HA: glb and lub for the y values End Behavior: Note that around ((2, 3) the graph does NOT imitate the horizontal asymptotes but far away from this point it does look like these lines. If you have a polynomial divided by a polynomial, you will have to identify and deal with all kinds of old ideas and new ideasold ideas include x and y intercepts, the domain and range; new ideas are vertical and horizontal asymptotes and the behavior there plus holes. Popper 05, Question 2 Vertical Asymptotes and Holes: Linear factors in the denominator create vertical asymptotes unless they are cancelled with a like factor in the numeratorin which case, they create a hole. The domain determines which x values have problems and the formula determines which kind of problem it is. Example: Whats the difference between  EMBED Equation.3  and  EMBED Equation.3  Well, x + 5 cancels out of the second function and not out of the first. The first graph has a vertical asymptote at x = "5 & a vertical asymptote at the disallowed value, ie the point we leave OUT of the domain& the second function has only a hole there& .(recall cancellation creates a hole at x = (5)  The first function  EMBED Equation.3  End Behavior at the VA: This graph has 2 branches arranged beside a vertical asymptote And the second function do the simplifying division and look at it again  EMBED Equation.3   ("5, "10) is missing from D and R   This graph looks like f (x) = x + 5 with a hole at ("5, "10). How did I know that the y value was "10? So when you have linear factors in the denominator you get a vertical asymptote or a hole. You get a vertical asymptote when it doesnt cancel and a hole when it does cancel. Popper 05, Question 3 Popper 05, Question 4 Now lets review at Horizontal Asymptotes: Basically you work off the leading terms. Divide the power of x in the numerator by the power of x in the denominator. See my handout on the website for more information and practice. answer > 1 no horizontal asymptote answer = 1 ha is ratio of coefficients n/d answer < 1 x axis is horizontal asymptote What is the HA for  EMBED Equation.3  What is the HA for  EMBED Equation.DSMT4  What is the HA for  EMBED Equation.DSMT4  Popper 05, Question 5 Popper 05, Question 6 Lets look at some functions and tell whats going on:  EMBED Equation.3 = [parabola over a parabola!] Domain: Vertical asymptotes: Holes: HA Range: x intercepts y intercept Sign chart  Graph: End Behavior: Behavior at the VA: Behavior at the HA Another one:  EMBED Equation.3  = Domain: Vertical asymptote: Holes: HA Range: Note crossing of HA! x intercepts y intercept  Graph End behavior: Behavior around the asymptotes: Note: crosses HA! Lets look more closely at rational functions that cross their horizontal asymptotes. It is possible to tell that this happens WITHOUT graphing. Lets first use the preceding function:  EMBED Equation.3  At the HA y = 1 and what is the y-value of a point on this asymptote? Set up the equation and solve for x. Now lets try some others:  EMBED Equation.DSMT4  Where is the HA? What is the y-value of a point on the HA? Set up the equation and solve  EMBED Equation.DSMT4  line/line form! Where is the HA? What is the y-value of a point on the HA? Set up the equation and solve  EMBED Equation.DSMT4  Where is the HA? What is the y-value of a point on the HA? Set up the equation and solve  EMBED Equation.DSMT4  Where is the HA? What is the y-value of a point on the HA? Set up the equation and solve Popper 05, Question 7 Lets look at a deceptively simple one:  EMBED Equation.3  This IS a line over line. SEE It doesnt cross its HA! Domain VA Holes HA Range x intercepts  y intercept graph End behavior: Behavior near asymptotes: Popper 05, Question 8 Hints and Summaries Zeros for the function are the zeros for the numerator after cancellation. x-axis intercepts are sometimes called zeros of the function because y is zero there. For example: What is the x-intercept for the following function?  EMBED Equation.DSMT4  Theres the LONG way or the SHORT way to do this! VAs come from the zeros in the denominator after cancellation. The VA is at the additive inverse of the number in the factor. Factors that cancel from the denominator create holes in the graph at the additive inverse of the number in the factor. For example:  EMBED Equation.DSMT4  HA. A function MAY cross its HA. Line/line graphs NEVER cross their HA (really!) and this makes for a restricted RANGE for these.  EMBED Equation.DSMT4  n/d > 1 none  EMBED Equation.DSMT4  Domain y-intercept HA n/d = 1 ratio of coefficients  EMBED Equation.DSMT4  Domain y-intercept HA n/d < 1 x axis, y = 0  EMBED Equation.DSMT4  Domain y-intercept HA Popper 05, Question 9 Tell me everything about  EMBED Equation.3  and sketch the graph. Rational function of the line/line type. Domain VA HA Range Intercepts: x x + 6 = 0 y f (0) Graph: Popper 05, Question 10 End Video 2 Polynomials A polynomial in x has one or more terms with a rational number coefficient and natural number power. In the etexts: Math 1310, College Algebra (4.1 and 4.3) Math 1330, Pre-calculus (2.2) Term: math expressions that are added Factor: math expressions that are multiplied  EMBED Equation.DSMT4  EXAMPLES: list leading term and constant term! a.  EMBED Equation.3 = 3 =  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  Samples of equations that are NOT polynomials:  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Back to Polynomials: The domain for every polynomial is all Real numbers. Always. The y-intercept of the graph is f (0)which is always the constant term (the term that has the  EMBED Equation.DSMT4  in it). Give the y-intercept of the polynomials in the example above: The x-intercepts happen when the y value is zero. Replace f (x) with zero and solve for x. This always involves factoring. Note that the zero is the opposite sign of the number in the factor. Give the x-intercepts of the examples above. Give the intercepts for:  EMBED Equation.3   EMBED Equation.DSMT4  Behavior at the ends of the graph of a polynomial: The Leading Term is  EMBED Equation.3 . The Leading Coefficient, a, can be negative or positive and it gives you graphing information. The Leading Power, n, can be even or odd and it gives you graphing information. The graphing information is about the ends of the graph way away from the origin. Heres a table of that information: Graphing Chart - Ends n is even n is odd a is positive  EMBED Equation.3    EMBED Equation.3  a is negative  EMBED Equation.3   EMBED Equation.3  Mnemonic: + even  EMBED Equation.DSMT4  + odd  EMBED Equation.DSMT4   EXAMPLE 1:  EMBED Equation.3  Leading term:  EMBED Equation.DSMT4  5 is the leading coefficient 7 is the leading power Domain: All Real numbers y-intercept: (3 as in (0, (3), the constant term Ends: a is positive, n is odd:  EMBED Equation.3  EXAMPLE 2:  EMBED Equation.3  Leading coefficient: 1 Leading power: 4 Constant term: "2 Domain: all Real numbers y-intercept: (0,(2) Ends: a is positive, n is even  EMBED Equation.3  Popper 06, Question 1 x-intercepts Note that both of the graphs above have x-intercepts. You find these by setting the polynomial to zero (y is always zero at an x-intercept) and factoring the polynomial to find the intercepts (aka: zeros). EXAMPLE 3:  EMBED Equation.3  Leading coefficient: 1 Leading power: 3 (+, odd) =  EMBED Equation.DSMT4  Constant term: 0 Domain: all Real numbers y-intercept: (0, 0) f (0)! Ends: positive, odd  EMBED Equation.3  x-intercepts:  EMBED Equation.3  Popper 06, Question 2 Behavior at x-intercepts: In example 3 above, the graph goes through each intercept locally just like a line. That happens because the exponent on each factor is 1. Lets look at the formula and graph more closely:   EMBED Equation.3  Local tiny circle The exponents on the factors tell you the local behavior of the graph at an intercept. exponent is 1 graph goes through like a line (x 1) exponent is 2 graph curves back tangent to the x-axis like a parabola (x 1)2 exponent is 3 graph squiggles through the point like x-cubed (x 1)3 Popper 06, Question 3 EXAMPLE 4:  EMBED Equation.3  Domain: all Real numbers Leading term: x6 Leading coefficient: 1 Ends: (+, even) =  EMBED Equation.3  y-intercept: 13("2)2((3)1= (12 use the last number in each factor constant term! f (0) x-intercepts: (1 squiggle (x + 1)3 2 parabola (x  2)2 3 line (x  3)1  Popper 06, Question 4 Working backwards from the graph to the equation EXAMPLE 5:  EMBED Equation.DSMT4  Leading term is positive and leading exponent is odd! (1 is an x-intercept and the graph is locally like a parabola there so the power on the factor is 2: 2 is an x-intercept and the graph is locally like a line there so the power on the factor is 1.  EMBED Equation.3   EXAMPLE 6:  EMBED Equation.DSMT4  Leading term is negative and even at (3 the graph is a squiggle the power on the factor is 3 at 1 the graph is a line so the power on the factor is 1  EMBED Equation.3  Popper 06, Question 5 Sequences* a good source: *Cohen, David Precalculus, 5th edition West Publishing isbn 0-314-06921-6 A numerical sequence is a function with a domain of the natural numbers. A single element of the range is called a term of the sequence. Some sequences are finite the number of terms is a given natural number and some are infinite. Very often the terms are indexed by the natural numbersthe indices are the domain numbers. EXAMPLE 1: A sequence with a formula  EMBED Equation.DSMT4  The SUBSCRIPTS are actually the domain elements. a1 gives the point (1, ). The subscript is the first coordinate! This is an infinite sequence bounded below by , which is a function point and above by 1 which is not. Each  EMBED Equation.DSMT4  is a y-value and each n is the associated x. [Sometimes the function has the whole numbers as its domain and the indexing starts with 0.] Lets write this sequence as point pairs and then graph them in 2 dimensions. Show where the bounds are and label those. Write out the domain and range in words. Is this a function? How do you know?   EXAMPLE 2:  EMBED Equation.DSMT4  What are the point pairs? Are there bounds? What is the domain and range (in words)? Is this a function? How do you know? Popper 06, Question 6 Recursive sequences Sometimes, the next term in a sequence is defined by a preceding term or two, these are called recursive sequences.  EMBED Equation.DSMT4  Note that the formula shows tnext in terms of the preceding t tnext is t preceding is Lets calculate the second and third ts: The sequence is: 3, 4, 6, 10, 18, 34, 66, 130, 258, 514 These are range elements. The domain elements are given in the inequality above. This is a finite sequence. Write the point pairs, graph and discuss bounds. 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These sequences are generally written as (a, d). The pattern is: a is given as the first term and d will be what is added to each succeeding term: a, a + d, a + 2d, a + 3d, a + 4d, The pattern allows us to come up with a formula for the nth term:  EMBED Equation.DSMT4  Distribute the d above EXAMPLE 4 Given the arithmetic sequence: 7, 10, 13, 16, . What is the 100th term? a = 7 d = 3  EMBED Equation.DSMT4  List point pairs, graph, discuss domain, range and bounds. 7, 10, 13, 16, . Domain: the natural numbers Range: a proper subset of the natural numbers = { 7, 10, 13, 16, 19, } Point pairs: (1, 7), (2, 10), (3, 13), (4, 16), (5, 19)(n,  EMBED Equation.DSMT4 ), Checking: a = 7 d = 3 (4, 3(4) + 7 3) = 16) Graph: Is it a function? Does it pass the vertical line test? Is it bounded below? Above? Popper 06, Question 8 EXAMPLE 5 Give the details of the arithmetic sequence with a second term of (2 and an eighth term of 40.  EMBED Equation.DSMT4  is the general formula I know (2= a + (2(1)d and 40 = a + (8(1)d So I can get a and d. (2 = a + d 40 = a + 7d multiply the second equation by (1add them.(42 = (6d so d = 7 and a = (9. This is all we need to specify to give the details. List the point pairs, graph and discuss domain, range, and bounds. Is it a function? How do you know? Popper 06, Question 9 Geometric Sequences A geometric sequence is a sequence of the form:  EMBED Equation.DSMT4  Note that the first term is really:  EMBED Equation.DSMT4  If you see it this way you can see the pattern The nth term is then:  EMBED Equation.DSMT4  Notice, too, where the ns are in this formula! And that were using whole numbers as exponents, but natural numbers to count the terms of the sequence! EXAMPLE 6 1, , , 1/8, , what are a and r? What are the domain elements? What are the point pairs? Graph in 2D and discuss boundaries! EXAMPLE 7 10, (100, 1000, (10,000 what are a and r? What are the point pairs? These graph as points in the plane. Discuss bounds. 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