ࡱ>  rtmnopql bjbj L&8;Bft!"!!!V@V@V@eeeeeee$gXjjeQV@;ZV@V@V@e!!Ge~L~L~LV@!!e~LV@e~L~LcJAe!+٭,Acqef0Bfc\jHj0AejAe0V@V@~LV@V@V@V@V@eeJV@V@V@BfV@V@V@V@jV@V@V@V@V@V@V@V@V@ :  Functions Increasing/decreasing graphs Basic Graphs Worksheets Linear Quadratic Cubic Cube Root Square Root Rational Absolute Value Exponential Logarithmic 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 Homework Increasing and decreasing graphs We often talk of a turn-around point, a place where a graph changes direction from increasing to decreasing or vice versaso it is important to be able to tell if a graph is increasing or decreasing. When looking at the xs we scan the graph left to right and we note that the smaller xs are on the left. In fact, if we number the first x and the 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. 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. For example: Here is the graph of y = .5x ( 6. It is increasing everywhereeven in Quadrant 3 and Quadrant 4 where the ys are negative numbers. Here is the graph of  EMBED Equation.3 . It is increasing on  EMBED Equation.DSMT4  and decreasing for xs beyond 0,  EMBED Equation.DSMT4 . 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. The turn-around point is the vertex.  Here is the graph of  EMBED Equation.3 , an exponential graph. Ive left the graph large so I can show you the motion on the y-axis. Lets work it together.   Heres a polynomial. There are two sections of increasing and one of decreasingwhere are they? Basic Graphs Worksheets 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. Example of a shifted graph:  EMBED Equation.DSMT4  shifting instructions: stretched by 3, left 2, down 1 new formula:  EMBED Equation.DSMT4  try to get this yourself by working with the shifting instructions!  Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: ((5/3, 0) y intercept: (0, 5) Linear Functions Review: What is the linear function with x-intercept 5 that has (6, 9) also on the graph? Is the graph increasing or decreasing? What is the slope of the function: 15x ( 3y = 12? Is the graph increasing or decreasing? Given (2, 3) and (9, 17), what is the equation of the line? Is the graph increasing or decreasing? 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  This function is not one-to-one. Example of a shifted graph:  EMBED Equation.DSMT4  shifting instructions: left 2, down 9 new formula:  EMBED Equation.DSMT4   Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercepts: ((5, 0) and (1, 0) y intercept: (0, (5) track the key point (0, 0) to ((2, (9) decreasing  EMBED Equation.3  increasing  EMBED Equation.3  Quadratic Functions Review: Graph the function:  EMBED Equation.3  using graph shifting. Where is the vertex? What are the intercepts? Where is the graphing increasing or decreasing? Graph:  EMBED Equation.3 . Where is the vertex? Where is the graph increasing or decreasing? 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: f(x ( 1) shifting instructions: shrink by , right 1 new formula:  EMBED Equation.3  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: (1, 0) y intercept: (0, (1/2) track the key point (0, 0) to (1, 0) Basic Cubic Review: Give the domain and range for  EMBED Equation.3 . Sketch the graph. Where is the graph increasing or decreasing? Graph:  EMBED Equation.3 . Where is the graph increasing or decreasing? 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) Basic Cube Root Review: Graph  EMBED Equation.3 . Where is the graph increasing or decreasing? 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. Example of a shifted graph: (f(3 ( x) 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 Basic Square Root Review: 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? 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: f(x + 2) - 3 shifting instructions: left 2, down 3 new formula:  EMBED Equation.3  try to get this yourself by working with the shifting instructions!  Domain:  EMBED Equation.3  Range:  EMBED Equation.3  x intercept: ("5/3, 0) y intercept: (0, "5/2) vertical asymptote: x = (2 horizontal asymptote y = (3 decreasing on it s domain Track the key point (1, 1) to ((1, (2). Basic Rational Function Review: 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: What are the x and y intercepts? Asymptotes: VA: Behavior at the VA: HA: End Behavior: Note that around ((2, 3) the graph does NOT imitate the 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, new ideas are vertical and horizontal asymptotes and holes. Vertical Asymptotes: 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. 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  First function  EMBED Equation.3  End Behavior: At VA: Second function do the simplifying division and look at it again  EMBED Equation.3    Lets talk about the range here: 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. 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  Lets look at some functions and tell whats going on:  EMBED Equation.3  Domain: Vertical asymptotes: Holes: HA x intercepts y intercept Sign chart  Graph: End Behavior: Behavior at the VA: Another one:  EMBED Equation.3  Domain: Vertical asymptote: Holes: HA x intercepts y intercept  Graph End behavior: Behavior around the asymptotes: Lets look at a deceptively simple one:  EMBED Equation.3  Domain VA Holes HA x intercepts  y intercept graph End behavior: Behavior near asymptotes: Hints and summaries Zeros for the function are the zeros for the numerator after cancellation. VA come from the zeros in the denominator after cancellation. Factors that cancel create holes in the graph at the additive inverse of the number in the factor. HA n/d > 1 none n/d = 1 ratio of coefficients n/d < 1 x axis, y = 0 Tell me everything about  EMBED Equation.3  and sketch the graph. 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  Basic Absolute Value Review: Where is the following function increasing and decreasing; show the graph.  EMBED Equation.3  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. (illustration is with b = 3) Example of a shifted graph: (f(x + 3) + 9 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. Be sure to know how to handle this if b = e. Basic Exponential Function Review: Tell me everything about:  EMBED Equation.3  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  try to get this yourself by working with the shifting instructions! Domain:  EMBED Equation.DSMT4  Range:  EMBED Equation.DSMT4  x intercept: (1, 0) y intercept:  EMBED Equation.DSMT4  vertical asymptote: x = 2 decreasing on its domain Basic Logarithmic Review: Tell me everything about  EMBED Equation.3  Polynomials A polynomial in x has one or more terms with a rational number coefficient and natural number power. EXAMPLES: a.  EMBED Equation.3 = 3 =  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  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. A. Behavior at the ends of the graph 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. Circle the leading term in each polynomial above. 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  Domain: All Real numbers y-intercept: (3 as in (0, (3) Ends: a is positive, n is odd:  EMBED Equation.3  EXAMPLE 2:  EMBED Equation.3  Domain: all Real numbers y-intercept: (2 Ends: a is positive, n is even  EMBED Equation.3  B. 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 it to find the intercepts (aka: zeros). EXAMPLE 3:  EMBED Equation.3  Domain: all Real numbers y-intercept: 0 (C is invisible!) Ends: positive, odd  EMBED Equation.3  x-intercepts:  EMBED Equation.3  C. Behavior at x-intercepts In example 3, the graph goes through each intercept locally just like a line. That happens because the exponent on each factor is 1. The exponents on the factors tell you the local behavior of the graph at a intercept. exponent is 1 graph goes through like a line exponent is 2 graph curves back tangent to the x-axis like a parabola exponent is 3 graph squiggles through the point like x-cubed EXAMPLE 4:  EMBED Equation.3  Domain: all Real numbers y-intercept: 1(4)((3)= (12 Ends: Positive, even  EMBED Equation.3  x-intercepts: (1 squiggle 2 parabola 3 line D. Working backwards from the graph to the equation EXAMPLE 5: Leading term is positive. (1 is an x-intercept and the graph is locally like a parabola: 2 is an x-intercept and the graph is locally like a line  EMBED Equation.3   EXAMPLE 6: Leading term is negative. at (3 the graph is a squiggle. at 1 the graph is a line.  EMBED Equation.3  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  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.] Write this sequence as point pairs and then graph them. 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? Recursive sequences Sometimes, the next term in a sequence is defined by a preceding term or two, these are called recursive sequences. B  EMBED Equation.DSMT4  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.  The Fibonacci sequence is an infinite recursive sequence. Arithmetic Sequences If you begin your sequence with a given number and then add a fixed number to each succeeding term, then youve made an arithmetic sequence. 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  Which gives some interesting problems. Show the point pairs, discuss if its a function. What are the domain, range and bounds? 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. EXAMPLE 5 Give the details of the arithmetic sequence with a second term of (2and 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. 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Is it a function? How do you know? Geometric Sequences A geometric sequence is a sequence of the form:  EMBED Equation.DSMT4  The nth term is then:  EMBED Equation.DSMT4  EXAMPLE 6 1, , , 1/8, , what are a and r? 10, (100, 1000, (10,000 what are a and r? These graph as points in the plane, in Quadrants 1 and sometimes IV.     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