TIPS4RM: MHF4U: Unit 1 Polynomial Functions

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1 TIPSRM: MHFU: Unit Polnomial Functions 008

2 .5.: Polnomial Concept Attainment Activit Compare and contrast the eamples and non-eamples of polnomial functions below. Through reasoning, identif attributes of ever polnomial function that distinguish them from non-polnomial functions: a. b. c. Eamples Non Eamples = = = f = = = 6 5 = + = 6 = + h = f = + = sin β = 0. + = = + + = TIPSRM: MHFU: Unit Polnomial Functions 008

3 .5.: Polnomial Concept Attainment Activit (continued) Eamples Non Eamples = = + h = + = + 0 = + TIPSRM: MHFU: Unit Polnomial Functions 008

4 .5.: Graphical Properties of Polnomial Functions. Use technolog to graph a large number of polnomial functions = a + k that illustrate changes in a, b, c, d, and k values. Certain patterns will emerge as ou tr different numerical values for a, b, c, d, and k. Remember that an of b, c, d, and k can have a value of 0. Your goal is to see patterns in overall shapes of linear, quadratic, cubic, and quartic polnomial functions, and to work with properties of the functions, rather than to precisel connect the equations and graphs of the functions the focus of later work. Answer the questions b referring to the graphs drawn using technolog. = a + b + k a 0 = a + b + c+ k a 0 = a + b + c + d + k a 0 Instructions Linear Quadratic Cubic Quartic Sketch, without aes, the standard curve (i.e., a =, and b, c, d, and k all = 0), and state the equation of the curve Sketch the basic shape, then place the - and -aes to illustrate the indicated numbers of zeros. Enter a sample equation for each sketch No zeros No zeros Wh is it impossible to have no zeros? Just zero Just zero Just zero (Hint: shapes possible) No zeros Just zero Eplain wh there can be no more than zeros Eactl zeros Eactl zeros Eactl zeros (Hint: aes placements possible) Eplain wh there can be no more than zeros Eactl zeros Eactl zeros Eactl zeros TIPSRM: MHFU: Unit Polnomial Functions 008

5 .5.: Graphical Properties of Polnomial Functions (continued). a) Read and reflect on the following description of the end behaviour of graphs. Graphs of functions come in from the left and go out to the right as ou read the graphs from left to right. A graph can come in high (large-sized negative -values correspond to large-sized positive -values) or low (large-sized negative -values correspond to large-sized negative -values). Similarl a graph can go out high (large-sized positive -values correspond to largesized positive -values) or low (large-sized positive -values correspond to large-sized negative -values). For eample, the standard linear function = comes in low and goes out high. b) Describe the end behaviours of: i) = ii) = iii) = iv) = v) = vi) = c) Contrast the directions of coming in and going out of the odd degreed polnomial functions (e.g., = and = ) and the even-degreed functions (e.g., = and = ). Recall that the domain of a function is the set of all -values in the relationship, and the range is the set of all -values. When the function is continuous, we cannot list the and -values. Rather, we use inequalities or indicate the set of values e.g., > 5, R. a) State the domain and range of each of the standard polnomial functions: i) = ii) = iii) = iv) = b) Sketch an eample of each of the following tpes of functions: i) quadratic with domain R. and range. ii) quartic with domain R. and range. TIPSRM: MHFU: Unit Polnomial Functions 008 5

6 .5.: Graphical Properties of Polnomial Functions (continued). Recall that a function is increasing if it rises upward as scanned from the left to the right. Similarl, a function is decreasing if it goes downward as scanned from the left to the right. A function can have intervals of increase as well as intervals of decrease. For eample, the standard quartic function = decreases to = 0, then increases. a) Sketch a cubic function that increases to =, then decreases to =, then increases. b) Use graphing technolog to graph = 5 +, sketch the graph, then describe the increasing and decreasing intervals. 5. Compare and contrast the following pairs of graphs: a) = and = b) = and = c) = and = d) = and = TIPSRM: MHFU: Unit Polnomial Functions 008 6

7 .5.: Numerical Properties of Polnomial Functions. Consider the function = a) What tpe of function is it? b) Complete the table of values. c) Calculate the first differences. d) In this case, the first differences were positive. How would the graph differ if the first differences were negative? First Differences 0. Consider the function = a) What tpe of function is it? b) Complete the table of values. c) Calculate the first and second differences. First Differences Second 0 TIPSRM: MHFU: Unit Polnomial Functions 008 7

8 .5.: Numerical Properties of Polnomial Functions (continued). Consider the function = a) What tpe of function is it? b) Complete the table of values. c) Calculate the first, second, and third differences. Differences First Second Third 0. Consider the function = a) What tpe of function is it? b) Complete the table of values. c) Calculate the first, second, third, and fourth differences. 0 Differences First Second Third Fourth 5. a) Summarize the patterns ou observe in Questions. b) Hpothesize as to whether or not our patterns hold when values for the b, c, d, and k parameters are not equal to 0 in = a+ k, = a + bc+ k, = a + b + c+ k, and = a + b + c + d+ k. c) Test our hpothesis on at least 6 different eamples. Eplain our findings TIPSRM: MHFU: Unit Polnomial Functions 008 8

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10 .7.: What Role Do Factors Pla?. Use technolog (Factored Polnomial.gsp sketch or graphing calculator) to determine the graph of each polnomial function. Sketch the graph, clearl identifing the -intercepts. a) f = ( )( + ) b) f = ( )( + )( + ) c) f = ( )( + )( + ) Degree of the function: Degree of the function: Degree of the function: -intercepts: -intercepts: -intercepts: d) f = ( + ) = ( + )( + ) e) f = ( ) ( + ) f) f = ( )( + )( + ) Degree of the function: -intercepts: Degree of the function: -intercepts: Degree of the function: -intercepts: g) f = ( ) h) f = ( + )( )( ) i) f = ( )( + ) Degree of the function: -intercepts: Degree of the function: -intercepts: Degree of the function: -intercepts:. Compare our graphs with the graphs generated on the previous da and make a conclusion about the degree of a polnomial when it is given in factored form.. Eplain how to determine the degree of a polnomial algebraicall if given in factored form. TIPSRM: MHFU: Unit Polnomial Functions 008 0

11 .7.: What Role Do Factors Pla?. What connection do ou observe between the factors of the polnomial function and the -intercepts? Wh does this make sense? (Hint: all co-ordinates on the -ais have = 0). 5. Use our conclusions from question to state the -intercepts of each of the following. Check b graphing with technolog, and correct, if necessar.! " f = ( )( + 5) $ % & ' = ( )( + 5)( ) f = ( )( + 5)( )( ) f -intercepts: Does this check? -intercepts: Does this check? -intercepts: Does this check? 6. What do ou notice about the graph when the polnomial function has a factor that occurs twice? Three times? TIPSRM: MHFU: Unit Polnomial Functions 008

12 .7.: Factoring in our Graphs Draw a sketch of each graph using the properties of polnomial functions. After ou complete each sketch, check with our partner, discuss our strategies, and make an corrections needed. a) f = ( )( + ) b) f = ( )( + )( ) c) f ( )( ) = d) f = ( ) e) f = ( ) ( + ) f) f = ( )( + )( + ) g) f = ( ) h) f = ( + ) ( ) i) f = ( + )( )( )( + ) TIPSRM: MHFU: Unit Polnomial Functions 008

13 .7.: What's M Polnomial Name?. Determine a possible equation for each polnomial function. a) f ( ) = b) f ( ) = c) f ( ) = d) f ( ) = e) f ( ) = f) f ( ) =. Determine an eample of an equation for a function with the following characteristics: a) Degree, a double root at, a root at b) Degree, an inflection point at, a root at 5 c) Degree, roots at,, d) Degree, starting in quadrant, ending in quadrant, root at and double root at e) Degree, starting in quadrant, ending in quadrant, double roots at 0 and 0 TIPSRM: MHFU: Unit Polnomial Functions 008

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15 .8.: What's the Change? If the graph of a function f change indicated. = is provided, choose the most appropriate description of the. = f ( + ). = f ( 5 ) a. shift right b. shift up c. slide left d. slide down a. vertical compression b. horizontal stretch c. horizontal compression d. shift up. = f. = f a. reflection about -ais b. reflection about -ais c. shift down d. slide left a. horizontal compression and shift right b. horizontal compression and shift left c. vertical stretch and shift right d. horizontal stretch and shift left 5. = f ( ) 6. = f ( 05. ( + )) a. horizontal stretch and reflection in -ais b. horizontal stretch and reflection in -ais c. vertical stretch and reflection in -ais d. horizontal stretch and reflection in -ais a. horizontal compression and shift right b. horizontal compression and shift left c. vertical stretch and shift right d. horizontal stretch and shift left 7. = f = f ( ) a. vertical shift down b. horizontal shift left c. vertical slide up d. horizontal shift right a. reflection in -ais and vertical compression b. reflection in -ais and horizontal compression c. reflection in both aes d. vertical and horizontal compression = + 9. = f ( ( + )) 0. f ( ) a. vertical compression and shift left b. reflection in -ais and shift right c. reflection in -ais and shift left d. reflection in -ais and shift left a. vertical compression, shift right, shift up b. vertical compression, slide left, shift up c. horizontal compression, shift right and shift down d. vertical stretch, shift right, shift up TIPSRM: MHFU: Unit Polnomial Functions 008 5

16 .8.: Transforming the Polnomials Using our knowledge of transformations and f = or f = as the base graphs, sketch the graphs of the following polnomial functions and confirm using technolog. f = f = f = = ( + ) f f = f ( ) = + f = f = f = = ( ) f f = + f = ( + ) Putting it all together: f = a k d + c For and c in terms of transformations., f = a k d + c describe the effects of changing a, k, d and TIPSRM: MHFU: Unit Polnomial Functions 008 6

17 .8.: Evens and Odds Graphicall What transformation reflects a function in the -ais? What transformation reflects a function in the -ais? For each function: Write the equation that will result in the specified transformation. Enter the equations into the graphing calculator (to make the individual graphs easier to view, change the line stle on the calculator). Confirm the equation provides the correct transformation; adjust if needed. Sketch a graph of the original function and the reflections. Record an observations ou have about the resulting graphs. Function Reflection in -ais Followed b reflection in -ais f = f ( ) = f ( ) = Observations f = f = f = a reflection in -ais makes no change in the graph a reflection in both aes makes a change in the graph f = f ( ) = f ( ) = f = f ( ) = f ( ) = TIPSRM: MHFU: Unit Polnomial Functions 008 7

18 .8.: Evens and Odds Graphicall (continued) Functions Reflection in -ais f = + Followed b Reflection in -ais f = f = Observations = ( ) f ( ) = f f = f = + f ( ) = f ( ) = f + f ( ) = f = f = + f ( ) = f = What conclusions can ou make between polnomial functions that have smmetr about the -ais? Both aes? Test other functions to confirm our hpothesis. TIPSRM: MHFU: Unit Polnomial Functions 008 8

19 .8.: Evens and Odds Algebraicall For each of the functions in the table below find the algebraic epressions for f and f ( ). Simplif our epressions and record an similarities and differences ou see in the algebraic epressions. The second one is done for ou as an eample. Function f f Observations f = f -f ( ) = -( ) =- = f(-)=(-) = The epressions for f f are the ( ) and same. The epressions for f f are and opposites. f = f = f = + TIPSRM: MHFU: Unit Polnomial Functions 008 9

20 .8.: Evens and Odds Algebraic (continued) Function f f Observations ( ) f = f = + f = + f = + What conclusions can ou make between polnomial functions that are the same and different f, f f? Test other functions to confirm our hpothesis. when comparing and TIPSRM: MHFU: Unit Polnomial Functions 008 0

21 .8.5: Evens and Odds Practice Determine whether each of the functions below is even, odd, or neither. Justif our answers f = + 6. f = f = + 8. f = + TIPSRM: MHFU: Unit Polnomial Functions 008