Chapter 5: Polynomial Functions

Transcription

1 Chapter : Polnomial Functions Section.1 Chapter : Polnomial Functions Section.1: Eploring the Graphs of Polnomial Functions Terminolog: Polnomial Function: A function that contains onl the operations of multiplication and addition with real-number coefficients, whole-number eponents, and two variables. E: f() = Tpes of Functions 1. Constant Function: A function of zero degree, whose greatest eponent is zero. E: f() = 7 2. Linear Function: A polnomial function of the first degree, whose greatest eponent is one. E: f() = Quadratic Function: A polnomial function of the second degree, whose greatest eponent is two. E: f() = Cubic Function: A polnomial function of the third degree, whose greatest eponent is three. E: f() =

2 Chapter : Polnomial Functions Section.1 Characteristics of a Polnomial Functions X-Intercepts: The point(s) at which a function crosses the -ais. Y-Intercepts: The point(s) at which a function crosses the -ais. Domain: All the possible values of which corresponds to a function. Range: All the possible values of which corresponds to a function. End Behaviour: The description of the shape of the graph, from left to right on the coordinate plane. NOTE: The Cartesian Plane is broken into four quadrants Quadrant II Quadrant III Quadrant I Quadrant IV E. Given this graph, the end behaviour is from Quadrant III to Quadrant I - Turning Point: An point where the graph of a function changes form increasing to decreasing of vice versa. - - This graph has two turning points since the -values change from increasing to decreasing to increasing again

3 Chapter : Polnomial Functions Section.1 For each polnomial function, identif: 1. -intercepts 2. -intercepts 3. end behaviour 4. domain. range 6. number of turning points (a) - - (b)

4 Chapter : Polnomial Functions Section.1 (c) - - (d) - - (e)

5 Chapter : Polnomial Functions Section.1 Summar of Functions Practice Questions: 1-3 pg

6 Chapter : Polnomial Functions Section.2 Section.2: Eploring the Graphs of Polnomial Functions Terminolog: Standard Form: The standard form for a linear function is: f() = a + b, 0 The standard form for a quadratic function is: f() = a 2 + b + c, 0 The standard form for a cubic function is: f() = a 3 + b 2 + c + d, 0 Leading Coefficient: The coefficient of the term with the greatest degree in a polnomial function in standard form. E. For the function f() = , the leading coefficient is 2. Constant Term: The term with no variable in a polnomial function. E. For the function f() = , the constant term is. Reasoning the Characteristics of a Graph of a Given Polnomial Function Using its Equation Determine the following characteristics of each function using its equation. - Number of possible -intercepts - -intercepts - end behaviour - domain - range - number of possible turning points (a) f() = 3 (b)f() =

7 Chapter : Polnomial Functions Section.2 (c) f() = (d) f() = (e) f() = (f) f() = NOTES: 123

8 Chapter : Polnomial Functions Section.2 Connecting Polnomial Functions to their Graphs Match each graph with the correct polnomial function. Justif our reasoning. g() = j() = p() = h() = k() = q() = 2 3 (i) (ii) (iii) 8 (iv) (v) (vi)

9 Chapter : Polnomial Functions Section.2 12

10 Chapter : Polnomial Functions Section.2 Sketching a Graph Given Some Polnomial Functions Sketch the graph of a possible polnomial function for each set of characteristics below. What can ou conclude about the equation of the function with these characteristics? WRITE A POSSIBLE EQUATION FOR EACH!!! (a) Range: { 2, R} -intercept: (b) Range: { R} Turning Points: One in Quadrant III and another in Quadrant I

11 Chapter : Polnomial Functions Section.2 (c) End Behaviour: Quadrant III to Quadrant IV X-intercept at (-3,0) and (1,0) - - (d) Two Turning Points End Behaviour: Quadrant II to Quadrant IV

12 Chapter : Polnomial Functions Section.2 Single, Double, and Triple Roots There are three different tpes of roots: When a graph passes directl through the -ais, we call that a single root When a graph touches the -ais and then bounces back, never passing through, we call that a double root When a graph levels off when approaching the -ais, passes through and then begins curving again, we call this a triple root (triple roots are onl possible in cubic functions and those with higher degrees) - - Practice Questions: 1, 2,3 pg

13 Chapter : Polnomial Functions Section.3 Section.3: Modelling Data with a Line of Best Fit Terminolog: Scatter Plot A set of points on a grid, used to visualize a relationship or possible trend in the data. E: Line of Best Fit: A straight line that best approimates the trend in a scatter plot. Regression Function: A line or curve of best fit, developed through a statistical analsis of data. Interpolate: The process used to estimate a value within the domain of a set of data based on a trend. Etrapolate: The process used to estimate a value outside the domain of a set of data based on a trend. Requires the etending of the line of best fit. 129

14 Chapter : Polnomial Functions Section.3 Determining a Linear Model for Continuous Data E. The one hour record is the farthest distance travelled b biccle in 1 h. The table below shows the world record distances and the dates the were accomplished. Year Distance (km) (a) (b) Use technolog to create a scatter plot and to determine the equation of the line of best fit Interpolate a possible world record distance for the ear 2006, to the nearest hundredth of a kilometre. (c) Compare our estimate with the actual world record distance of 8.99 in (d) Interpolate a possible world record distance in the ear 2000, what would ou epect this distance to have been? 130

15 Chapter : Polnomial Functions Section.3 Using Linear Regression to Solve a Problem that Involves Discrete Data Matt bus T-shirts for a compan that prints art on T-shirts and then resells them. When buing the T-shirts, the price Matt must pa is related to the size of the order. Five of Matt s past orders are listed in the table below. Number of Shirts Cost Per Shirt ($) Matt has misplaced the information from his supplier about price discount on bulk orders. He would like to get the price per shirt below $1.0 on his net order. (a) Use technolog to create a scatter plot and determine an equation for the linear regression function that models the data. (b) What does the slope and -intercept of the equation of the linear regression function represent in this contet? (c) Use the linear regression function to etrapolate the price per shirt of a 900 shirt order. 131

16 Chapter : Polnomial Functions Section.3 Using a Ruler to Estimate the Equation that Best Fit Equation Using a ruler, determine the slope and -intercept of the line of best fit for each data. Use this information to create an equation for the data. (a) (b) Practice Questions: 1,2,3,4,,6,7 pg

17 Chapter : Polnomial Functions Section.4 Section.4: Modelling Data with a Curve of Best Fit Terminolog: Curve of Best Fit A curve that best approimates the trend of a scatter plot. Using Technolog to Solve a Quadratic Problem Audre is interested in how speed plas a role in car accidents. She knows that there is a relationship between the speed of a car and the distance needed to stop. She has found the following eperimental data on a reputable website, and she would like to write a summar from the graduation class website. (a) Using Technolog determine if this data is best modelled b a quadratic or cubic function. How do ou know? (b) What is the equation of the curve of best fit? Speed (km/h) Distance (m) (c) Use our equation to compare the stopping distance at 30 km/h, to the nearest tenth of a metre. (d) Determine the stopping distance associated with the speed of 90 km/h. 133

18 Chapter : Polnomial Functions Section.4 Using Observation Skills to Answer Questions about Data E1. From the data, determine the best tpe of regression that should be used

19 Chapter : Polnomial Functions Section.4 2. Given the regression for a set of data is given as: Cost ($) Time (h) This is a cubic function with a regression equation of: = (a) Determine the cost associated with hours of service. (b) Determine the cost associated with 1 hours of service. (c) Using the graph, determine the hours of service that would produce a cost of $0 (d) Using the graph, determine the hours of service that would produce a cost of $90 (e) What is the constant term from the equation? How does this relate to the graph? 13

20 Chapter : Polnomial Functions Section.4 (f) When Jesse was tring to fit the data to the graph, he ran three different regressions: Linear Regression: r 2 = Quadratic Regression: r 2 = Cubic Regression: r 2 = Do ou think his decision to use a cubic equation was the best decision for this data. Wh or Wh not. Eplain. 136