POLYNOMIAL FUNCTIONS. Chapter 5

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1 POLYNOMIAL FUNCTIONS Chapter 5

2 5.1 EXPLORING THE GRAPHS OF POLYNOMIAL FUNCTIONS 5.2 CHARACTERISTICS OF THE EQUATIONS OF POLYNOMIAL FUNCTIONS Chapter 5

3 POLYNOMIAL FUNCTIONS What s a polynomial? A polynomial is any algebraic expression that features addition, subtraction, multiplication and division (but not division by a variable). It has real number coefficients, whole number exponents, and two variables x and y or f(x). In general, a polynomial equation is: P( x) ax n bx n 1 cx n 2... k Where n=0 or any positive integer (no fractional or negative exponents).

4 Examples of Polynomials y 2x 7 2 f ( x) 2x 6x 8 P( x) 10x 7x 6x 3 f ( x) 90x Not Polynomials f ( x) 2 x 6x 8 y 2 x P( x) 2x 6x 1 3

5 VOCABULARY Degree: The value of the highest exponent of a polynomial function. f x x x 3 ( ) Leading Coefficient: The coefficient of the term with the greatest degree (exponent) in a polynomial function. Constant Term: The term in a polynomial function with no variable.

6 POLYNOMIAL FUNCTIONS ARE NAMED ACCORDING TO THEIR DEGREE AND THEIR DEGREE DETERMINES THE SHAPE OF THE FUNCTION. DEGREE NAME EXAMPLE 0 Constant 1 Linear 2 Quadratic 3 Cubic y y 7 5x 11 2 y x 7x 6 3 y 7x 2x

7 EXAMPLE 1. DETERMINE THE DEGREE, THE LEADING COEFFICIENT, AND THE CONSTANT TERM OF EACH POLYNOMIAL FUNCTION. DEGREE LEADING COEFFICIENT CONSTANT TERM 2 f ( x) x 4x 5 g( x) 2x 4 h( x) 3 j( x) x 3

8 EXAMPLE 2. WRITE A POLYNOMIAL FUNCTION IN DESCENDING ORDER THAT SATISFIES THE FOLLOWING CONDITIONS. A. Degree 2, leading coef ficient of -3 B. Degree 2, leading coef ficient of 7, two terms C. Degree 1, leading coefficient of 1 D. Degree 0 E. Degree 3, constant term -8

9 PAGE 287 # 1, 4 Independent practice

10 VOCABULARY CONTINUED Domain: the set of all possible x-values which will make the function work and will output real y -values. Range: The complete set of all possible resulting y -values of the dependent variable. End Behaviour: This is the description of the shape of the graph, from left to right, on the coordinate plane. It is the behaviour of the y-values as x becomes large in the positive or negative direction. i.e. as ). x x approaches ± Turning Point: Any point where the graph of a function changes from increasing (y-values) to decreasing (y -values) or from decreasing to increasing.

A Cartesian grid (the x/yaxis) has four quadrants.

11 HOW GRAPHS WORK End behaviour: The description of the shape of the graph, from left to right, on the coordinate plane. A Cartesian grid (the x/yaxis) has four quadrants. Example: the graph of f(x) = x + 1 begins in quadrant III and extends to quadrant I.

12 DOMAIN AND RANGE Domain is how much of the x-axis is spanned by the graph. Range is how much of the y-axis is spanned by the graph.

13 TURNING POINTS A turning point is any point where the graph changes from increasing to decreasing, or from decreasing to increasing.

14 INVESTIGATING GRAPHS OF CONSTANT FUNCTIONS DEGREE 0 Function Graph of Function y 3 f ( x) 2 h( x) 0 # of x-intercepts # of y-intercepts End Behaviour Domain Range Turning Points

15 SUMMARY y b constant function Degree # of x-intercepts # of y-intercepts Domain Range # of Turning Points

16 INVESTIGATING GRAPHS OF LINEAR FUNCTIONS DEGREE 1 Function y 3x 1 f ( x) 2x 3 h x x ( ) 1 2 Graph of Function # of x-intercepts # of y-intercepts End Behaviour Domain Range Turning Points

17 SUMMARY y mx b linear function Degree # of x-intercepts # of y-intercepts Domain Range # of Turning Points What s the relationship between slope of a line and its end behaviour?

18 INVESTIGATING GRAPHS OF QUADRATIC FUNCTIONS DEGREE 2 Function y x 2x 8 f ( x) x 2x 2 y x 4x 4 Graph of Function # of x-intercepts # of y-intercepts End Behaviour Domain Range Turning Points

20 SUMMARY 2 y ax bx c Degree # of x-intercepts # of y-intercepts Domain # of Turning Points quadratic function

21 How does the sign of the leading coefficient help us to determine the end behaviour of a quadratic? If the leading coefficient is positive, the graph goes from Quadrant 2 to Quadrant 1 (opens up). If the leading coefficient is negative, the graph goes from Quadrant 3 to Quadrant 4 (opens down). How does the sign of the leading coefficient help us to determine the range of a quadratic? If the leading coefficient is positive, {y/y> minimum} If the leading coefficient is negative, {y/y< maximum}

22 INVESTIGATING GRAPHS OF CUBIC FUNCTIONS DEGREE 3 Function y x 3 x x y x 3x 2 y x 3x 3x 1 Graph of Function # of x-intercepts # of y-intercepts End Behaviour Domain Range Turning Points

23 SUMMARY 3 2 y ax bx cx d cubic function Degree # of x-intercepts # of y-intercepts Domain Range # of Turning Points

24 How does the sign of the leading coefficient help us to determine the end behaviour of a cubic polynomial function?

25 POLYNOMIAL FUNCTIONS Pull out a graphing calculator. Equation Name / Degree f(x) = 4 Constant Deg 0 End Behaviour Max # of x- intercepts # of y- intercepts Q2 Q1 0 1 f(x) = x + 1 Linear Deg 1 Q3 Q1 1 1 f(x) = x 2 + 3x 4 Quadratic Deg 2 Q2 Q1 2 1 f(x) = 5x 3 + 5x 2 4x 2 Cubic Deg 3 Q3 Q1 3 1

26 How would a negative leading coefficient change the above graphs?

27 PG # 2, 3, 5-13 Independent practice

28 5.3 MODELLING DATA WITH A LINE OF BEST FIT 5.4 MODELLING DATA WITH A CURVE OF BEST FIT Chapter 5

29 LINE OF BEST FIT A line of best fit is a straight line that best approximates the trend in a scatter plot. A regression function is a line or curve of best fit, developed through statistical analysis of data.

30 EXAMPLE 1 The one-hour record is the farthest distance travelled by bicycle in 1 h. The table below shows the world-record distances and the dates they were accomplished. a) Use technology to create a scatter plot to find the equation of the line of best fit. b) Interpolate a possible world-record distance for the year 2006, to the nearest hundredth of a kilometre. c) Compare your estimate with the actual world-record distance of km in 2006.

31 A) USING YOUR CALCULATOR TO CREATE A SCATTER PLOT/LINE OF BEST FIT. Entering data: STAT EDIT Enter your x-values in under L1 and your y-values in under L2

32 OR USING DESMOS

33 A) USING YOUR CALCULATOR TO CREATE A SCATTER PLOT/LINE OF BEST FIT. Making a scatter plot: 2 nd Y= ENTER ZOOM 9

34 TO MAKE A LINE OF BEST FIT: STAT CALC 4 ENTER Write down your equation in the form: y = ax + b y = x

35 GRAPH YOUR LINE OF BEST FIT:

36 B) INTERPOLATE A POSSIBLE WORLD-RECORD DISTANCE FOR THE YEAR 2006, TO THE NEAREST HUNDREDTH OF A KILOMETRE. Is 2006 an x-value or a y-value? y = x y = (2006) y = km c) Compare your estimate with the actual world -record distance of km in It s very close to the real data. If I subtract them I find that it is only 0.39 km greater than the real data.

EXAMPLE 2 Matt buys t-shirts for a company that prints art on t-shirts and then resells them. When buying the t-shirts, the price Matt must pay is related to the size of the order.

37 EXAMPLE 2 Matt buys t-shirts for a company that prints art on t-shirts and then resells them. When buying the t-shirts, the price Matt must pay is related to the size of the order. Five of Matt s past orders are listed in the table below. a) Create a scatter plot and determine an equation for the linear regression function. b) What do the slope and y-intercept represent? c) Use the linear regression to extrapolate the size of the order necessary to achieve the price of $1.50 per shirt.

38 x- and y-values line of best fit scatter plot y=-.0065x+6.5

39 B) WHAT DO THE SLOPE AND Y-INTERCEPT REPRESENT?

40 C) USE THE LINEAR REGRESSION TO EXTRAPOLATE THE SIZE OF THE ORDER NECESSARY TO ACHIEVE THE PRICE OF $1.50 PER SHIRT.

41 MODELLING DATA WITH A QUADRATIC CURVE OF BEST FIT Audrey is interested in how speed plays 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 would like to write a summary of this data for the graduation class website.

43 a) Plot the data on a scatter plot. Determine the equation of a quadratic regression function that models the data. y x x

44 B) USE YOUR EQUATION TO COMPARE THE STOPPING DISTANCE AT 30 KM/H WITH THE STOPPING DISTANCE AT 50 KM/H, LENGTH OF A CAR.

45 C) DETERMINE THE MAXIMUM SPEED THAT A CAR SHOULD BE TRAVELLING IN ORDER TO STOP WITHIN 4 M, THE AVERAGE LENGTH OF A CAR.

46 MODELLING DATA WITH A CUBIC CURVE OF BEST FIT The following table shows the average retail price of gasoline, per litre, for a selection of years in a 30-year period beginning in 1979.

47 a) USE TECHNOLOGY TO GRAPH THE DATA AS A SCATTER PLOT. WHAT POLYNOMIAL FUNCTION COULD BE USE TO MODEL THE DATA? EXPLAIN.

48 B) DETERMINE THE CUBIC REGRESSION EQUATION THAT MODELS THE DATA. USE YOUR EQUATION TO ESTIMATE THE AVERAGE PRICE OF GAS IN 1984 AND P n n n

49 c) Estimate the year in which the average price of gas was 56.0 /L.

50 PG , #3, 4, 6, 7, 8, 11, 14 Independent practice PG , #1, 2, 3, 5, 7, 8, 9

51 Practice Assignment

Review of Section 1.1. Mathematical Models. Review of Section 1.1. Review of Section 1.1. Functions. Domain and range. Piecewise functions

Review of Section 1.1. Mathematical Models. Review of Section 1.1. Review of Section 1.1. Functions. Domain and range. Piecewise functions Review of Section 1.1 Functions Mathematical Models Domain and range Piecewise functions January 19, 2017 Even and odd functions Increasing and decreasing functions Mathematical Models January 19, 2017