3.1 Quadratic Functions and Their Models. Quadratic Functions. Graphing a Quadratic Function Using Transformations

Transcription

1 3.1 Quadratic Functions and Their Models Quadratic Functions Graphing a Quadratic Function Using Transformations

2 Graphing a Quadratic Function Using Transformations from Basic Parent Function, ) ( f Equations must be in verte form, k h a f ) ( ) ( Complete the square on the right side of the equation. Apply transformations to obtain 3 new points for your parabola. EX) Graph the function 5 8 ) ( f. Find the verte and ais of symmetry. EX) Graph the function 3 3 ) ( f. Find the verte and ais of symmetry.

3 Graphing By Using the Verte, Ais of Symmetry, and Intercept(s) Not the best method to use if the quadratic function cannot be factored. EX) Graph f ( ) 5 3using the verte, ais of symmetry, and intercepts. Verte Ais of Symmetry -intercept(s) y-intercept(s)

4 EX) Graph f ( ) 3 6 1using the verte, ais of symmetry, and intercepts. Verte Ais of Symmetry -intercept(s) y-intercept(s) EX) Graph f ( ) 4 4 using the verte, ais of symmetry, and intercepts. Verte Ais of Symmetry -intercept(s) y-intercept(s) EX) Graph f ( ) 1using the verte, ais of symmetry, and intercepts. Verte Ais of Symmetry -intercept(s) y-intercept(s)

5 Finding a Quadratic Function EX) Determining Maimum and Minimum Values of a Function Maimum and Minimum values will occur at the of a quadratic function. EX) Determine the maimum or minimum value of f ( ) 4 5without graphing. EX)

6 EX) (a) Find the maimum height of the projectile. (b) How far from the base of the cliff will the projectile strike the water? Fitting a Quadratic Function to Data When a data set appears to model quadratic behavior use QuadReg in doing regression. EX) A farmer collected data, which shows crop yields Y for various amounts of fertilizer used,. A) Use a graphing calculator to find the quadratic function that best fits the data. B) Use the function to determine the optimal amount of fertilizer to apply. C) Use the function to predict crop yield when the optimal amount of fertilizer is applied. HW pg all, 9, 17, 3, 5, 33, 41, 43, 47, 51, 65, 73, 75, 85, 87

7 3. Power Functions and Models A power function of degree n is a function of the form, f n ( ) a, where a is a real number, 0 a, and n > 0. Eamples of power functions are: f ( ) 3 f ( ) 5 f ( ) 8 3 f ( ) 5 4 degree degree degree degree Properties of Power Functions, f ( ) n, n is an Even Integer 1. The graph is symmetric with respect to the y-ais; f is even.. The domain is the set of all real numbers. The range is the set of nonnegative real numbers. 3. The graph always contains the points (0, 0), (1, 1), and (-1, 1). 4. As the eponent n increases in magnitude, the graph becomes more vertical when < -1 or > 1; but for near the origin, the graph tends to flatten out and lie closer to the -ais. Properties of Power Functions, f ( ) n, n is an Odd Integer 5. The graph is symmetric with respect to the origin; f is odd. 6. The domain and range are the set of all real numbers. 7. The graph always contains the points (0, 0), (1, 1), and (-1, -1). 8. As the eponent n increases in magnitude, the graph becomes more vertical when < -1 or > 1; but for near the origin, the graph tends to flatten out and lie closer to the -ais. Eample Graph the following transformations using Transformations. Identify 3 points on each graph. f ( ) 1 5 ( ) f f ( ) 6 7

8 Fitting Data to a Power Function Use PwrReg to find a power regression model to fit a data set. EX) Scott drops a ball from various heights and records the time it takes for the ball to hit the ground. He collects the following data: A) Using your graphing calculator find the power function that best fits the data. Time t (seconds) Distance s (meters) B) Predict how long it will take the ball to fall 100 meters C) Predict the distance the ball would fall after 5 seconds HW pg. 00 1, 5, 9, 13, 17 a-d 3.3 Polynomial Functions and Models Identifying Polynomial Functions The degree of a polynomial function is the degree of the largest power of that appears.

9 Summary of the Properties of the Graphs of Polynomial Functions The graph of every polynomial function is both smooth and continuous. By smooth we mean the graph contains no sharp corners or cusps; by continuous we mean the graph has no gaps or holes and can be drawn without lifting your pencil. Identifying the Zeros of a Polynomial Function and Their Multiplicity EX) Find a polynomial of degree 3 whose zeros are -4, -, and 3.

10 EX) For the polynomial list all zeros and their multiplicities. f 4 1 ( ) 5( )( 3) 3 EX) For the polynomial f ( ) ( 3) ( ) A) Find the and y intercepts of the graph of f. B) Graph the polynomial on your calculator C) For each -intercept determine whether it is of odd or even multiplicity. Eploration

11 Turning Points End Behavior Eploration

12 Analyzing the Graph of a Polynomial Function Modeling Polynomial Functions Most statisticians do not recommend finding polynomials of best fit higher than degree 3. A Cubic Function of Best Fit Use CubicReg to find the cubic model that best fits your data if it appears to be cubic. EX) The data in the table represents the average fuel consumption for cars (in gallons) in the US from , where 1 represents 1991, represents 199, and so on. A) Find the cubic function of best fit that models this data. B) Use the cubic function to predict the average fuel consumption in C) Predict the year in which average miles per gallon is 18. HW pg , 5, 7, 11, 17, 19, 5, 55 a-d, 65, 67, 69, 71

13 3.4 Rational Functions I Rational Functions EX) Find the domain of the following rational functions. A) C) 4 R ( ) 5 B) 3 R ( ) 1 D) 1 R ( ) 4 R ( ) 3 E) 1 1 R ( ) F) R( ) 1 If p( ) R( ) is a rational function and if p and q have no common factors, then the rational function R is said to be q( ) in lowest terms. For a rational function in lowest terms, the zeros, if any, of the numerator are the -intercepts of the graph of R. The zeros of the denominator are values that are not in the the domain of R. They create vertical asymptotes or holes. Graphing Rational Functions y 1 1 y

14 1 EX) Use transformations to graph the rational function, y 1 ( ) -intercept(s) y-intercept(s) domain range symmetry vertical asymptote(s) horizontal asymptote(s) lim f ( ) lim f ( ) lim f ( ) lim f ( ) Horizontal and Vertical Asymptotes

15 EX) Find the vertical asymptotes of the following functions. Eploration R H F 3 4 G

16 Horizontal Asymptotes A horizontal asymptote is the y value the function approaches as or. To find Horizontal Asymptotes use the following guidelines: If the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptote. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the two leading coefficients. (y = #) If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is zero. (y = 0) Oblique Asymptotes (Slant Asymptotes) An oblique asymptote is the diagonal line the function approaches as If the degree of the numerator is greater than the degree of the denominator by one then there is an oblique asymptote. To find the oblique asymptote we must use long division. The asymptote is the part that goes evenly using long division. or. Can a rational function have an oblique asymptote and a horizontal asymptote? Can a rational function have a vertical asymptote and an oblique asymptote? Can a rational function have a vertical asymptote and a horizontal asymptote?

17 Eamples

18 HW pg , 5, 7, 9, 13, 15, 19, 3, 7, 39, 41, 43, 45

19 3.5 Rational Functions II: Analyzing Graphs

20 1 EX) Sketch the graph and complete the analysis of y 4 Domain Range X-Intercept(s) Y-Intercept(s) H.A. V.A. O.A. Symmetry End Behavior EX) Sketch the graph and complete the analysis of y 1 Domain Range X-Intercept(s) Y-Intercept(s) H.A. V.A. O.A. Symmetry End Behavior

21 EX) Sketch the graph and complete the analysis of y Domain Range X-Intercept(s) Y-Intercept(s) H.A. V.A. O.A. Symmetry End Behavior EX) Sketch the graph and complete the analysis of y 5 4 Domain Range X-Intercept(s) Y-Intercept(s) H.A. V.A. O.A. Symmetry End Behavior

22 EX) Make up a rational function that might have the graph shown. Applications of Rational Functions HW pg , 3, 9, 13, 17, 7, 39, 47, 49, 51

23 3.6 Polynomial and Rational Inequalities EX) EX)

24 EX) EX) Tami is considering leaving her $30,000 a year job and buying a cookie company. According to the financial records of the firm, the relationship between pounds of cookies sold and profit is shown in the table. A) Using your graphing calculator, find the quadratic function of best fit. (QuadReg) B) Using the function found in part A, determine the number of cookies Tami must sell in order for the profits to eceed $30,000 a year, and therefore make it worthwhile for her to quit her job. C) Using the function found in part A, determine the number of pounds of cookies Tami should sell in order to maimize profits. D) Using the function found in part A, determine the maimum profit that Tami can epect to earn. HW pg , 7, 19, 37, 41, 53, 59, 61, 63, 65