Precalculus Chapter 2A Practice Guide Name

Transcription

1 Precalculus Chapter A Practice Guide Name Day 1 Day.1 (page 96). (page 108 ).3 (page 1) 15,1,,3,7,33 37,4,49,50,5,55 17,30,38,47,53,61 67,85 Day 3 43,48,51,68 1,4,6,7,13,16,18,19.4 Worksheets.5 (page 145) Day 4 16,40,46,59,6 6abd,9abcefg Day 5 44,58,69 7abd Day 6 39,56,63 30 Day 7 QUIZ ,11abc, 13abc,1,3 17,4,37,38,39,55,57 67,69 Semester #1 Cumulative Review Worksheet #1 Day ,40,54 Worksheet.4a Day 9 18abcf 41,58,70 Worksheet.4b Day 10 Day 11 Day 1 Day 13 QUIZ.4-.5 Day 14 Day 15 Day 16 SEMESTER #1 TARGET TEST #1 (Practice TBA) Chapter A Review Worksheet (all) Prepare for Chapter A Test Chapter A Test 1-0,,3 Worksheet.5b Worksheet.5c

2 Precalculus Unit A: Functions and Their Graphs Mrs. Flood s office hours: 4,7,8 This is an outline. The practice/quizzes/tests are subject to change Day Section New Learning Targets and Assessments Students will be able to determine if an equation is a function Students will be able to find values of a function Students will be able to find the domain and range of a function.1 Students will be able to evaluate the graph of a function Students will be able to answer questions about a function. Students will be able to determine if a relation is linear or nonlinear Students will be able to use a graphing utility to find a linear equation of best fit 4. Students will be able to solve application problems involving a linear equation of best fit Students will be able to determine the intervals at which a function is increasing or decreasing Students will be able to determine local maximum and local minimum points.3 Students will be able to determine if a function is even, odd, or neither Students will be able to use a graphing utility to analyze a function QUIZ.1-.3 (Practice days 1-6 due in class) 8.4 Students will be able to evaluate piece-wise defined functions Students will be able to graph piece-wise defined functions Students will be able to define a piece-wise function for a given graph 9.4 Students will be able to use piece-wise functions to solve application problems SEMESTER 1 TARGET TEST #1 (Chapter 1,.1-.3) 10 Semester 1 Target Review #1 due in class Students will be able to recognize the quadratic, cubic, and absolute value functions Students will be able to recognize transformations to the quadratic, cubic, and absolute value functions.5 Students will be able to recognize the square root and reciprocal functions Students will be able to graph functions using transformations Students will be able to define functions given the transformations 13.5 Students will be able to 14 QUIZ.1-.3 (Practice days 8-13 due in class) Review CHAPTER TEST

3 PreCalculus NOTES.1A Functions Relation, Value, Domain Students will be able to determine if an equation is a function. RELATIONS A relation is a correspondence between two sets. y depends on x. For a relation to represent a function there can only be 1 y value for every x. EXAMPLES: Determine whether each relation represents a function. For each function, state the domain and range. 8) { ( 0, ), ( 1, 3), (, 3), ( 3, 7)} 10) { ( -4, 4), ( -3, 3), ( -, ), ( -1, 1),( -4,0)} VALUE Instead of y =, function notation is f(x) =, or g(x) =, or h(x) =.. f, g or h is the name of the function and x is the variable used in the function. f(x) is read f of x meaning that the function f is dependent on the variable x. g(m) reads Students will be able to find values of a function. EXAMPLES: 1) Find the following values for the function x 3x 4 a) f(0) c) f( -1) versus d) f( -x) e) - f(x) f) f( x + 1)

4 EXAMPLE: Determine whether the equation is a function. 9) y x x 3 4 5) y 4 x Students will be able to find the domain and range of a function. DOMAIN AND RANGE The domain of a function is the set of all allowable values. Non-allowable x values could be values of x that would make the denominator = 0, or would make a negative value in the radical. The range of a function is the set of all allowable values. EXAMPLES: Find the domain of each function. 34) f x ( ) x 36) f( x) x x 1 x ex) f( x) x 9 4) g( x) 1 x VERTICAL LINE TEST: A graph can represent a function or not a function. Remember to be a function there can only be one y for every x. The Vertical-line test is an easy way to determine if the graph represents a function. A vertical line drawn through a graph should only intersect the graph in at most one point. EXAMPLES: Determine whether the graph is that of a function by using the vertical-line test. If it use the graph to find a) domain and range b) The intercepts, if any. a) b) c) d)

5 PreCalculus NOTES.1B Functions and Graphs- Interpreting the Graph Students will be able to evaluate the graph of a function. Ex) Use the following graph of the function f to answer the questions: a) Find f(0) b) Find f(5) c) Find f() d) Find f(-3) e) Is f(3) positive? f) Is f( -1) positive? g) For what numbers is f(x) = 0? h) For what numbers is f(x) < 0? i) What is the domain of f? j) What is the range of f? k) What are the x-intercepts? l) What are the y- intercepts? m) How often does the line y=1 intersect the graph? n) How often does the line x=1 intersect the graph? o) For what value of x does f(x) = 3? p) For what value of x does f(x) = -? Students will be able to answer questions about a function. x Ex 66) Use the function f( x) to solve. x 1 a) Is the point, on the graph? 3 b) If x= 1, what is f(x)? What point is on the graph of f? c) If f(x) = 1, what is x? What point is on the graph of f? d) What is the domain of f? e) List the x- & y- intercepts, if any, of the graph of f.

6 Ex) The effect of gravity on Jupiter. If a rock falls from a height of 0 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately: H(x) = 0 13x a) graph H(x) by hand graph H(x)w/graphing utility X Y b) What is the height of the rock when: x = 1 second? x = 1.1 second? x = 1. second? Check by calculator c) When is the height of the rock: 15 meters? When is it 10 m? When is it 5 m? Check by calculator d) When does the rock strike the ground?

7 PreCalculus NOTES.A Linear Functions and Models LINEAR FUNCTIONS A linear function is of the form EXAMPLES Graph each linear function by hand. 6) h( x) x 4 8) G(x) = - 3 Students will be able to determine if a relation is linear or nonlinear. SCATTER DIAGRAMS What is a scatter diagram(plot)? Examine the scatter diagrams and determine whether the type of relation, if any, that may exist is linear or nonlinear?

8 Students will be able to use a graphing utility to find a linear equation of best fit. LINE OF BEST FIT The line that best fits liinearly related data is called the line of best fit. Directions for finding the line of best fit (a) Draw a scatter diagram (b) Select two points from the scatter diagram and find the equation of the line containing the points selected. (c) Graph the line found in part (b) on the scatter diagram. (d) Use a graphing utility to find the line of best fit. (e) Use a graphing utility to graph the line of best fit on the scatter diagram. 0) X Y (a) (b) (d) 4) X Y (a) (b) (d)

9 PreCalculus NOTES.B Linear Functions and Models: Line of Best Fit, Direct Variation Students will be able to solve application problems involving a linear equation of best fit. EXAMPLES Graph each linear function by hand. 8. Mortgage Qualification The amount of money that a lending institution will allow you to borrow mainly depends on the interest rate and your annual income. The folliowing data represent the annual income I, required by a bank in order to lend L dollars at an interest rate of 7.5% for 30 years. Annual Income, I ($) Loan Amount, L ($) 15,000 44,600 0,000 59,500 5,000 74,500 30,000 89,400 35, ,300 40, ,00 45, ,100 50, ,000 55, ,900 60, ,800 75, ,700 70,000 08,600 Let I represent the independent variaible and L the dependent variable. a) Use a graphing utility to draw a scatter diagram of the data. b) Use a graphing utility to find the line of best fit to the data. c) Graph the line of best fit on the scatter diagram drawn in part (a) d) Interpret the slope of the line of best fit. e) Determine the loan amount that an individual would qualify for if her income is $4,000.

10

11 PreCalculus NOTES.3a Properties of Functions Students will be able to determine the intervals at which a function is increasing or decreasing. PROPERTIES OF FUNCTIONS Increasing function- Decreasing function- Constant function- DETERMINE IF A FUNCTION IS INCREASING, DECREASING OR CONSTANT Ex 1) Using the graph of the function f given below. Find the interval(s) where: a) f is increasing: b) f is decreasing : c) f is constant: d) Is f decreasing on the interval (-8, -4)? ( -, 6) (, 10) ( 10, 8) ( 15, 8) ( -10, 0) ( -5, 0) (0, 0) ( 5, 0) ( -8, - 4) Students will be able to determine local maximum and local minimum points. LOCAL MINIMUMS AND MAXIMUMS Local minimum- Local maximum- x- value tells where the local minimum or maximum is y- value tells what the local minimum or maximum is Ex ) Use the graph from ex) 1 to find the local minima and maxima. a. local minimum: b. local maximum:

12

13 PreCalculus NOTES.3b Properties of Functions Students will be able to determine if a function is even, odd or neither. Example 1: Let A be (3,4). Plot it on the graph and plot the following points Bsymmetry in x-axis Csymmetry in the y-axis Dsymmetry in the origin Are the following graphs symmetric in the x-axis, y-axis, or origin? Even function: Odd Function: f( -x ) = f(x) f( -x) = - f(x) (When plugging in x, function remains same.) (When plugging in x, function is exact opposite.) graph is to an even power of, 4, 6, 8. graph is to an odd power of 3, 5, 7, 9, graph is symmetric over the y- axis graph is symmetric about the origin GRAPHICALLY ALGEBRAICALLY EVEN ODD Examples: Algebraically verify if the following functions are odd, even or neither. 1) f ( x) 3x x 4 ) f x 3 ( ) 8x 1 3) x 46) hx ( ) x 1 3 f ( x) 4x x

14 Students will be able to use a graphing utility to analyze a function Use a graphing calculator to graph the function over the indicated interval. 5) f x x x 3 ( ) 3 5 ( 1,3) a) Using the calculator find any local minima Using the calculator find any local maxima b) Determine where the function is increasing Determine where the function is decreasing c) State the domain of the function State the range of the function Minimize the Material Needed to Make a Box. 68. An open box with a square base is required to have a volume of 10 cubic feet. The amount A of material used to make such a box as a function of the length x of a side of the square base is A x A( x) 40 x x Graph A and determine where A is smallest.

15 PreCalculus NOTES.4A Piecewise-defined functions Students will be able to evaluate piecewise-defined functions. When functions are defined by more than one equation, they are called piecewise-defined functions. Each equation is true only in its unique domain. Evaluate the piecewise function at the given x-values. Ex 1) 3x, x1 f( x) 4, x 1 ex) x, x 0 f( x) x, x 0 f (0) f ( ) f ( ) f (0) f (1) f (5) Students will be able to graph piece-wise defined functions. ex 3) x, x0 f( x) x1, x0 x-intercepts: y-intercepts: Find f( -): Find f( 3): ex 4) x3, x f( x) x1, x x-intercepts: y-intercepts: f( -): f( 3):

16 Students will be able to define a piece-wise defined functions for a given graph. The graph of a piecewise-defined function is given. Write a definition for each function. Ex 3) (, 1) ( -1, 1) Ex)

17 PreCalculus NOTES.4B Library of Functions, Piecewise-defined Functions Continued NAME Linear Function EQUATION f(x) = DOMAIN RANGE TYPE Odd,Even,Neither Constant Function Square Function Absolute Value Function

18 Minimum Payment Payment More Piece-Wise Functions Examples: Ex ) ****3 intervals**** 3 pieces****** a) x 3, 4 x1 f ( x) 5, x 1 x, x1 f(1): Study each graph and then write a piecewise-defined function. 5. f( x) 6. The minimum amount due on a credit card each month depends on the current balance. If your current balance is less than or equal to $0, then your minimum payment will be your current balance. If your current balance is between $0 and $40, then your minimum payment will be $0. If your current balance is $40 or more, then your minimum payment will be $40. Write a piecewise defined function where x represents the current balance and f(x) represents the minimum payment. Then graph it below. f( x) Current Balance

19 PreCaluculus Notes.5A Library of Functions/Transformations Students will be able to recognize the quadratic, cubic and absolute value functions. Students will be able to recognize transformations to the quadratic, cubic and absolute value functions. Square Function f ( x) x f x ( ) x 1 f () x x f ( x) ( x ) f ( 1) f (0) f (1) f ( 1) f (0) f (1) f ( 1) f (0) f (1) f (3) f () f (1) f ( x) x f ( 1) f (0) f (1) f ( x) ( x 1) f ( x) 4x f ( x) ( x 3) f x ( ) ( x ) 1 f x ( ) 3x 1 f ( x) ( x 1) f x ( ) 4x 5

20 Cube Function f ( x) x 3 f ( 1) f (0) f (1) f x 3 ( ) x 1 f ( x) x 3 f ( x) ( x ) 3 Abs. Value Function f ( x) x f ( 1) f (0) f (1) f ( ) f () f ( x) x 1 f ( x) x f ( x) x Can you write the equation if you are given a graph?

21 PreCalculus NOTES.5B Graphing Techniques; Transformations Students will be able to recognize the square root and reciprocal functions. NAME EQUATION f(x) = DOMAIN RANGE TYPE Odd,Even,Neither Linear Function Constant Function Identity Function Square Function Absolute Value Function Cube Function Square Root Function Reciprocal Function

22 f(x) = afnc[b(x-h)] +k a b h k Transformation How does the equation change? Give an example using the function f ( x) x Sketch the change in the function from the parent function. (Shift up units) Vertical Shift (Up) y = f(x) + k (Shift down units) Vertical Shift (Down) y = f(x) - k (Shift right units) Horizontal Shift (Right) y = f(x - h) (Shift left units) Horizontal Shift (Left) y = f(x +h) Reflection about the x-axis y = - f(x) Reflection about the y-axis y = f(- x) (Vert stretch by a factor of ) Vertical Stretch y = af(x) a > 1 (Vert comp by a factor of ½) Vertical Compression y = f(ax) 0 < a < 1 Horizontal Stretch y = f(ax) 0 < a < 1 (Horiz strectch by a factor of ½) (Horiz comp by factor of ) Horizontal Compression y = f(ax) a > 1

23 f(x) = afnc[b(x-h)] +k vertical horizontal horizontal shift vertical shift stretch or compression stretch or compression Graph each function using transformations. Remember the parent graph and 3 key points. 1) f x ( ) x 4 ) 1 f( x) 3 x 3) f ( x) 4 x 1 `4) f ( x) x 3 Students will be able to define functions given the transformations. ex) Rewrite the function to represent the indicated transformation. f ( x) x a) Shift up 3 units b) Shift to the left 4 units c) Reflect about the y-axis d) Shift down 5 e) Reflect about the x-axis f) Stretch vertically by 3 and right 4 and shift down 3

24 PreCalculus NOTES.5C Graphing Techniques; Transformations NAME EQUATION f(x) = Greatest f(x) = int(x) Integer Function = the greatest integer less than or equal to x. i.e. x = 1.8 f(x) = x = f(x) = DOMAIN RANGE ( -, ) set of integers TYPE Odd,Even,Neither neither f(x) = afnc[b(x-h)] +k vertical horizontal horizontal shift vertical shift stretch or compression stretch or compression Graph each function using transformations. Remember the parent graph and 3 key points. 1) f ( x) ( x 1) 3 ) f ( x) x 1 3) f ( x) x `4) f x x 3 ( ) 1 1