GRAPHING CALCULATOR - WINDOW SIZING

Transcription

1 Section 1.1 GRAPHING CALCULATOR - WINDOW SIZING WINDOW BUTTON. Xmin= Xmax= Xscl= Ymin= Ymax= Yscl= Xres=resolution, smaller number= clearer graph Larger number=quicker graphing Xscl=5, Yscal=1 Xscl=10, Yscal=100 Xmin=, Xmax= Xmin=, Xmax= Ymin=, Ymax= Ymin=, Ymax= Set your viewing window to the following specifications and draw the results: Xscl 5, Yscl 10 10,10, 30,30 Xscl, Yscl 10,10, 10,0 1

2 RAPHING CALCULATOR - GRAPHING STANDARD WINDOW Graph each equation in a standard viewing window. " Zoom 6" gives you a standard window. APPROPRIATE WINDOW Graph the following, then 1) to change the window for the given interval. 1) Play with the ymin and ymax and the scalers. ) "Zoom 0" will fit the graph within you xmin and xmax. 3) Use the table to find appropriate window limits. y x 300x 500 y 800 x FINDING AN UNKNOWN X -VALUE Use the trace and zoom keys, then use the intersect key to find the matching coordinates.,,0 y 8 x x x 3x 5 y, x,9 POINT OF INTERSECTION Graph the following and then use the intersect command to find the point of intersection. y x 10x, y 1 5x y 9x 0, y 0.x 10 SOLVING AN EQUATION Use the intersect command to solve the equation 0. 1 x x x x x 7x 11 Use the zero command to solve the equation 0.x x x x x 7x 11 0

3 FUNCTIONS Section 1. Ordered pair Relation Function Determine if the following are functions:,3, 4, 1, 5,3 1,0,, 1,,3 1,0,, 1, 3,3 f (x) means the value of the function f at x. Be careful f (x) doesn t mean f times x. x is also called the "INPUT." Let s start with a function f ( x) x 4 f (1) f (5) f ( ) f (h) f ( x h) 3

4 Difference Quotient Find for the following: i) ii) Functions Part II f ( x) y y x 5 f ( x) x f ( ) x, y,1 1) Write as a function. Plug it into your calculator and find f(-3). ) Given g ( x) x 4, Find: a) g() b) x when g(x) = 5 c) Fill in the blank (, 4) 3) A rental company's daily charges are calculated using the function, where x is the number of miles. Translate this algebraic statement into a verbal statement explaining the rental companies daily charges. 4

5 Domain and Range Domain: Range: Determine the domain and range for the following:,3, 4, 1, 5,3 1,0,, 1,,3 1,0,, 1, 3,3 y 1 x 3x 5 y 3 x 1 y x y x 4 y x 3 x y x 5

6 GRAPHS - PROPERTIES Section 1.3 Find the following: A) DOMAIN B) RANGE C) X-INTERCEPTS D) Y-INTERCEPTS E) INTERVALS OF INCREASING F) INTERVALS OF DECREASING G)Local Extrema a) c) e) b) d) f) g) a) a) b) b) c) c) d) d) e) e) f) f) g) g) 6

7 Using your calculator draw a quick sketch and mark any points of interest you can find, then find the intervals or points for the following: a) domain b) range c) x-intercepts (calc.) d) y-intercepts (calc.) e) intervals of increasing f) intervals of decreasing g) local extrema h) domain and range 3 f ( x) x 5 m ( x) 0.05x 0.5x 1.5x 1. 5 a) a) b) b) c) c) d) d) e) e) f) f) g) g) h) h) DISCONTINUITY Breaks of gaps in the graph. Find the domain and locate any points of discontinuity. 7

8 Piece-Wise Graphs 3x1 if x 1 Graph the function f x x if 1 x 4, and evaluate f 3, f 1, f 0, f 4, f 7 x 5 if x 4 x y x y x y APPLICATIONS i) The profit (in dollars) from the sale of x car seats for infants is given by Find the number of car seats that must be sold to maximize the profit. What is the maximum profit (to the nearest dollar)? ii) A box with no top is to be made from a 10 by 0 inch cardboard by cutting equal size squares from each corner and folding up the sides. Let x be the length of the side of the square to be cut from each corner. Answer the following: a) What is the restriction on x? b) Find the value of x that will maximize the volume of the box. What is the maximum volume? c) Find the value of x at which the volume of the box will be greater than 150 cubic inches? 8

9 Section 1.4 GRAPHS AND TRANSFORMATIONS PARENT GRAPHS Sketch the shape of each in the box below. There is no grid. I just want the shape. 3 f x x 1 f x x f x x f x x f x 3 x f x x VERTICAL SHIFTS Graph the following and look at the parent graphs to compare. f x x f x x 3 f x x 1 What is happening to the graph of the original function when you add or subtract to the original function? y f x c y f x c HORIZONTAL SHIFTS Graph the following and look at the parent graphs to compare. f x x 1 f x x f x x 1 What is happening to the graph of the original function when you add or subtract to the x to the original function? y f x c y f x c 9

10 Graph without using a calculator, just move the original function. f x x 1 f x x f x x 1 f x 1 x f x x 3 f x x f x 1 x 1 f x x 3 f x x 1 Give the function of the following: The graph is shifted four units to the left and five units down. 10

11 REFLECTION ABOUT THE X-AXIS Graph the following and look at the parent graphs to compare. f 3 x x f x x f x x What is happening to the graph of the original function? REFLECTION ABOUT THE Y-AXIS Graph the following and look at the parent graphs to compare. f x x f x 3 x 3 f x x What is happening to the graph of the original function when you multiply a negative to the original function? STRETCHING AND SHRINKING GRAPHS VERTICAL STRETCHING AND SHRINKING GRAPHS Graph the following and look at the parent graphs to compare. 3 x x f x 3x f x 0. 5 x f x 1/3x f f x 0.5 f x HORIZONTALY STRETCHING AND SHRINKING GRAPHS Graph the following and look at the parent graphs to compare. f x x f x 3x f x 0.5 f x f 0.5 x x 1 f x x 3 3 Find: 0)= 11

12 PUTING IT ALL TOGETHER af ( x h) k If a is negative = h= k= Graph without using a calculator, just move the original function. f x x 1 f x 1 x f x x f x x 1 1 Find an equation for the function g. Check your work with a calculator. The graph of is shifted five units to the right and four units up. The graph of up. is vertically stretched by a factor of, reflected in the x-axis, and then shifted 6 units The graph of is shifted 6 units up, vertically stretched by a factor of, and then reflected in the x- axis. The graph of down. is shifted to the left 3, vertically stretched by a factor of 0.5, and then shifted units The graph of 3. is shifted units down, vertically shrunk by a factor of 0.5, and then shifted to the left The graph of is shifted 4 units up, vertically shrunk by a factor of 1/3, shifted to the right 1 unit, and then reflected in the x-axis. 1

13 EVEN-ODD FUNCTIONS EVEN FUNCTION-- f x f x Why are the two functions equal? --so symmetric about the y-axis a) f x x and gx x b) f x x and gx x ODD FUNCTION-- f x f x Why are the two functions equal? --so symmetric about the origin 3 a) f x x and gx x 3 b) f x 5 x and gx 5 x Prove analytically that the following function is even or symmetric about the y-axis: f x x 3 f x x x 4 Start with: f x x 3 f x x 3 Prove analytically that the following function is odd or symmetric about the origin: x x 5 x 3 x f x f 3 Start with: 1) f x x 5 x 3 3x 1 x ) f x x 5 x 3 3 x 13

14 Section 1.5 OPERATIONS ON FUNCTIONS f gx f x gx f gx f x gx fgx f x gx f x f ( x) x 4, g ( x ) x 4 Find: f f f fg g f gx gx g3 x x g f ( x) g( x) FIND THE DOMAIN OF f gx f x gx f gx f x gx fgx f x gx f x x 3 g x x 4 f g x f ( x) g( x) 14

15 Given: ( x) x 4 f, ( x) 3x 4 COMPOSITE FUNCTIONS f gx f gx g, p x 4 x Find: g f x f px g p Find f g f g4 g f 3 g f 1 f g3 f g f g 4 g f g f f f f g g g Find a functions f(x) and g(x) such that the 15

16 DOMAIN OF COMPOSITE FUNCTIONS Remember, we are looking for the real number values of x that we can input an get real number outputs. f ( x ) x 4, g(x) 4 x Find the domain of composite functions f(g(x)) 1) f(g(x)) Find the domain of composite functions g(f(x)) 1) g(f(x)) Find the domain of composite functions f(g(x)) 1) f(g(x)) Find the domain of composite functions g(f(x)) 1) g(f(x)) 16

17 ONE-TO-ONE FUNCTIONS Section 1.6 A function in which each element in the range corresponds to one and only one element in the domain. Determine if the following are One-to-one functions:,3, 4, 1, 5,3 1,0,, 1,,3 1,0,, 1, 3,3 If implies, then is one-to-one Determine if the following are One-to-one functions: y 3x y x 3 17

18 INVERSE FUNCTIONS If is one-to-one, then an Inverse function,, exists Find the inverse by switching the x and the y of the following:,3, 4, 1, 5,4 1,0,, 1, 3,3 Find the inverse of the following: Ex/ y 3x Ex/ y x 3 3 y x, Ex/ y x 3 x Finding the inverse of a graph: f ( x) x 18

19 INVERSE FUNCTIONS - DETERMINING IF A FUNCTION IS AN INVERSE 1 1 f f x x for domain of f x, and f 1 f x x for domain of f x Determine are inverses. Determine are inverses. Why are we looking for x? = Why do you need to check both? APPLICATIONS OF INVERSES The number q of cd players a retail chain is willing to supply at a price of $p is given approximately by a) Find the range of S using your calculator. b) Find, and find it's domain and range. HINT: DO NOT SWAP THE LETTERS, THIS WILL CAUSE CONFUSION. 19