Advanced Functions Unit 4

Transcription

1 Advanced Functions Unit 4

2 Absolute Value Functions

3 Absolute Value is defined by:, 0, if if (), if 0

4 The graph of this piecewise function consists of rays, is V-shaped and opens up. To the left of =0 the line is y = - To the right of = 0 the line is y = Notice that the graph is symmetric in the y-ais because every point (,y) on the graph, the point (-,y) is also on it.

5 y = a - h + k Verte (h,k) & is symmetrical in the line =h V-shaped If a< 0 the graph opens down (a is negative) If a>0 the graph opens up (a is positive) The graph is stretched vertically if a > 1 The graph is compressed vertically if a < 1(fraction < 1) a is the slope to the right of the verte ( -a is the slope to the left of the verte)

6 Graph y = V = (-,3). Apply the slope a=-1 to that point 3. Use the line of symmetry =- to plot the 3rd point. 4. Complete the graph

7 Write the equation for:

8 The verte (0,-3) It has the form: y = a So the equation is: y = -3 To find a: substitute the coordinate of a point (,1) in and solve (or count the slope from the verte to another point to the right) Remember: a is positive if the graph goes up a is negative if the graph goes down

9 Domain and Range Domain : all real # Range y: y verte or y verte depending on a

10 Piecewise Functions

11 Up to now, we ve been looking at functions represented by a single equation. In real life, however, functions are represented by a combination of equations, each corresponding to a part of the domain. These are called piecewise functions.

12 f 3 1, if 1, if 1 1 One equation gives the value of f() when 1 And the other when >1

13 Evaluate f() when =0, =, =4 f ( ), if 1, if First you have to figure out which equation to use You NEVER use both X=0 This one fits Into the top equation So: 0+= f(0)= X= This one fits here So: () + 1 = 5 f() = 5 X=4 So: This one fits here (4) + 1 = 9 f(4) = 9

14 Graph: f 1, if ( ) 3, if For all s < 1, use the top graph (to the left of 1) For all s 1, use the bottom graph (to the right of 1)

15 f 1 3, if ( ) 3, if 1 1 =1 is the breaking point of the graph. To the left is the top equation. To the right is the bottom equation.

16 Graph: f ( ) 3, if 3 1, if

17 Step Functions f ( ) 1, if, if 3, if 4, if

18 f ( ) 1, if 0 1, if 1 3, if 3 4, if 3 4

19 Graph : f ( ) 1, if 4 3, if 3 3, if 1 4, if 1 0

20

21 Graphing Simple Rational Functions

22 Hyperbola A type of rational function. Has 1 vertical asymptote and 1 horizontal asymptote. Has parts called branches. (blue parts) They are symmetrical. We ll discuss different forms. =0 y=0

23 Hyperbola (continued) One form: a y k h Has asymptotes: =h (vert.) and y=k (horiz.) Graph points on either side of the vertical asymptote. Draw the branches.

24 y 3 1 E: Graph State the domain & range. Vertical Asymptote: =1 Horizontal Asymptote: y= y Left of vert. asymp. Right of vert. asymp. Domain: all real # s ecept 1. Range: all real # s ecept.

25 Domain and Range Domain : all real # asymptote Range y: all real # asymptote

26 Graphing Square Root & Cube Root Functions

27 First, let s look at the parent graphs. y y , 4 3 7, , 3 9, , 1 4, , , , 1-8, , ,

28 Now, what happens when there is a number in front of the radical? 3 y y , 10-7, , , 8 4-1, , 4 9, 6 0 0, , , -6 8, , * Notice the graph goes thru the points (0,0) and (1,) * Notice the graph goes thru the points (-1, 3), (0,0), & (1,-3). 7, -9

29 Generalization y a Always goes thru the points (0,0) and (1,a). y a 3 Always goes thru the points (-1,-a), (0,0), and (1,a).

30 Now, what happens when there are numbers added or subtracted inside and/or outside the radical? y 3 a h k or y a h k Step 1: Find points on the parent graph y a or Step : Shift these points h units horizontally (use opposite sign) and k units vertically (use same sign). y a 3

31 Domain and Range , 4 3 7, , 3 9, , 1 4, , , , 1-8, , , Domain : h Range y: y k Domain : all real # Range y: all real #

32 Inverse & Joint Variation

33 Direct Variation Use y=k. Means y varies directly with. k is called the constant of variation. Note: k = y

34 Inverse Variation y k y varies inversely with. k is the constant of variation. Note: k = y

35 How to tell whether & y show direct variation, inverse variation, or neither. a. y=4.8 y 4.8 b. y=+4 Inverse Variation Neither Hint: Solve the equation for y and take notice of the relationship. c. y 1.5 y 1. 5 Direct Variation

36 E: The variables & y vary inversely, and y=8 when =3. Write an equation that relates & y. y 8 k k 3 y 4 k=4 Find y when = -4. use: y k 4 y 4 y= -6

37 Joint Variation When a quantity varies directly as the product of or more other quantities. For eample: if z varies jointly with & y, then z=ky. E: if y varies inversely with the square of, then y=k/. E: if z varies directly with y and inversely with, then z=ky/.

38 Eamples: Write an equation. y varies directly with and inversely with z. k y y varies inversely with 3. z y varies directly with and inversely with z. y k z z varies jointly with and y. y varies inversely with and z. y k 3 z k y y k z

39 Solving Rational Equations

40 When there is only fraction on each side of the =, just cross multiply as if you are solving a proportion ( ( 4)( 3) or 3 4 or 3 4) Check your solutions! 0

41 Hint for solving: A quick way to solve a rational equation is to multiply everything through by the LCD. This will get rid of all the fractions! Beware of etraneous solutions!

42 Eample: Solve *3 * LCD: Multiply each fraction through by the LCD Check your solution! 18

43 Solve LCD:? LCD: (+1) 5( 1) ( 1) 4( 1) 5( 1) ( 1) ( 1) Check your solution!? No Solution!

44 3 6 Solve ( 6 )( ) (3 Factor 1 st! )( )( ) ( ) ( 3)( 1) or 1 0 6( )( ) ( )( ) LCD: (+)(-) 4 ( 3 or )( 1 Check your solutions! )