At right: Closeups of the graphs of. with WINDOW settings Xmin=-1, Xmax=1, Xscl=0.1, Ymin=-1, Ymax=1, Yscl=0.1

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Nigel Willis
5 years ago
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1 Know the graphs of f(x) = x n for n = odd, positive: 1, 3, 5, The Domain is All Real Numbers, (, ), or R. The Range is All Real Numbers, (, ), or R. It is an Odd function because f( x) = f(x). It is symmetric with respect to the origin: for each point (x, y) has a twin at ( x, y). For example, for f(x) = x 3, f( 7) = 343, f(7) = +343 As the exponent gets larger, o the graph becomes narrower and steeper when x < 1 and x > +1 o but the graph flattens more around the origin, where 1 < x < +1 y = x y = x 3 y = x 5 At right: Closeups of the graphs of y = x 3 and y = x 5 near the origin, with WINDOW settings Xmin=-1, Xmax=1, Xscl=0.1, Ymin=-1, Ymax=1, Yscl=0.1 Compare: What s the difference between y = x 3 and y = x 5 near the points x = 0.5 and x = +0.5?

2 Know the graphs of f(x) = x n for n = even, positive: 2, 4, 6, The Domain is All Real Numbers, (, ), or R. The Range is All Non-Negative Real Numbers, [0, ). It is an Even function because f( x) = f(x). It is symmetric with respect to the y-axis: for each point (x, y) has a twin at ( x, y). For example, for f(x) = x 2, f( 7) = +49, f(7) = +49 As the exponent gets larger, o the graph becomes narrower and steeper when x < 1 and x > +1 o but the graph flattens more around the origin, where 1 < x < +1 y = x 2 y = x 4 y = x 6 At left: Closeups of the graphs of y = x 2 and y = x 4 near the origin, with WINDOW settings Xmin=-1, Xmax=1, Xscl=0.1, Ymin=-1, Ymax=1, Yscl=0.1 Compare: What s the difference between y = x 2 and y = x 4 near the points x = 0.5 and x = +0.5?

3 n Know the graphs of f(x) = x, x o 2 Square Root y = x = x 3 Cube Root y = x even 4 6 and similarly for x, even roots, x, x, etc. odd 5 7 and similarly for x, odd roots,, x, x, etc. Domain [0, ) and Range [0, ) Domain (, ) and Range (, ) Symmetric with respect to the origin, (0,0) As the index (the little number in the nest) gets larger, the graph goes lower, closer to the x-axis when x > 1 but the graph goes higher, closer to y = 1 when x < 1 4 Example: > The 4 th root is larger. 4 But < The 2 nd (square) root is larger. 4 Here are graphs of y = x and in bold, y = x in the window Xmin=0, Xmax=2, Xscl=0.5, Ymin=0, Ymax=1.5, Yscl=0.5. Observe that they cross at (1,1) because for any n th n root, 1 = 1.

4 Know the graphs of f(x) = 1 x n for n = odd, positive: 1, 3, 5, The Domain is All Real Numbers except x = 0, (, 0) (0, ). The Range is All Real Numbers except x = 0, (, 0) (0, ). It is an Odd function because f( x) = f(x). It is symmetric with respect to the origin: for each point (x, y) has a twin at ( x, y). For example, for f(x) = 1/x 3, f( 2) = 1, f(2) = 1. ( 2, 1 ) and (+2, + 1 ) are both on the graph As the exponent gets larger, o the curve seems flatter when x < 1 and x > +1 o but it gets steeper when 1 < x < +1 These graphs were produced with WINDOW Xmin=-5, Xmax=5, Ymin=-5, Ymax=5. y = 1 x y = 1 x 3 y = 1 x 5

5 Know the graphs of f(x) = 1 x n for n = even, positive: 2, 4, 6, The Domain is All Real Numbers except x = 0, (, 0) (0, ). The Range is All Positive Real Numbers, (0, ). It is an Even function because f( x) = f(x). It is symmetric with respect to the y-axis: Each point (x, y) has a twin at ( x, y). For example, for f(x) = 1/x 2, f( 2) = 1, f(2) = 1. ( 2, 1 ) and (+2, 1 ) are both on the graph As the exponent gets larger, o the curve seems flatter when x < 1 and x > +1 o but it gets steeper when 1 < x < +1 These graphs were produced with WINDOW Xmin=-5, Xmax=5, Ymin=-5, Ymax=5. y = 1 x 2 y = 1 x 4 y = 1 x 6

6 Know the graph of f(x) = x, the absolute value function It is a V shape. It has a corner at the Origin, (0,0). The Domain is All Real Numbers, or R. The Range is All Non-Negative Real Numbers, [0, ). To put this function into your TI-84, press the MATH key, then right arrow to the NUM submenu, and choose 1:abs( It is an Even function because f( x) = f(x). It is symmetric with respect to the y-axis: Each point (x, y) has a twin at ( x, y). For example, for f(x) = x, f( 7) = 7, f(7) = 7. ( 7,7) and (+7, 7) are both on the graph. These graphs were produced with WINDOW Xmin=-5, Xmax=5, Ymin=-5, Ymax=5. y = x y = 3 x y = 1 3 x This is the standard building block graph for absolute value Multiplying by a > 1 makes it steeper and narrower. Multiplying by 0 < a < 1 makes it broader and of gentler slope.

7 Multiplying by 1 flips the graph vertically, across the x-axis. Each (x, y) in the original graph is transformed into (x, y). For example, consider f(x) = x 2 at x = 5. f(5) = 25. The point (5,25) is on the graph. But when f(x) = x 2, f(5) = 25. The point (5, 25) is on the graph. y = x 2 and y = x 2 y = x 3 and y = x 3 y = 1 x and y = 1 x y = x and y = x

8 Know the graph of f(x) = x the Floor function It is a Step function because it looks like the steps in a flight of stairs. It means What is the largest integer x? Examples: 4.7 = 4 and 4.7 = 5. The Domain is All Real Numbers, or R. The Range is the set of Integers, Z, or {, 3, 2, 1, 0, 1, 2, 3 } To put this function into your TI-84, press the MATH key, then right arrow to the NUM submenu, and choose 5:int( In better resolution, the left endpoints of each segment are solid dots (because int(5) = 5, for example) and the right endpoints are open dots (because int(6)=6, not 5, so the segment from x = 5 to x = 6 has a solid dot at the left end where x = 5 and an open dot at the right end where x = 6. y = x y = 3 x y = x/2 Step height is 1. Step length is 1. Multiplying x outside the int() sets the height of each step. The range is multiples of 3: {, 9, 6, 3, 0, 3, 6, 9, } Multiplying x inside the int() sets the length of each step to the reciprocal of the multiplier. Here, multiplier 1/2 sets step length to 2.

9 Piecewise Functions are really easy Example: f(x) = { 2x 2 x 2 when x 1 when x > 1 The function is made up of two pieces. That s why we call it a piecewise function. When x 1, that is, on the interval (, 1], the rule = 2x 2 is in effect. When x > 1, that is, on the interval (1, ), the rule = x 2 is in effect. There is a closed dot at ( 1,0) and an open dot at ( 1,1) That s how the graph tells us which of the pieces rules that point in the domain. In this example, at x = 1, the straight line 2x 2 determines the value at x = 1. And immediately to the right of x = 1 is where the parabola x 2 takes over.

( ) ( ) SECTION 1.1, Page ( x 3) 5 = 4( x 5) = 7. x = = = x x+ 0.12(4000 x) = 432

( ) ( ) SECTION 1.1, Page ( x 3) 5 = 4( x 5) = 7. x = = = x x+ 0.12(4000 x) = 432 CHAPTER Functions and Graphs SECTION., Page. x + x + x x x. x + x x x x x. ( x ) ( x ) x 6 x x x x x + x x 7. x + x + x + 6 8 x 8 6 x x. x x 6 x 6 x x x 8 x x 8 + x..x +..6.x. x 6 ( n + ) ( n ) n + n.