Chapter 2. jlwtical. .me#fiw. is worth. 7) The Origin (0,0) 8) (-2, 3) 9) (-2, -3) 10) (2, 3) 11) (2, -3) 12) Open points vs.

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1 Chapter 2 21 Graphs Ex: Identify the following on this graph: ± I 1) Axis =L jlwtical me#fiw 2) xaxis and yaxis I I 3) Quadrants 4) Ordered Pair 5) Scale ( x What, y ) k,i ) each square Size is worth 6) Rectangular Coordinates or Cartesian Coordinates 7) The Origin (0,0) 8) (2, 3) 9) (2, 3) 10) (2, 3) I 11) (2, 3) 12) Open points vs closed points n i 't e,?3 ( < iiiii?,, > ia ;D C 53 ) I axis one 3) axes = two axees ' ' # I

2 Midpoints Midpoint Formula: The Midpoint between (x1, y1) and (x2, y2) is given by! +! $ 2, ( + ( $ 2 Ex: Find the midpoint of the line segment with endpoints at (1, 3) and (2, 45) ( I, 3) C 2 4, 5) ;, l Y *, s 4i =#tf4 ' = Midpoint f 05, 375 )

3 22 Graphing Linear Equations in Two Variables An ordered pair is a solution of an equation if substituting it into the equation makes the equation true Ex: : e checking a) Is (2, 3) a solution of the equation 3y = 45x? 3y= 45 3 C 3) I 45k ) 9 t 9 No ' n' b) Is (4, 6) a solution to that equation? 3y= 4 5x (4) 18 = 18 Yes Graphing an equation means to plot the entire solution set for that equation if an ordered pair is a solution of that equation, then it appears in the graph Intercepts are the points where the graph of an equation intercepts one of the axes xintercept: (a, #e s to 0) yintercept: (0, b)

4 y= Ex: Graph 2x 3y = 12 by finding the intercepts 6= * int : y=o ( 6,0) 2 301=12 yn2 34=12 she = % 6 > X 2x=l2 _ x=6 : x=o = yint =12 ) 34=12 ( 6,0 ) 3,1=12 4 (054)

5 Ex: Graph 1) y = 5 and 2) x = 5 t Y = 5 iffy ' ff!i Ef * 5

6 23 Rate of Change and the Slope of a Line Rate of change = )*+, /,012 34*50+ * Ex: At 4 pm, there were 20 tribbles in the grain bin By 6 pm, there were 35 What s the rate of change of tribbles in this example in tribbles per hour? Tribble s per hour = Trek ' = I = u I± ; How about in tribbles per minute? 7,5 Tribble s Slope per III Lets slope = m = 8792 /, : = = : 850 /, ; ; hour =? Tribble s per =IYnbd min umnitath Tagqopro#rohIIes=totrmodwmI=TiiYY=EFoFE8slope=m=rE=chnI8gYx=F* Delta = charge

7 Ex: Find the slope of these lines: Y Tm Mt > 0 m,, ppyi 5 ( 3,5 ) % m< 0 unread I m=0 am! 11 It X,, In undefined ( 5,5 ) m=z an mid} c (,z) using triangles m fza w rm = ) me 22=1,=' III =L 1 trormua,f mz=i x=it?,= = = a Ex: Find the slope of the line passing through (3, 4) and (5, 1) m=ryne=ey A = YI ( 3,4 ) (51) m=tfi5d=e Xz X, or 4 ) 4+1 EI m=fs===e= HE

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9 Positive slopes: Negative slopes: Parallel lines have the same slope Perpendicular lines have slopes that are opposite reciprocals m Parallel m Perpendicular m ± z ± 3 } 2 3= Undefined The slope of a vertical line is: undefined Mris The slope of a horizontal MFOline is:

10 ' ' 24 Writing Equations of Lines What does the equation of a line look like? Standard Form: Ax + By = C Ex: Solve for y: 3x 4y = = =3 +12 = y= x +±, 3 SlopeIntercept Form: y = mx + b m = slope b = yintercept (x = 0) Ex: Find the slope and yintercept for the graph of y = 3/4x 3 y= x MIZ, 3 [ yint : 3g;) Ex: Write an equation of the line passing through (0, 4) with slope = 5 m= 5 T y= y t ( E 4) 5 +6 y= int 5 +4

11 PointSlope Form: y y1 = m ( x x1 ) Ex: Write an equation of the line passing through the points (3, 4) and (1, 8) Step : Find m mie#=t85=i=z Step 2 : 2 choices AO Use Point y y,=m( x xd Slope form ( 3,4 ) 4=243 y ) or stop keep going, = 2*+4 yiy Into Use Slope Intercept Y=mx+b y= 2 +6 m= 2 ( 3,4 ),fij6xµy b 4=2 (3)

12 25 An Introduction to Functions Example: Student Domain Birth Date Range Function: Every element of the Domain corresponds to a unique element of the Range One question one answer Ex: Are these functions? List the domain and range for each 1) A B Monkeys Function? Domain: Range: Set Yes Monkeys of Set of bananas Bananas

' 2) a function Notpg A Monkeys Bananas Function? No Domain: Set of Monkeys Range: Set of Bananas 3) Domain: Students Range: Colors Correspondence: Color of clothes Function?

13 ' 2) a function Notpg A Monkeys Bananas Function? No Domain: Set of Monkeys Range: Set of Bananas 3) Domain: Students Range: Colors Correspondence: Color of clothes Function? Domain: Range: No Students But its not afa Colors so we don't use these 4) {(1, 2), (2, 4), (3, 6)} 1 2 Function? Domain: Range: 2 4 Yes 3 6 { 1,33 } {2,4/6} Dothan 99 ( x )

14 5) {(1, 2), (2, 4), (1, 6)} Function? Domain: Range: No { 1,2 2 6 } chaos! { 34,644 AAAA! Graphs and Functions (7, 5) y, ' : fo one point (2, 1) Is this a function? Yes : I What s the domain and range? D :[ 2,7 ] x R :[ 1,5 ] y

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16 Vertical Line Test Is this a function? Why or why not? 3 different ;, is P ovalue µ y : vi \ < went > ^ : I Function Notation Ex: If f(x) = 2x, then f( )= f of Vertical line f(3) = 23=6 f( D) =2D f(0) = 20=0 Ex: If g(x) = x, then f(? )=Z? g(1) = g(2) = 1 2 g(0) = g(p) = : ort Ex: h(x) = 12, find h(1) = 12 YI,lz ( 1,12 ) h(0) = h(12) = 12 ( 0,12 ) 12 ( 12,12 )

17 C) Not a FI c f± µ f ± <g F_ Not at # fa

18 Ex: f(x) = x 2, g(x) = x + 1 Find: f(0) = f(1) = f(2) = g(0) = g(1) = g(2) = 02=0 8 i I i 22=4 K 4 0+1= =2 FIL 2+1=3 x=2 y =3 f(g(2)) = f (2+1) = f G) =3 2=9 f ( x )= 2g(x)= xtl More About Domains Ex: For f(x) = atria, find the domain Domain : the set of inputs F This would be bad 0 +3 x= 3 =) D :X # 3 Ex: For g(x) = MH, find the domain Mntga This Is bad ( it's not Real ) Fin D: xzo

19 26 Graphs of Functions Ex: Graph and then determine the domain and range of each: ( x ) ( y ) 1) f(x) = x absolute value f ( x )= 1 1 ] a yx±i Yes 0 101= = =2 IM#nsxIzftIEtD:RYz:yzo 3 131=3 : ER

20 2) g(x) = x 2 jf ; ; ; of 3 f t 32=9 < ; ; 3) h(x) = x 3 < 1/45=1 1 2 E D :R R :Y20 :a si as > ( 25=4 ;3 ty'=9 z ( ' = E glasses h(x)=x t= ;=t ; =33327 IEEE Edi y 43=44464, ' ydihhhtt D: R R :R 2 a 3=8

MAC College Algebra

MAC College Algebra MAC 05 - College Algebra Name Review for Test 2 - Chapter 2 Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact distance between the