Winter Session 1 Completed

C Ired c ~ e:-c.. t.-.. L, '< .,.(,_ h - lu:, ...,e J 1> .,. k/ Pbf f€r.J , v~ e-~l~ ~l""-1-o, o ex a ~ ,V '\ MATH 1310 Session capt..+~ @ fl1 i-;..-l-h. al½ j MATH 1310 C-<'I' f,,..f- 't> @_ tA 1 &...~, edc._ ~ • '2._/4.,._ o_,.,.. Section 1.1: Points, Regions, Distance and Midpoints vr:JJU]lng ro1111.s 110d Regions "J -- V .-< /"1.-€... .' 4bv -f'a-- lA P/ d(o~n Section 1.1: Pointst Region· , Distance and Midpoints Cpo._O (~e~ Section 1.1: Points, Regions, Distance and Midpoints «;ropJHng Polnlnod ll,eg,.onF As we see the plane consists of two perpendicular lines, the x-axis and the v-axis. These two lines separate them into four regions, or quadrants. -- - The pair, (x, y), is called an ordered pair. It corresponds to a single unique point in the coordinate plane. The first number is called the x coordinate, and the second number is called t he v coordinate. The ordered !}air (O, O) is referred to as the origi'n. ll1AlC71f Section 1.1: Points, Region~ Distance and Midpoints e~--t' . ( t.1 'i CYe_ ~ t:?l><_J't i '"'1:t,-,.1 The x coordrnate tells us the honzontal d1st-ance a pornt is from the origin. They coordinate tells us the v ertical distance a point rs from the origin. You'll move right or up for positive coordinates and left or down for negative coordinates. Example: Plot the foUoWing points. A (8,6) ~ ~ U b B. (-2,4) 1.,.). I/ Lf Q 11 v fi" Ql c. c2.sJ R. J. D. (-3,7) ('... ~ E. (2.0 3) {l;2. F. (-5,3) r?7 Q I ~111 C7,3 Gt V L. ,S-- t.1'3 QI I f I QI/ -9 1 QJ \ -s -1 -c! ..$ -i -3 -J, -1 -I Qlll -2 - "C • D Slide4 Graphing Regions in the Coordinate Plane The set of all points in the coordinate plane withycoordinate kis the bortz.ontalliney= k The set of all points in the coordinate plane with xcoordinate kis the vertical line X = k K ...5 ::=- Y~ --Z 'tA vlbJ. "iJ/ Y) {?/ y) "v O _-, 4 - 7 _, ➔ -.) -t -3 -1 - I -3 .... I ~ .a , 6 "' t " Y=-). /kl/ Cr,7) I.J,.'1./p\. ~e Cx1 -2) Graphing Regions in the Coordinate Plane Graphing Regions in the Coordinate Plane r:?t>++~~L1 ~ L:. e>~ > (:; 'P"-,/.~ }2 'I..\-- If} Q-f' ~(Aj;{l n iry' 5 0 t ·, ~ l- I~ : (J t> "'.., ~r, G.J &,t. -L '"7 ;-.t I e-t.S ~ 0 ~~,.., l iry Graphing Regions in the C-0ordinate Plane S 1 r'"" ~'b +-o (51>/~fj~ L.t -s cit j 51 t-.r./€. The Distance Formula For ~nytwo points (x,. y,) and (x,, y1), tl1e distance between them is given by a.) (- 3,1) & (1,3) ~ '-"'\, (x,.1 '-I,) f:11 X1 ::-j '1, -;, .+ t?{; .P..f-ere-,Le. 0\-" Example: F1111l lhe distance between th~ following pair of polnlS. ro Y- ~ ~; V.e=r I✓ ,.f b+ ri._ k.-v--,,/.; i.. P.fe,,,,.e:, ~< .of- t.. Y;.) X '). :: ( J ~j. - J16-t- '-I ~rio ~3 A tJ,-.::: J (-;JJ __ -:1.r&)L -r (-R:-sJ?:jz :: j{-3 ~)2 -t {-'1"6) 2The Distance F~ r~ f t:f ,':J) t ( 16 ; b) :::. Fi Jr- [2 rs-/ The Midpoint Formula Midpoint Form ula The midpoint of the line segment jolnlng the two points (,c,, y,) and (x.._ y,J 1s given by _(½ +2 Xi • M - Y2 + J/1) 2 Example: Find the midpoint between the following pair of points. a) (-3,1) & (1.3) f1-: ( The Midpoint Formula 1;-J J o ,,, t?'l J-,~ oi V . -C rJ 1 ~ y-- VA I,.,,es- Lines In this section, we'll review slope and different equations of lines. We will also talk about x-intercept and y-in tercept, parallel and perpendicular lines. S+rv..i;1hf L '~¾ J+l)("l.-z.(l'Vl+-"'-1 L V-e,h c_-./ L 1• J. Yl€S '/r:: )< _· >< ~ k Y1e.r- : Y= rn>< +b "/ - '/, ::=. v>, ( K - X,) a..~ --t- k:>y -=- C Lines ...?5.- -r..Z:-=.. / ~ b ( s )(Jp~ -,"-if-) .S)ope) (s~ ~ o(( -e-<) ( pD,•-if- - { ,· -,f - ,·-;+) Slope Dellnitlon: The slope of a line measures the steepness of a line or the rate of change of the line. To llnd the slope of a line you need two points. You can find the slope ofa line between two points (XLY>) and (x,, y,) by using this formula. Y2 -Y1 sJupe (m) = - - x2 - x1 Example 1: Find the slope of the line containing the following points a rri C hAi14-e t,, c1,,;,,.-,e b& IV) X ,. -t-b ~ slbpe; Y-, ·.,, f- 1/ -:: '!~ - ~ X-t' --t.t _!_.. X :;. l- Slide12 ...,.,.., E --L.J - -sLf v b ; V""} - ,..{I - -.r b - 'I Example 3: Write an equation of the line that satisfies the given conditions. a. m = ½ and the y-interceptis 3 . Y= WJ x rb Y~ ~ x +3 b. m = -3 and the line passes through (-2. 1). (Y-y 1)-=- (>') (x-xJ ~1 - I ~ - .J (. x: - - .:z) y - / :: -- 3 ( X. +-:2) c. line passes through (-6, 10) and (-2, 2). ').-JD (>J :; Slide13 - '2 - -' -'B = l-f- ~ Y- J:: -.3 (><-t2) Y,:. .J-=-3x-C --rJ -t~ --y ::= -3 X -6- y - / D-:: - J_ ( X -r {:.) I.\Ji~j (-~Jo) "... ( .,\ y _ ::: -- ;2. ( X -t;i_) J J i ;J - ')., >I 2 ~ ~ Y- 1o == -;i (x-t- t,) Y-2 = - ;2. (x +..2) Y-ya -2x -I) Y-y -f/b t')_, J-\o Public Page 1 == ~ ~2x -1 +2 Parallel and Perpendicular Lines Definition: Parallel llnes are lines with slopes rn, and m; such that they are equal. in other words Nev~, l1'1terJec.+ Definition: Perpendicular lines are lines in which the product of the slopes equal -1. t,. i'v/.e.J' Also known as the negative l'ecip1'ocal. m z = :1. ;-~t >'v, f-erfec;.,fc.e;f- r i.14-1- ~1.:f IGf mt ~vi,p(_ Y2 (A. y'.e,. 6'"x-r2 f ~ y"' l 1,e. I 0- Parallel and Perpendicular Li~ y::: -I-g-X -r J_ ?\f'.e. p-e..,,,.p-e..., o< - I) ] Slide15 'i ~+- -J} Y:_t_ ; ~ x -~\ [Y= ½x~~/ c 1·p / " CA/ x-intercept an ,d y-intercept When graphing an equation, it is usually very helpful to find thex intercept(s) and the y-intercepts of the graph. An x intercept is the first coordinate of the ordered pair of a point where the graph of the equation crosses the x axis. To find an x intercept, lety = 0 and solve the equation for x The y·intercept is the second coordinate of the ordered pair of a point where the graph of the equation crosses they axis. To find ayintercept, let x = 0 and solve the equation for y. 5e_ t -t ~ .5 i. lvt x-intercept and y-intercept -to v b ff' -r~ t:. .J ,· +( ti vter<-intercept: a. (-2,0) b. (0.5, 0) c.. (8/3, 0) d. (3/8~ 0) Question 2: y-intercept: a. (0, 2) b. (0, -2) c.. (0, 4) d. (D, -4) 1,,,/~{.'""' Example 6: Find the x andyintercepts of the graph of the equationy= X2 • 9 . Question 3 : x-intercept: ot\ ~le a:. ~ 4 c..re rt> a. (3,0) b. (4.5, 0) c. (±3, O) d. {±4.5, 0) a. (0, -9) b. (01 9) c. (O, ±9) d. (0, ±2) Question 4: Slide11'1tercept: y j of y~4.. e+ ~ ~ 11/1./~ Graphing Equations Y~/1/a 11/o Example 1: Determine which of rhe points (3. 2), ( •l , 3) and (0, 2) are on the graph of the equation 4x - ~ ~ 6, '-/ (-I) - 3 ( ;5) =C IQ -(, C ~ -1-9 =- 6 -=--6 c../ Example 2: Determine w hich of rhe points (-~ -13 2,~ ( -/)'2- -r 3(- l)l l)-r J - 3 -rJ. Graphing Equation~ 7- i --;:;. 0 o ...,,,/ -=-o and(~ (Z.I -u 2..-z. -t 3 2 x +3x~~2{:~) Lf { oJ - ·3 (z) =-( ;r!-& ) are on the graph of lhe equation ( -Z; -/) (2-)(-J) '-( - '- -t CJ -r- 2 :::.. == 0 / 2 --o 0 c--2.).._~3(-2.){-l) ,2:::.o ~ rt-6 r 2. .,.. o -, l r- ;zC 1 D When we graphing an equation, it will be helpful to have more points than just the x and y intercepts of the graph. We can create a table of values with more choices for xand find the con-espondingyvalues. Example 3: Sketch tbe graph oftlie equation by plotting points:y~ -3x + 2 . X Y=-:3x+2. -2 -J (-2)-1- 2. Slide19 8 <-218) -( -J(-() -t-2 0 -3{0) -t- .2 .2 I -s (1)-r.2. -I { I_,, -I) -3(:2) t-2 -7 ,2 )( (~>:} ( - 1/s;) ( 0/.2) (~-~ Example 4: Sketch the graph ort:he equation by plotting points: y = x .- 3 Y~ )( -2 -2 - ) -/ 0 0 ( 2 Slide20 l .,. X-r.J (x Y (-;,, u -t.J -f3 2 {-1~ 2) -r .:s ..J ( o/JJ 3 1 ( I' s (2js) 2. TJ ~