"Warm-Up 2.1: Polynomial, Power, & R a ronal I Functions!X~-11 1. Solve l + 6x = 7 (b~ )~ m ~le:ling thi ij, re ~l't~x-t:2 t "l-tlo~ t, ~ 110I-t 9 'A"'3,"iL!.tED ~-1.-y---y-= - m-(x - -:- x)--- Mat~h~:e~2criPtion~o the correction equation. 2. Ax + By + C= 0 2. L.y a) General Form 3. tj.x b) Point-Slope Form 4. ax + bx + c = 0 2 cl Slope Intercept Form '5 S. Y =mx + b, (b)2 dl Quadratic Equation Lj 6. x' + bx +G) ~ c + ;; e) Slope 3~_ **Polynomial Fu~n~c~ti:o:ns~=-~;:::~~Fu~~~ l,... P I omial Function DEF INITION 0 yn be real numbers Letln be a nonnegative.integ.erl and let ao. at> a2,..., lln - 1> a,.. h -"- 0 The function given by WIt an -r-. 2 + a f(x)=a~ n +an -l,r't - 1 +... + a~ + alx. 0. mial f1ll1ction of degree n. The leadin~a coefficient IS a,.. IS The a polyno zero function. f(x) _. - 0 I mial function. It has no degree and no Iead- IS a po yno. ing coefficient. IdentifyIng Polynomial Functions. EXAMPLE 1.? F r those that are polynomial fimc. 1 mial fimctlolls. o. h t Which of the foll owmg are po yno " F those that are not, explam w y no.. tate the degree an d I ea d illt;> 0 coefficient. or. ' tlons, S. 1 (b) g (x) = 6x-4 + 7 (a) lex) = 4x 3-5x - -i (c) hex) = Y9x 4 + 16x 2 (d) k (x) = 15x - 2x4 d) \bi~nom\q I, oog(\qq 3. leudl1'l9 b) '(\ct Q coeff\c\ef'\l po\~nom \Q \ becqusq Of lhe exponent c) nat 'J 1 Q po\~n6miu \ I :JX -t ix. d) r()\~n~ml all lot fn Citnt -2.
"""Linear Functions & Their Graphs Linear Function: 5(x) :: ax 1b Degree: ~\8htst t'x\a>oen1 af a Pol ~DOmIQl. Slant Line: \jud Dn\\j be Q \in-ea( functloo \t \1 ~ \) noi Q noyllonta\ O( 'Jert 'ICQ~ \ln~. Example 2: Write an equation for the linear functionfsuch thatf(-l) = 2 andf(3) = -2. j(-i) -=2 5(3)= -2 \l-~:rn()(-x) (_ \ I 2 ) t3 I 2 ) 'i -1 ~ - \ (X ~ \) 0 ' -2:: - l\ - y- y - 2.' 2 ~_ ~ ~ _ I r'i ffi ':. -.~ A; I Y ~ -\-til 'X -.x ~ \ Example 3: Write an equation forthe linearfunctionfsuch thatf(-3) = 5 andf(6) = -2. J (-3) ~ 5 5 ( 10 ) -:. - Z "i -~ -=- m (x " f-3,5) l~/-2) 'i t2 -= ~ (\-,lo) t 2=-]. l<. t ~ - '(-Y _ -2-5 -L f1 \J m X-X - p bt 3 - q J 9 3 """Average Rate of Change (a.k.a "slope) THEOREM Constant Rate of Change p, 'i ~ ~ Xt ~ \ A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph. ) The amount of change divided by the time it takes. Example 4: Find the average rate of change of f(x) :: x 3 - x over the interval [1,3]. j(b)~ ~/-3:-14 b-q 5(,,) :: \~- I: 0 b-:.;3 q :. I q j'tb
... Linear Correlation & Modeling Correlation Coefficient, r: A number that measure the strength and direction of the linear correlation of Example 5: a data set. Properties of the Correlation Coefficient, r 1. -1 ::; r ::; l. 2. When r > 0, there is a positive linear correlatior. 3. When r < 0, there is a negative linear correlation. 4. When Irl = 1, there is a strong linear correlation. S. When r = 0, there is weak or no linear correlation. Age (years) Ufe Expectancy (years) 10 67.4 20 57.7 30 48.2 40 38.8 50 29.8 60 21.5 70 14.3 80 8.6 50Y~" 5OY1c2 Q 40 40 : 40 50 30 ~O 30 20 20 20 " LO :. x 10.< 10, 10 20 30 40 50 10 20 :>0 40 50 10 2Jl 30 40 50 Strong p03ilivc Ilne3f Weak positive. linear Linle or no lincaf oorrelatk>o c.orrelatioo correlarioo (aj (cj y y 5O~.. Q 5O.'. 40 40.. ~... m 20 20 10 ",10 ", lo 20 30 40 50 lo 2!) ~ 40 50 StrOllS oega[ lvt': IUle.ar Weak oegallve hnear v;;j-ill (..()Ill'; LauOfl correlauoo Y I \KJ \)\iji\os11\oh _:;;l (a) How to view the correlation on your calc. TER -4~ ~ (b) Draw a scatter plot EN (c) Describe the strength and direction of the correlation. l) 31l0n9 (\e~ Q1\ ve COfYe\ Ql\O\'\
**Quadratic Functions & Their Graphs Quadratic Function: A polynomial function of degree 2. 2 Example 6: Describe how to transform the graph of f(x) =x into the graph of the given function. Sketch its graph by hand. (a) g(x) = -(1/2)i + 3 X x 2 Y x (1/2)x 2 y Q - -(1/2)i x y 0 0 ~ -, 5 1-1. 3.. ~ -(1/2)x 2 + 3 x y 0 0 Q 0 ~ I \ 5 i 1 5 Z 'i '2 2. z I 3 q 3 Ii S 3 -i.5 " J t j f/p3 L" -1 \1 P CiC;(CSS (til ~'~ <.ut ih liz verilcai S~rm~ Standard Quadratic Form (ax 2 + bx + c) -4 Vertex Form x- QXl) Vertex Form of a Quadratic Function Any quadratic function I(x) = ax 2 + bx + c, a '" 0, can be written in the vertex form If(x) = a(x - h)2 + k. \ The graph of I is a parabola with vertex (h, k) and axis x = h. where h = -b/(2a) and k = C - ah 2. If a> O. the parabola opens upward, and if a < 0, it opens downward. (See Figure 2.6.) CPf\)Si10 S\<j(\ Vertex: Axis of Symmetry: b - - 10 Example 7: Find the vertex and axis of the graph of the function. (a) f(x) = 3(x - 1)2 + 5 h (b) g(x) = -3(x + 2f-l (-2,-1 ) h~-2 I I
Example 8: Find the vertex & axis of the graph of the function. Rewrite the equation for the function in vertex form. (b) -2i-7x-4 (. ~ - 'J Example 9: Use completing the square to describe the graph of fix) =x 2-4x + 6. 2. 2 X2 -L\x Tl1l - -~ 1 (1 L 4 i I - ~ X 2 - IX,. i - L.. ( X-2) 2 -:; - 2 (X-2.)2 12 ~rqbo\q opens \.\p vertex ( 2,Z) Example 10: Write an equation for the parabola shown. using the fact that one of the given points is the vertex. h -~ - \ 5 ~ Q ( \ + \) 2. -- 3 (- 1,-3) K ~ - 3 5 ~ LtQ - 3 '" ~ \ ~ ~ ~ q ~ -= ~ ~~2J 5(x ):: 2 (X t \) - 3
**Applications of Quadratic Functions A square of side x inches is cut out of each comer of an 5 in. by 7 in. piece of cardboard, and the sides are folded up to form an open-topped box. Use your graphing calculator to determine the dimensions of the cut-out squares that will produce the box of maximum volume. (Hint: you will first need to write the volume of the box Vas a function of x.) x ( 5-2X ) ( I -2)1 ) mox w X';,,9~ Ve rtical Free -F all Motion The height s and vertical velocity v of an object in free fall are gi~n by 1 s(t) = -2gt2 + vot + So and v(t) = -gt + vo, where t is time (in seconds), g "'" 32 ft/sec 2 "'" 9.8 m/sec 2 is the acceleration due to gravity. Vo is the initial vertical velocity of the object, and So is its initial height. Free-Fall Motion As a promotion for the Houston Astros downtown ballpark, a competition is held to see who can throw a baseball the highest from the front row of the upper deck of seats, 83 ft above field level. The winner throws the ball with an initial vel1ical velocity of 92 ftlsec and it lands on the infield grass. (a) Find the maximum height of the base ball. (b) How much time is the ball in the air? (c) Detenninc its vcl1ical velocity when it hits the ground. ------ -t (32)t~ t q2t -+ 83 go= -\It;t? tq2t t 83 ~3\ v(t) :; - 32 1:. ; q2 q)~ :. -92 : 2.<61S z~ 2 (-.110) S(ZS15)" 115 **Homework Assignment: Quick Review, 1, 21 7, 10, 13, 16, 20, 24, 27, 32, 34, 37, 40, 43, 48, 50, 54, 57,58,61,66,71,73,74,77,78
b) j= 0 OJ - 1 q3 ~ I b'5 flhd 2e.ms US1h3 ~ 5.s.tcond S Colc or iuad fuhdid1. c) V! f:) :::.-3 2-1: t 92-32 ( ~,5') t q2 t tfid i ImE' -ihe bail hlis, fhe JrovlYJ. ~ - ~ \Il ~t (Stc.