y = f(x + 4) a) Example: A repeating X by using two linear equations y = ±x. b) Example: y = f(x - 3). The translation is

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1 Answers Chapter Function Transformations. Horizontal and Vertical Translations, pages to. a h, k h, k - c h -, k d h 7, k - e h -, k. a A (-,, B (-,, C (-,, D (,, E (, A (-, -, B (-,, C (,, D (, -, E (, - h( f( - g( f( c A (-, -, B (-7,, C (-,, D (-, -, E (-, - s( f( d A (-, -, B (-, -, C (-, -, D (, -, E (, t( f( -. a (, ( -, (, (, - c (, ( + 7, + d (, ( +, +. a a vertical translation r( f( + - of units down and a horizontal translation of - units left; (, ( -, - a vertical translation s( f( - - of units down and a - horizontal translation of units right; - (, ( +, - c d t( f( v( f( a vertical translation of units up and a horizontal translation of units right; (, ( +, + a vertical translation of units up and a horizontal translation of units left; (, ( -, +. a h -, k ; - f( + h, k ; - f( - c h, k -; + f( - d h -7, k -; + f( + 7. It has been translated units up. 7. It has been translated unit right.. Transformed Transformation of Translation Function Points vertical f( + (, (, + horizontal f( + 7 (, ( - 7, horizontal f( - (, ( +, vertical f( - (, (, - horizontal and vertical + 9 f( + (, ( -, - 9 horizontal and vertical f( - - (, ( +, - horizontal and vertical f( + + (, ( -, + horizontal and vertical f( - h + k (, ( + h, + k 9. a ( + + { R}, {, R} c To determine the image function s domain and, add the horizontal and vertical translations to the domain and of the base function. Since the domain is the set of real numbers, nothing changes, but the does change.. a g( The new graph is a vertical and horizontal translation of the original b units up and 9 units right. c Eample: (,, (,, (, (9,, (,, (, 7 d Eample: (,, (,, (, (9,, (,, (, 7 e The coordinates of the image points from parts c and d are the same. The order that the translations are made does not matter.. a f( - + f( -. a Eample: It takes her h to ccle to the lake, km awa. She rests at the lake for h and then returns home in h. This translation shows what would happen if she left the house at a later time. c f( -. a Eample: Translated units right. Eample: f( -, f( - +., f( a Eample: A repeating X b using two linear equations ±. Eample: f( -. The translation is horizontal b units right.. a The transformed function starts with a higher number of trout in 97. f(t + The transformed function starts in 97 instead of 97. f(t -. The first case, n f(a +, represents the number of gallons he needs for a given area plus more gallons. The second case, n f(a +, represents how man gallons he needs to cover an area A less units of area. 7. a ( - 7( - or ( Horizontal translation of units right and vertical translation of 9 units down. c -intercept 7 MHR Answers

2 . a The original function is units lower. The original function is units to the right. c The original function is units lower and units left. d The original function is units higher and units right. 9. a The new graph will be translated units right and units down ( - -( - - C a f( f( - h f( - h + k. Looking at the problem in small steps, it is eas to see that it does not matter which wa the translations are done since the do not affect the other translation. The domain is shifted b h and the is shifted b k. C a f( ( + ; horizontal translation of unit left g( ( - - ; horizontal translation of units right and unit down C The roots are and 9. C The can be taken as h or k in this problem. If it is h then it is -, which makes it in the left direction.. Reflections and Stretches, pages to. a f( + g( -f( h( f( g( -f( h( f(- - - f( + - c The -coordinates of g( have changed sign. The invariant point is (-.,. The -coordinates of h( have changed sign. The invariant point is (,. d The graph of g( is the reflection of the graph of f( in the -ais, while the graph of h( is the reflection of the graph of f( in the -ais.. a f( g( f( h( f( f( g( f( h( f( - - c The -coordinates of g( are three times larger. The invariant point is (,. The -coordinates of h( are three times smaller. The invariant point is (,. d The graph of g( is a vertical stretch b a factor of of the graph of f(, while the graph of h( is a vertical stretch b a factor of of the graph of f(.. a g( - f(: domain { R}, f( { R} g(: domain { R}, { R} - g( - c. a c - - g( + h( - - h( - - k( - f( g( - g( + h( + h( - - k( - h( - - g(: domain { R}, {, R} h(: domain { R}, { -, R} k( - h(: domain {, R}, {, R} k(: domain {, R}, {, R} g( - f(: domain { R}, { R} g(: domain { R}, { R} h( + g(: domain { R}, {, R} h(: domain { R}, {, R} k( - h(: domain {, R}, {, R} k(: domain {, R}, {, R} Answers MHR

3 . a The graph of f( is a vertical stretch b a factor of of the graph of f(. (, (, The graph of f( is a horizontal stretch b a factor of of the graph of f(. (, ( _, c The graph of -f( is a reflection in the -ais of the graph of f(. (, (, - d The graph of f(- is a reflection in the -ais of the graph of f(. (, (-,. a domain { -, R}, { -, R} The vertical stretch affects the b increasing it b the stretch factor of. 7. a The graph of g( is a vertical stretch b a factor of of the graph of f(. f( The graph of g( is a reflection in the -ais of the graph of f(. -f( c The graph of g( is a horizontal stretch b a factor of of the graph of f(. f( d The graph of g( is a reflection in the -ais of the graph of f(. f(-. f( a horizontall stretched b a factor of horizontall stretched b a factor of c verticall stretched b a factor of d verticall stretched b a factor of e horizontall stretched b a factor of and reflected in the -ais f verticall stretched b a factor of and reflected in the -ais. a c The are both incorrect. It does not matter in which order ou proceed.. a d Both the functions - t are reflections of - d -.t the base function in the t-ais. The d -.9t - object falling on Earth is stretched verticall more than - the object falling on the moon.. Eample: When the graph of f( is transformed to the graph of f(, it undergoes a horizontal stretch about the -ais b a factor of and onl the b -coordinates are affected. When the graph of f( is transformed to the graph of af(, it undergoes a vertical stretch about the -ais b a factor of a and onl the -coordinates are affected.. a D As the drag factor decreases, the length of the skid mark increases for the same speed. D S D S D S D S 7 D S D S. D S 7. S. a -,, - c -, d -,.. a I III c IV d IV. a f( f( - g( g( C Eample: When the input values for g( are b times the input values for f(, the scale factor must be b for the same output values. g( f ( b ( f( C Eamples: a a vertical stretch or a reflection in the -ais a horizontal stretch or a reflection in the -ais C f( g( Transformation (, (, - reflection in the -ais (, (-, reflection in the -ais (, (, vertical stretch b a factor of (, - (, - horizontal stretch b a factor of and vertical stretch b a factor of MHR Answers

4 C - f ( f( f( C a t n n - t n -n + c The are reflections of each other in the -ais.. Combining Transformations, pages to. a -f( or - f(- or. The function f( is transformed to the function g( b a horizontal stretch about the -ais b a factor of. It is verticall stretched about the -ais b a factor of. It is reflected in the -ais, and then translated units right and units down.. Function Reflections Vertical Stretch Factor Horizontal Stretch Factor Vertical Translation Horizontal Translation - f ( - none none none + f( none f ( ( - none + -f(( + -ais - none none - -. a f(-( + - f(( + -. a f(( f( f( f( 7. a vertical stretch b a factor of and translation of units right and units up; (, ( +, + horizontal stretch b a factor of, reflection in the -ais, and translation of units down; (, (, - - c reflection in the -ais, reflection in the -ais, vertical stretch b a factor of, and translation of units left; (, (- -, - d horizontal stretch b a factor of, reflection in the -ais, and translation of units right and units up; (, ( +, - + e reflection in the -ais, horizontal stretch b a factor of _, reflection in the -ais, and vertical stretch b a factor of _ ; (, _ (-, - _ f reflection in the -ais, horizontal stretch b a factor of, vertical stretch b a factor of, and translation of units right and units up; (, ( - +, +. a + -f( _ f(-( - 9. a f( - - c f(- -f(( - + _ f ( f( _ f( _ f( ( - -. a (-, (-, 7 c (-, - d (9, - e (-, -9 d f ( Answers MHR 7

5 e f f( f ( - ( + -. a -f( - + -f( - +. a c - c f(-( g( -f(( g( -f( g( - f(-( a A (-, -, B (-7,, C (-,, D (-,, E (, - -f ( ( + +. a The graphs are in two locations because the transformations performed to obtain Graph do not match those in - +. Gil forgot to factor out the coefficient of the -term,, from -. The horizontal translation should have been units right, not units. He should have rewritten the function as ( a ( ( ( ( + +. a (-a,, (,- (a,, (, c and d There is not enough information to determine the locations of the new intercepts. When a transformation involves translations, the locations of the new intercepts will var with different base functions.. a A - + A - c For (,, the area of the rectangle in part a is square units. A - + A -( + ( A + For (,, the area of the rectangle in part is square units. A - + A - ( + ( A 7. ( - + ( - -. Eample: vertical stretches and horizontal stretches followed b reflections C Step The are reflections in the aes. : +, : - -, : - Step The are vertical translations coupled with reflections. : +, : -, : -, : - - C a The cost of making b + bracelets, and it is a horizontal translation. The cost of making b bracelets plus more dollars, and it is a vertical translation. c Triple the cost of making b bracelets, and it is a vertical stretch. d The cost of making b_ bracelets, and it is a horizontal stretch. C ( - + ; a vertical stretch b a factor of and a translation of units right and unit up C a H is repeated; J is transposed; K is repeated and transposed H is in retrograde; J is inverted; K is in retrograde and inverted c H is inverted, repeated, and transposed; J is in retrograde inversion and repeated; K is in retrograde and transposed. Inverse of a Relation, pages to. a f( - - f( f( MHR Answers - f(

6 . a f( a The graph is a function but the inverse will be a relation. The graph and its inverse are functions. c The graph and its inverse are relations.. Eamples: a {, R} or {, R} { -, R} or { -, R} c {, R} or {, R} d { -, R} or { -, R}. a f - ( 7 f - ( - ( - c f - ( - d f - ( + e f - ( - ( - f f - ( -. a E C c B d A e D 7. a. a function: domain {-, -, -,, }, {,,, } inverse: domain {,,, }, {-, -, -,, } f( The inverse is a function; it passes the vertical line test. The inverse f( is not a function; it does not pass the vertical line test. - - c f( The inverse is not a function; it does not pass the vertical line test. - - function: domain {-, -,,, }, {-,,, 7, } inverse: domain {-,,, 7, }, {-, -,,, } Answers MHR 9

7 9. a f - ( ( - f( f - ( ( - f(: domain { R}, { R} f - (: domain { R}, { R}. a i f( ( + -, inverse of f( ± + - ii f( f - ( (- + f - ( ( c f - ( + f - ( + f( - f(: domain { R}, { R} f - (: domain { R}, { R} f( - f(: domain { R}, { R} f - (: domain { R}, { R} d f - ( - - f( +, f - ( e f - ( - f - ( - - f( -, f(: domain {, R}, {, R} f - (: domain {, R}, {, R} f(: domain {, R}, {, R} f - (: domain {, R}, {, R} inverse of f( - - i ( - -, ± + + ii f( inverse of f( Yes, the graphs are reflections of each other in the line.. a ± - restricted domain {, R} f( + ± _ ± restricted domain {, R} f( ± - - f( +, f - ( - f(, f - ( MHR Answers

8 c ± _ - restricted domain {, R} ± - f - ( f( f( -, - d ± - restricted domain { -, R} - - f( ( + - ± - _ e ± - + restricted domain {, R} ± f( -( - f ± f( ( - - ± f( ( +, - f - ( - f - ( - + f( -( -, - - restricted domain {, R} f - ( f( ( - -,. a inverses inverses c not inverses d inverses e not inverses. Eamples: a or or c or d - or -. a _ c _ d. a approimatel. C + ; represents temperatures in degrees Celsius and represents temperatures in degrees Fahrenheit 9_ c 9. F d C F 9_ C F - C _ (F The temperature is the same in both scales (- C - F. 7. a male height 7. cm, female height. cm i male femur.7 cm ii female femur 9. cm. a. +.; is finger circumference and is ring size c. mm,. mm, 9. mm 9. Eamples: a i ii - f - ( f(, - f(, - f - - ( i ii - - f - ( f(, f(, - f - (. a 7 c. a (, (, c (-, -9 C a Subtract and divide b. Add, take the positive and negative square root, subtract. C a f( - + f - ( Eample: The graph of the original linear function is perpendicular to, thus after a reflection the graph of the inverse is the same. c The are perpendicular to the line. Answers MHR

9 C Eample: If the original function passes the vertical line test, then it is a function. If the original function passes the horizontal line test, then the inverse is a function. C Step f(: (,, (,, (-, -, and (a, a + ; g(: (,, (,, (-, -, and ( a +, a The output values for g( are the same as the input values for f(. Eample: Since the functions are inverses of each other, giving one of them a value and then taking the inverse will alwas return the initial value. A good wa to determine if functions are inverses is to see if this effect takes place. Step The order in which ou appl the functions does not change the final result. Step The statement is saing that if ou have a function that when given a outputs b and another that when given b outputs a, then the functions are inverses of each other. 7. a f( g( f( h( f( - - If the coefficient is greater than, then the function moves closer to the -ais. The opposite is true for when the coefficient is between and.. a In this case, it could be either. It could be a vertical stretch b a factor of or a horizontal stretch b a factor of. Eample: g( f( 9. a Chapter Review, pages to 7. a c f( h( f( + f( f ( - f( -. The are both horizontal stretches b a factor of. The difference is in the horizontal translation, the first being unit left and the second being unit left.. g( f(( - -. a f(-( + f(. Translation of units left and units down: + +. { 9, R}. No, it should be (a +, b -.. a -ais, (, - -ais, (-,. a f( f( c f( + + f(-: domain { R}, {, R} (, -f(: domain -f( { -, R}, {, R} (,, (-, f( -f(( f( - f( MHR Answers

10 . a - - f(, (-, - c f(: domain { R}, { R} f(: domain { R}, { R} d f(. a Using the horizontal line test, if a horizontal line passes through the function more than once the inverse _ is not a function. ± c Eample: restricted domain { -, R} ; f ( ( - f( Chapter Radical Functions. f( f - ( a The relation and its The relation is a function. inverse are functions. The inverse is not a function.. - +, restricted domain {, R} 7. a not inverses inverses Chapter Practice Test, pages to 9. D. D. B. B. B. C 7. C. domain { -, R} 9. - f ( ( + +. a To transform it point b point, switch the position of the - and the -coordinate. c (-, -. ( -. f (- ( -. a It is a translation of units left and 7 units down. g( c (-, -7 d No. Invariant points are points that remain unchanged after a transformation.. a f( g( f(; a vertical stretch b a factor of c g( f ( ; a horizontal stretch b a factor of. Radical Functions and Transformations, pages 7 to 77. a c d - (, (, (, (, domain {, R}, {, R} - (-, (-, (, + (-, - - domain { -, R}, {, R} (-, - (-, (, (, domain {, R}, {, R} - - (-7, (- 9 _, (-, (- _, domain { - _, R }, {, R}. a a 7 vertical stretch b a factor of 7 h 9 horizontal translation 9 units right domain { 9, R}, {, R} b - reflected in -ais k vertical translation up units domain {, R}, {, R} c a - reflected in -ais b horizontal stretch factor of domain {, R}, {, R} vertical stretch factor of d a h - horizontal translation units left k - vertical translation units down domain { -, R}, { -, R}. a B A c D d C. a + _ - Answers MHR

11 c -( - + or _ d -.. or - _. a domain f( - - {, R}, { -, R} - r( + domain { -, R}, {, R} d - All graphs are the same. c d e - - p( ( - +. a a m( + domain {, R}, {, R} domain {, R}, {, R} domain {, R}, {, R} f domain { -, R} { -, R} - -( + - vertical stretch factor of b horizontal stretch factor of _, c a _ vertical stretch factor of _ b _ horizontal stretch factor of 7. a r(a _ A_ π A r.... r r(a A_ π A. a b. horizontal stretch factor of or _. d. h Eample: I prefer the original function because the values are eact. c approimatel. miles 9. a domain {, R}, { -, R} h no horizontal translation k vertical translation down units. a c -( - - or d - -( - + or Eamples: a - - or c d - _ -( + +. a a 7 vertical stretch factor of 7 k vertical translation up Y c domain {n n, n R} {Y Y, Y R} Y(n 7 n + n d The minimum ield is kg/hectare. Eample: The domain and impl that the more nitrogen added, the greater the ield without end. This is not realistic.. a domain {d - d, d R} {P P, P R} The domain is negative indicating das remaining, and the maimum value of P is million. a - reflected in d-ais, vertical stretch factor of ; b - reflected in P-ais; k vertical translation up units. MHR Answers

12 c P(d - -d P d Since d is negative, then d represents the number of das remaining before release and the function has a maimum of million pre-orders. d 9. million or 9 9 pre-orders.. a Polling errors reduce as the election approaches. _.9 - There are no translations since the graph starts on the origin. The graph is reflected in the -ais then b -. Develop the equation b using the point (-, and substituting in the equation a, solving for a, then a.9. c a.9 vertical stretch factor of.9 b - reflected in the -ais Eamples a _ a China, India, and USA (The larger the countr the more unfair the one nation - one vote sstem becomes. Tuvalu, Nauru, Vatican Cit (The smaller the nation the more unfair the one person - one vote sstem becomes. Nation Percentage d Nation Percentage China.% China.% India 7.% India.% US.% US.% Canada.% Canada.77% Tuvalu. % Tuvalu.% Nauru. 7% Nauru.% Vatican Cit. % Vatican Cit.% c V( e The Penrose sstem gives larger nations votes based on population but also provides an opportunit for smaller nations to provide influence.. Answers will var. 9. a The positive f - (, domain of the inverse is the same as the of the original f( function. i g - ( +, ii h - ( -( -, iii j - ( ( + + 7_, -. Vertical stretch b a factor of _. Horizontal stretch b a factor of 7_. Reflect in both the and aes. 7 Horizontal translation of units left. Vertical translation of units down. C The parameters b and h affect the domain. For eample, has domain but ( - has domain. The parameters a and k affect the. For eample, has but - has -. _ C Yes. For eample, 9 can be simplified to. C The processes are similar because the parameters a, b, h, and k have the same effect on radical functions and quadratic functions. The processes are different because the base functions are different: one is the shape of a parabola and the other is the shape of half of a parabola. C Step ; Step Step Triangle Number, n Length of Hpotenuse, L First. Second.7 Third Step L n + Yes, the equation involves a horizontal translation of unit left.. Square Root of a Function, pages to 9. f( f( undefined... a (,. (-,. c does not eist d (.9, e (-, f (m, n. a C D c A d B. a When - < then - is undefined; when < - < then - > - ; when - > then - > - ; - - when and c The function f( - is undefined when - <, therefore the domain is {, R} whereas the function f( - has a domain of { R}. Since f( is undefined when f( <, the of f( is {f( f(, f( R}, whereas the of f( - is {f( f( R}.. a For -, - domain { R}, { R}; for -, domain - {, R}, {, R}. The domains differ since - is undefined when <. The of - is, when -. Answers MHR

13 c d - For +, + domain { R}, _ { R}. For +, domain + { -, R}, { _, R}. + is undefined when + <, therefore - and For - + 9, domain {_ R}, { R}; for - + 9, domain _ { 9, R}, {, R} is undefined when <, therefore 9 and _ 7. a Since - is undefined when - <, the domain changes from { R} to { - and, R} and the changes from { -, R} to {, R}. Since + is undefined when + <, the changes from {, R} to {, R}. _ c Since - is undefined when - <, the domain changes from { R} to { -, R} and the changes _ from {, R} to {, R} or {, R}. _ d Since + is undefined when + <, the changes _ from {, R} to {, R} or {, R}.. a f( c - f( - - f( f( f( f( For -. -, domain { R}, { R}; for -. -, domain { -, R}, {, R} is undefined when -. - <, therefore - and.. a For - 9, domain { R}, { -9, R}. For - 9, domain { - and, R}, {, R}. - 9 is undefined when - 9 <, therefore - and and. For -, domain { R}, {, R}. For -, domain { -, R}, {, R}. - is undefined when - <, therefore and - and. c For +, domain { R}, {, R}. For +, domain { R}, {, R}. + is undefined when + <, therefore R and. d For. +, domain { R}, { _, R}. For. +, domain { _ R}, {, R}.. + is undefined when. + <, therefore R and a and i For +, domain { R}, {, R} ii For -, domain { R}, { -, R} iii For - +, domain { R}, {, R} iv For - -, domain { R}, { -, R}. MHR Answers

14 c The graph of j( does not eist because all of the points on the graph j( are below the -ais. Since all values of j( <, then j( is undefined and produces no graph in the real number sstem. d The domains of the square root of a function are the same as the domains of the function when the value of the function. The domains of the square root of a function do not eist when the value of the function <. The s of the square root of a function are the square root of the of the original function, ecept when the value of the function < then the is undefined.. a For -, domain { R}, { -, R}; for -, domain { - and, R}, {, R}. The value of in the interval (-, is negative therefore the domain of - is undefined and has no values in the interval (-,.. a I sketched the graph f( b locating ke points, including invariant points, and determining the f( image points on the graph of the square root of the function. - For f(, domain { R}, { -, R}; for f(, domain { -. and., R}, {, R} The domain of f(, consists of all values in the domain of f( for which f(, and the of f(, consists of the square roots of all values in the of f( for which f( is defined.. a d h + 7h domain {h h, h R}, {d d, d R} c Find the point of intersection between the graph of the function and h. The distance will be epressed as the d value of the ordered pair (h, d. In this case, d is approimatel equal to 9. d Yes, if h could be an real number then the domain is {h h - 7 or h, h R} and the would remain the same- since all square root values must be greater than or equal to.. a No, since a, a < is undefined, then f( will be undefined when f( <, but f( represents values of the not the domain as Chris stated. If the consists of negative values, then ou know that the graph represents f( and not f(. _. a v. - h domain {h h., h R}, {v v., v R} since both h and v represent distances. c approimatel. m. Step Step The parameter a determines the minimum value of the domain (-a and the maimum value of the domain (a; therefore the domain is { -a a, R}. The parameter a also determines the maimum value of the, where the minimum value of the is ; therefore the is { a, _ R}. Step Eample: - the reflection _ of the graph in the -ais is the equation - -. The graph forms a - circle a (-7, (-, - c (, - 7. a f( c d f( f( - f( f( -f( + f(- - f( Answers MHR 7

15 . Eample: Sketch the graph in the following order: f( Stretch verticall b a factor of. f( - Translate horizontall units right. _ f( - Plot invariant points and sketch a smooth curve above the -ais. - _ f( - Reflect _ f( - in the -ais. 9. a r A_ r A _ π π( + 7 C Eample: Choose to ke points on the graph of f(. Transform the points (, (,. Plot the new points and smooth out the graph. If ou cannot get an idea of the general shape of the graph, choose more points to graph. C The graph of - is a linear function spanning from quadrant II to quadrant IV with an -intercept of and a -intercept of. The graph of - onl eists when the graph of _ - is on or above the -ais. The -intercept is at while the -intercept stas the same. -values for are the same for both functions and the -values for - are the square root of values for -. C No, it is not possible, because the graph of f( ma eist when < but the graph of f( does not eist when <. C a ( ( - - The graph of ( - - is a quadratic function with a verte of (, -, -intercept of -, and -intercepts of - and. It is above the -ais when > and < -. The graph of _ ( - - has the same -intercepts but no -intercept. The graph onl eists when > and < -.. Solving Radical Equations Graphicall, pages 9 to 9. a B A c D d C. a 9 (9, c The roots of the equation are the same as the -intercept on the graph.. a c ±.79 d no solution. a.. a, 9_, c -.9, -. d -9.,.. a 7_, -, -, -_ or _ _ c., -, -_ d,, - _ 7. a -.7, no real roots, 7 c, _ or -_ d, or -. a a. a -. MHR Answers

16 c no solution d a -., a. 9. a _ Left Side + _ Right Side Left Side Right Side Since, there is no solution. Yes, if ou isolate the radical epression like + -, if the radical is equal to a negative value then _ there is no solution.. Greg N(t. t million, Yolanda N(t. t million Greg is correct, it will take more than ears for the entire population to be affected.. approimatel 99 cm. a Yes kg. No, 9 has two possible solutions ±9, whereas ( 9 has onl one solution a m/s 7. kg. c - or - c + c -. (-, (, If the function -( + c + c passes through the point (.,.7, what is the value of c? 7. Lengths of sides are. cm, cm, and. cm or.7 cm, cm, and. cm. C The -intercepts of the graph of a function are the solutions to the corresponding equation. Eample: A graph of the function - - would show that the -intercept has a value of. The equation that corresponds to this function is - - and the solution _ to the equation is. C a s 9.d where s represents speed in metres per second and d represents depth in metres. _ s 9.d s (9. m/s ( m s m /s s. m/s c approimatel. m d Eample: I prefer the algebraic method because it is faster and I do not have to adjust window settings. C Radical equations onl have a solution in the real number sstem if the graph of the corresponding function has an -intercept. For eample, + has no real solutions because there is no -intercept. C Etraneous roots occur when solving equations algebraicall. Etraneous roots of a radical equation ma occur antime an epression is squared. For eample, has two possible solutions, ±. You can identif etraneous roots b graphing and b substituting into the original equation. Chapter Review, pages 99 to. a c (9, (, (, domain {, R} {, R} All values in the table lie on the smooth curve graph of. (-, (-, - (, (, domain {, R} {, R} All points in the table lie on the graph of -. (-., - (, (-., + 7 (-, - - domain { -., R} {, R} _ All points in the table lie on the graph of Use a b( - h + k to describe transformations. a a vertical stretch factor of h - horizontal translation left units; domain { -, R}; {, R} b - horizontal stretch factor of, then reflected on -ais: k - vertical translation of units down. domain {, R}; { -, R} Answers MHR 9

17 c a - reflect in -ais b horizontal stretch factor of h horizontal translation right units; domain {, R}, {, R}.. a _ +, domain {, R}, {, R} domain { -9, R}, {, R} c - _ ( - 7 -, domain { 7, R}, { -, R}. a domain (, {, R}, {, R} c (, - - (-, - - domain {, R}, { -, R} ( + + domain { -, R}, {, R}. The domain is affected b a horizontal translation of units right and b no reflection on the -ais. The domain will have values of greater than or equal to, due to a translation of the graph units right. The is affected b vertical translation of 9 units up and a reflection on the -ais. The will be less than or equal to 9, because the graph has been moved up 9 units and reflected on the -ais, therefore the is less than or equal to 9, instead of greater than or equal to 9.. a Given the general equation a b( - h + k to describe transformations, a indicates a vertical stretch b a factor of, k indicates a vertical translation up units. S S(t + t (, t Since the minimum value of the graph is, the minimum estimated sales will be units. c domain {t t, t R} The domain means that time is positive in this situation. {S(t S(t, S(t W}. The means that the minimum sales are units. d about 7 units 7. a _ ( c _ -( - -. a For -, domain { R}, { R}; for -, domain {, R}, {, R}. The domain changes because the square root function has restrictions. The changes because the function onl eists on or above the -ais. For -, domain { _ R}, { R}; for -, domain {, R}, {, R} The domain changes because the square root function has restrictions. The changes because the function onl eists on or above the -ais. c For +, domain { R}, { R}; for +, domain { - _, R }, {, R}. The domain changes because the square root function has restrictions. The changes because the function onl eists on or above the -ais. 9. a Plot invariant points at the intersection of the graph and lines f( and. Plot an points (, where the value of is a perfect square. Sketch f( a smooth curve through the invariant points and points satisfing (,. f( is positive when f( >, f( does not eist when f( <. f( > f( when < f( < and f( > f( when f( > c For f(: domain { R}, { R}; for f(, domain { -, R}, {, R}, since f( is undefined when f( <.. a - domain { R}, {, R} for - domain { -, R}, {, R}, since - > onl between - and then the domain of - is -. In the domain of - the maimum value of - is, so the maimum value of - is then the of the function - will be. 7 MHR Answers

18 + domain { R}, { _, R} for + _ domain { R}, {, R}. The domain does not change since the entire graph of + is above the -ais. The changes since the entire graph moves up units and the graph _ itself opens up, so the becomes. c - domain { R}, _ { -9, R} for - domain { or, R}, {, R}, since - < between and, then the domain is undefined in the interval (, and eists when or. The changes because the function onl eists above the _ -ais.. a h(d - d h(d -d h (,. a c -7. and 7... m. a. -7 (-, (, d c.7 domain {d - d, d R} {h h, h R} c In this situation, the values of h and d must be positive to epress a positive distance. Therefore the domain changes to {d d, d R}. Since the of the _ original function h(d - d is alwas positive then the does not change.. a f( f( f( f( c f( f( d - and 7. a Jaime found two possible answers which are determined b solving a quadratic equation. Carl found onl one intersection at (, or -intercept (, determined b possibl graphing. c Atid found an etraneous root of.. a m m Chapter Practice Test, pages to. B. A. A. C. D. B a c The root of the equation and the value of the -intercept are the same.. or 9. For 7 - domain { R}, { R}. Since 7 - is the square root of the -values for the function 7 -, then the domain and s of 7 - will differ. Since 7 - < when > 7, then the domain of 7 - will be { 7, R} and since 7 - indicates positive values onl, then the of 7 - is {, R}. Answers MHR 7

19 . The domain of f( is { R}, and the of f( is {, R}. The domain of f( is { -, R} and the of f( is {, R} , ( b ± b - ac a _ -(-9 ± (-9 - (( (. or. B checking,. is an etraneous root, therefore.... a Given the general equation a b( - h + k to describe transformations, b indicating a horizontal stretch b a factor of. To sketch _ the graph of S d, graph the function S d and appl a horizontal stretch of, ever point on the graph of S d will become ( d_, S. S S (9, S d d d 9 m The skid mark of the vehicle will be approimatel 9 m.. a Given the general equation a b( - h + k to describe transformations, a - reflection of the graph in the -ais, b horizontal stretch b a factor of, k vertical translation up units. c domain {, R}, {, R}. d The domain remains the same because there was no horizontal translation or reflection on the -ais. But since the graph was reflected on the -ais and moved up units and then the becomes. _ e The equation + _ can be rewritten as - +. Therefore _ the -intercept of the graph - _ + is the solution of the equation f( f( Step Plot invariant points at the intersection of f( and functions and. Step Plot points at ma value and _ perfect square value of f( Step Join all points with a smooth curve, remember that the graph of f( is above the original graph for the interval. Note that for the interval where f( <, the function f( is undefined and has no graph. _. a ( -( - domain {, R}, {, R} Domain: cannot be negative nor greater than half the diameter of the base, or. Range: cannot be negative nor greater than the height of the roof, or. c The height of the roof m from the centre is about. m. Chapter Polnomial Functions. Characteristics of Polnomial Functions, pages to 7. a No, this is a square root function. Yes, this is a polnomial function of degree. c No, this is an eponential function. d Yes, this is a polnomial function of degree. e No, this function has a variable with a negative eponent. f Yes, this is a polnomial function of degree. 7 MHR Answers

20 . a degree, linear, -, degree, quadratic, 9, c degree, quartic,, d degree, cubic, -, e degree, quintic, -, 9 f degree, constant,, -. a odd degree, positive leading coefficient, -intercepts, domain { R} and { R} odd degree, positive leading coefficient, -intercepts, domain { R} and { R} c even degree, negative leading coefficient, -intercepts, domain { R} and {.9, R} d even degree, negative leading coefficient, -intercepts, domain { R} and { -, R}. a degree with positive leading coefficient, parabola opens upward, maimum of -intercepts, -intercept of - degree with negative leading coefficient, etends from quadrant II to IV, maimum of -intercept, -intercept of c degree with negative leading coefficient, opens downward, maimum of -intercepts, -intercept of d degree with positive leading coefficient, etends from quadrant III to I, maimum of -intercepts, -intercept of e degree with negative leading coefficient, etends from quadrant II to IV, -intercept, -intercept of f degree with positive leading coefficient, opens upward, maimum of -intercepts, -intercept of. Eample: Jake is right as long as the leading coefficient a is a positive integer. The simplest eample would be a quadratic function with a, b, and n.. a degree The leading coefficient is and the constant is -. The constant represents the initial cost. c degree with a positive leading coefficient, opens upward, -intercepts, -intercept of - d The domain is {, R}, since it is impossible to have negative snowboard sales. e The positive -intercept is the breakeven point. f Let, then P(. 7. a cubic function The leading coefficient is - and the constant is. c d The domain is {d d, d R} because ou cannot give negative drug amounts and ou must have positive reaction times.. a For ring, the total number of heagons is given b f(. For rings, the total number of heagons is given b f( 7. For rings, the total number of heagons is given b f( heagons 9. a End behaviour: the curve etends up in quadrants I and II; domain {t t R}; {P P 7, P R}; the for the period {t t, t R} that the population model can be used is {P P 7, P R}. t-intercepts: none; P-intercept: people c people d ears. a From the graph, the height of a single bo must be greater than and cannot be between cm and cm. V( ( - ( -. The factored form clearl shows the three possible -intercepts.. a The graphs in each pair are the same. Let n represent a whole number, then n represents an even whole number. (- n (- n n n n n The graphs in each pair are reflections of each other in the -ais. Let n represent a whole number, then n + represents an odd whole number. (- n + (- n + n + (- n (- n + -( n n + - n + c For even whole numbers, the graph of the functions are unchanged. For odd whole numbers, the graph of the functions are reflected in the -ais.. a vertical stretch b a factor of and translation of units right and units up vertical stretch b a factor of and translation of units right and units up c. If there is onl one root, ( - a n, then the function will onl cross the -ais once in the case of an odd-degree function and it will onl touch the -ais once if it is an even-degree function. C Eample: Odd degree: At least one -intercept and up to a maimum of n -intercepts, where n is the degree of the function. No maimum or minimum points. Domain is { R} and is { R}. Even degree: From zero to a maimum of n -intercepts, where n is the degree of the function. Domain is { R} and the depends on the maimum or minimum value of the function. C a Eamples: i ii iii - iv - Answers MHR 7

21 Eample: Parts i and ii have positive leading coefficients, while parts iii and iv have negative leading coefficients. Parts i and iii are odd-degree functions, while parts ii and iv are even-degree functions. C Eample: The line and polnomial functions with odd degree greater than one and positive leading coefficient etend from quadrant III to quadrant I. Both have no maimums or minimums. Both have the same domain and. Odd degree polnomial functions have at least one -intercept. C Step Function Degree End Behaviour + etends from quadrant III to I - + etends from quadrant II to IV - opens upward opens downward - etends from quadrant III to I etends from quadrant II to IV + etends from quadrant III to I - - etends from quadrant II to IV - + opens upward opens downward + + opens upward etends from quadrant III to I - etends from quadrant III to I etends from quadrant II to IV ( + ( + etends from quadrant III to I Step The leading coefficient determines if it opens upward or downward; in the case of odd functions it determines if it is increasing or decreasing. Step Alwas have at least one minimum or maimum. Not all functions will have the same. Either opens upward or downward. Step. Alwas have the same domain and. Either etends from quadrant III to I or from quadrant II to IV. No maimum or minimum.. Remainder Theorem, pages to _ + -. a + - c ( - ( + d Multipling the statement in part c ields + -. _ a c ( + ( d Epanding the statement in part c ields a Q( + + Q( + + c Q(w w - w + d Q(m 9m + m + e Q(t t + t - t + 7 f Q( a Q( - + Q(m m + m + c Q( d Q(s s + 7s + e Q(h h - h f Q( a , - t _ - t - 7 t - -t - t - t t -, t c , - n _ + 7n - d n - + n + n +, n - n _ - n + e n n - + n + + n -, n f , -. a c - d -7 e - f 7. a 9 - c d -. a - c d and -. a + 9, it represents the rest of the width that cannot be simplified an more.. a n n - - and -.. a 9π + π + π, represents the area of the base π( + ( + c cm r cm and h. m -_, n 9. a -_, b - _. Divide using the binomial - _. 7. Eamples: a c C Eample: The process is the same. Long division of polnomials results in a restriction. C a ( - a is a factor of b + c + d. d + ac + a b C a c The remainder is the height of the cable at the given horizontal distance.. The Factor Theorem, pages to. a - + c - d - a. a Yes No c No d Yes e Yes f No. a No No c No d No e Yes f No. a ±, ±, ±, ± ±, ±, ±, ±, ±9, ± c ±, ±, ±, ±, ±, ±, ±, ± d ±, ±, ± e ±, ±, ±, ± f ±, ±, ±. a ( - ( - ( - ( - ( + ( + c (v - (v + (v + d ( + ( + ( - ( + e (k - (k - (k + (k + (k + 7 MHR Answers

22 . a ( + ( - ( - (t - (t + (t + c (h - (h + h - d e (q - (q + (q + q + 7. a k - k, -7 c k - d k. h, h -, and h - 9. l - and l +. - cm, + cm, and + cm. + and +. a - is a possible factor because it is the corresponding factor for. Since f(, - is a factor of the polnomial function. -ft sections would be weak b the same principle applied in part a.. +, +, and +. Snthetic division ields a remainder of a + b + c + d + e, which must equal as given. Therefore, - is a possible divisor.. m - 7_, n - _. a i ( - ( + + ii ( - ( iii ( + ( - + iv ( + ( - + +, - + c -, + + d ( + ( - + C Eample: Looking at the -intercepts of the graph, ou can determine at least one binomial factor, - or +. The factored form of the polnomial is ( - ( + ( +. C Eample: Using the integral zero theorem, ou have both ± and ± as possible integer values. The -intercepts of the graph of the corresponding function will also give the factors. C Eample: Start b using the integral zero theorem to check for a first possible integer value. Appl the factor theorem using the value found from the integral zero theorem. Use snthetic division to confirm that the remainder is and determine the remaining factor. Repeat the process until all factors are found.. Equations and Graphs of Polnomial Functions, pages 7 to. a -,, -,, c -,. a -, - c -, -. a ( + ( + ( -, roots are -, - and -( + ( - ( -, roots are -, and c -( + ( - ( -, roots are -, and. a i -, -, and ii positive for - < < - and >, negative for < - and - < < iii all three zeros are of multiplicit, the sign of the function changes i - and ii negative for all values of, -, iii both zeros are of multiplicit, the sign of the function does not change c i - and ii positive for < - and >, negative for - < < iii - (multiplicit and (multiplicit, at both the function changes sign but is flatter at d i - and ii negative for - < < and >, positive for < - iii - (multiplicit and (multiplicit, at - the function changes sign but not at. a B D c C d A. a a. vertical stretch b a factor of., b - horizontal stretch b a factor of and a reflection in the -ais, h translation of unit right, k translation of units up (-.(-.(-( - +, (, (, (, (, (, ( -, ( -, ( _ 9_, (, ( - _, ( - _, (, c. (-( - + (-, - ( _, - ( _, - ( _ (-, - (, - (, - ( _, 7_ a i -,, and 9 ii degree from quadrant III to I iii -,, and 9 each of multiplicit iv v positive for - < < and > 9, negative for < - and < < 9 i -9, and 9 ii degree opening upwards iii (multiplicit, -9 and 9 each of multiplicit iv v positive for < -9 and > 9, negative for -9 < < 9, c i -, -, and ii degree from quadrant III to I iii -, -, and each of multiplicit iv - v positive for - < < - and >, negative for < - and - < < d i -, -,, and ii degree opening downwards iii -, -,, and each of multiplicit iv - v positive for - < < - and < <, negative for < - and - < < and > Answers MHR 7

23 . a c d - h( - h( a f( (-, (, (, - - (, - f( + + c d h( (, (-, (, (, e f g( (-, (-, (, - (, - - g( ( ( + ( + f( k( - - f( - k( (-, (-, (, (, (, (, (, - h( (-, - - f( - - (, - - f( (,. a positive leading coefficient, -intercepts: - and, positive for - < < and >, negative for < -, ( + ( - negative leading coefficient, -intercepts: -, -, and, positive for < - and - < <, negative for - < < - and >, -( + ( + ( - c negative leading coefficient, -intercepts: -, -,, and, positive for - < < - and < <, negative for < - and - < < and >, -( + ( + ( - ( - d positive leading coefficient, -intercepts: -,, and, positive for < - and < < and >, negative for - < <, ( + ( - ( -, h, k - Horizontal stretch b a factor of, translation of units right, and translation of units down c domain { R}, { R}. m b m b m. ft. a ( + ( - ( + ( -. a a, b ( + ( c - ( + ( - - ( + ( ( + ( - -. cm b cm b cm. -7, -, and - 7. The side lengths of the two cubes are m and m.. a ( - - in. b in. c in. b in. 9.,,, and 7 or -7, -, -, and -. - ( - ( + ( -. roots: -.,, and ; ( +.( - ( -. a translation of units right c - ( - : and, ( - - ( - ( - ( - : and 7 MHR Answers

24 . When., or when the sphere is at a depth of approimatel. m. C Eample: It is easier to identif the roots. C Eample: A root of an equation is a solution of the equation. A zero of a function is a value of for which f(. An -intercept of a graph is the -coordinate of the point where a line or curve crosses or touches the -ais. The all represent the same thing. C Eample: If the multiplicit of a zero is, the function changes sign. If the multiplicit of a zero is even, the function does not change sign. The shape of a graph close to a zero of a (order n is similar to the shape of the graph of a function with degree equal to n of the form ( - a n. C Step Set A Set F a The graph of ( is stretched horizontall b a factor of b relative to the graph of. When b is negative, the graph is reflected in the -ais. When b is < b < or < b <, the graph of ( is stretched horizontall b a factor of b relative to the graph of. When b is negative, the graph is reflected in the -ais. Step a: vertical stretch; reflection in the -ais; b: horizontal stretch; reflection in the -ais Chapter Review, pages to Set B a The graph of + k is translated verticall k units compared to the graph of. The graph of ( - h is translated horizontall h units compared to the graph of. Step h: horizontal translation; k: vertical translation Step Set C Set D a The graph of a is stretched verticall b a factor of a relative to the graph of. When a is negative, the graph is reflected in the -ais. When a is < a < or < a <, the graph of a is stretched verticall b a factor of a relative to the graph of. When a is negative, the graph is reflected in the -ais. Step Set E. a No, this is a square root function. Yes, this is a polnomial function of degree. c Yes, this is a polnomial function of degree. d Yes, this is a polnomial function of degree.. a degree with positive leading coefficient, opens upward, maimum of -intercepts, -intercept of degree with negative leading coefficient, etends from quadrant II to quadrant IV, maimum of -intercepts, -intercept of c degree with positive leading coefficient, etends from quadrant III to quadrant I, -intercept, -intercept of - d degree with positive leading coefficient, opens upward, maimum of -intercepts, -intercept of - e degree with positive leading coefficient, etends from quadrant III to quadrant I, maimum of -intercepts, -intercept of. a quadratic function 99 ft c s d. s 7 _ 9 _. a 7, ,, , - c 9, , d, , -. a a Yes, P(. No, P(-. c Yes, P(-. d Yes, P(.. a ( - ( - ( + -( - ( + ( + c ( - ( - ( - ( + d ( + ( - ( - 9. a +, -, and + m b m b m Answers MHR 77

25 . k -. a -intercepts: -, -, and ; degree etending from quadrant III to quadrant I; -, -, and each of multiplicit ; -intercept of -; - - positive for - ( + ( - ( + - < < - and >, negative for < - and - < < -intercepts: - and ; degree etending from ( - ( + quadrant III to quadrant I; - - (multiplicit and (multiplicit ; - -intercept of -; positive for >, - negative for <, - c -intercepts: -,, and ; degree opening upwards; (multiplicit, - and each of multiplicit ; -intercept of ; positive for < - and >, negative for - < <, g( g( - d -intercepts: -,, and ; degree etending from quadrant II to quadrant IV; -,, and each of multiplicit ; -intercept of ; positive for < - and < <, negative for - < < and > - g( g( c (-( a ( + ( + -( + ( -. a Eamples: ( + ( + ( - and -( + ( + ( - ( + ( + ( -. a V l (l - cm b cm b cm Chapter Practice Test, pages to. C. B. D. B. C. a - and - and c -,, and d - and 7. a P( ( + ( + P( ( - ( - - c P( -( - d P( ( + ( - +. a B C c A 9. a V ( - ( - cm b cm b cm, vertical stretch b a factor of ; b, no horizontal stretch; h -, translation of units left; k -, translation of units down domain { R}, { R} c. a a ( Cumulative Review, Chapters, pages to 9. a -. a a vertical stretch b a factor of, b - horizontal stretch b and reflection in the -ais, h translation of unit right, k translation of units up Parameter Transformation Value Equation horizontal stretch/ reflection in -ais - (- vertical stretch/ reflection in -ais (- translation right (-( - translation up (-( f( MHR Answers

26 c d f( f(- - - f(-. + f( -. a translation of unit left and units down vertical stretch b a factor of, reflection in the -ais, and translation of units right c reflection in the -ais and translation of unit right and units up. a (9, (, - c (-, 9 and, -intercept: - -intercepts: - and, -intercept:. a Yes - ( c Eample: No; restrict domain of + to {, R}.. a -intercepts: - _ 7. g( ( + -. _ -( + -( _ 9. a g( _ 9 c 9 9 ( ( domain { -, R}, {, R} g(. a The -intercepts are invariant points for square roots of functions, since. f(: domain { R}, { -, R}; g(: domain { - or, R}, {, R}; The square root function has a restricted domain.. a No, substituting -.7 back into the equation does not satisf the equation. Onl one solution, -.. a f( The -intercept is. same. f( - - c The are the - _ a ; P( ; P(-. ±, ±, ±, ±; P(, P(- -, P( -, P(-, P(, P(- 9, P(, P(- 7. a ( + ( - ( - ( - ( + ( + c -( - ( +. a f( f( g( g( intercepts: -,, and ; -intercept: - -intercepts: - and, -intercept: 7. a ( + and ( -. m b 7. m b. m. (-( - Unit Test, pages to. D. C. D. A. D. C 7. A {, R}. g( a ( - (( - c The both have the same shape but one of them is shifted right further. Answers MHR 79

27 . a f( - - f( c ± + 9 ± d for part a: domain { R}, { -9, R}; for part : domain { -9, R}, { R}; for part c: domain { - or, R}, {, R}. Quadrant II: reflection in the -ais, f(-; quadrant III: reflection in the -ais and then the -ais, -f(-; quadrant IV: reflection in the -ais, -f(. a Mar should have subtracted from both sides in step. She also incorrectl squared the epression on the right side in step. The correct solution follows: + + Step : ( - ( + Step : Step : Step : ( - ( - Step : - or - Step : _ Step 7: A check determines that is the solution. Yes, the point of intersection of the two graphs will ield the possible solution,.. c -; P( ( + ( + ( - 7. a ±, ±, ±, ± P( ( - ( + ( + c -intercepts: -, - and ; -intercept: - d - - and Chapter Trigonometr and the Unit Circle. Angles and Angle Measure, pages 7 to 79. a clockwise counterclockwise c clockwise d counterclockwise. a c - - π - π _ e 9 _ - 9 d _ π 9 π _ 9 f 7 7π. a or. or. c - or -.7 d or. e -_ 7π or -. f π or 9.. a c 7. d _ _ e π or 7. f 9 π or 7. _. a or.9 or _ c π or.97 d 9 π or e π or -.79 f π or.9. a quadrant I quadrant II - c quadrant II d quadrant IV 7π _ e quadrant III f quadrant IV - π - MHR Answers