Chapter 1 transformations answers

Answers Chapter 1 Function Transformations 1.1 Horizontal and Vertical Translations , pages 12 to 15 1. a ) h = 0 , k = 5 b ) d) h =7 , k = −3 e) h = 0 , k = −4 c ) h =−1 , k =0 h = −2 , k = 4 2. a) A '( - 4 , 1), B ′ ( — 3, 4 ) , b) A'( - 2 , −2) , C' ( - 1, 4), D'(1 , 2) , E '(2, 2) УЛ B '( - 1, 1), C′(1, 1) , D ' ( 3, −1 ) , E’ (4, −1) - - 5. a ) h = -5 , k = 4; y − 4 = f (x +5) b) h = 8 , k = 6 ; y − 6 = f (x — 8) c) h = 10, k = -8; y + 8 = f ( x - 10) d) h -7 , k = −12 ; y + 12 = f ( x + 7 ) 6. It has been translated 3 units up. = 7. It has been translated 1 unit right. Transformed Function Transformation of Points (x, y) → ( x, y + 5) 8 . Translation vertical y = f(x ) + 5 4 21 h ( x ) = f( x − 2) horizontal y = f(x + 7) ( x, y ) → (x - 7 , y ) horizontal y = f(x - 3 ) ( x, y) → ( x + 3, y) 2 7 0 N X y = f( x) — 6 (x, y) → ( x , y - 6) | g( x ) = f( x )

+3 4 -2 0 c)_A ′( — 8, −2) , B ′( − 7, 1) , C'( − 5, 1) , D′( − 3, −1) , E( -2 , -1) s (x) = f(x + 4 ) YɅ 2+ -8 -6 -4 -2 0 × 3. a) (x, y) ( x - 10, y ) N d) A '( - 4,-4 ), B' ( − 3, −1 ) , C '( - 1 , -1 ) , D'( 1 , −3) , E'( 2, −3) У -4-20 ( x , y) → (x + 1, y +3) a vertical translation - 21 | t(x) = f( x) − 2 → b) → (x, y ) (x, y — 6 ) - c) ( x, y) → (x + 7, y + 4 ) d ) 4. a) y 20 x of 3 units down and a horizontal translation of - 2+ b) c) d) ♫ ( x ) = f( x + 4) - 3

y +2) vertical y + 9 = f( x + 4) y = f(x - 4) – 6 y = f(x + 2) + 3 y = f(x - h) + k | ( x , y ) → ( x − 4 , y − 9 ) ( x, y ) → (x + 4, y − 6 ) ( x , y ) → ( x − 2, y + 3 ) ( x , y) → (x + h , y + k) R ) , ( y y≥ 5 , ye R } horizontal and vertical horizontal and vertical horizontal and vertical horizontal and vertical 9. a) y = (x + 4) 2 + 5 b ) ( xxЄ c) To determine the image function's domain and range, add the horizontal and vertical translations to the domain and range of the base function. Since the domain is the set of real numbers , nothing changes , but the range does change. 10. a) g (x) = |x − 9❘ + 5 b) The new graph is a vertical and horizontal translation of the original by 5 units up and 9 units right . c) Example: ( 0 , 0) , ( 1 , 1), ( 2, 2) → (9 , 5 ) , ( 10 , 6 ) , ( 11 , 7 ) d) Example: (0 , 0) , (1 , 1), ( 2, 2) → ( 9 , 5 ), ( 10, 6) , (11 , 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 . y = f (x − 3) 11. a) 12. a) b) y + 5 = f( x − 6) Example : It takes her 2 h to cycle to the lake, 25 km away. She rests at the lake for 2 h and then returns home in 3 h . b) This translation shows what would happen if she left the house at a later time . c) y = f ( x − 3) 13. a) Example: Translated 8 units right . b) Example: y = f ( x − 8) , y = f (x − 4 ) + 3.5 , y = f(x +4) + 3.5 14. a) Example : A repeating X by using two linear equations y = ± x . b) Example : y = f( x − 3) . The translation is - horizontal by 3 units right. 15. a) The transformed function starts with a higher number of trout in 1970. y = f( t) + 2 b ) The transformed function starts in 1974 instead of 1971. y = f ( t − 3 ) 16. The first case, n = ƒ (A ) + 10 , represents the number

of gallons he needs for a given area plus 10 more gallons. The second case , n = f(A + 10) , represents how many gallons he needs to cover an area A less 10 units of area . 17. a) y = ( x − 7) (x − 1) or y - = (x - 4 )2 - 9 b ) Horizontal translation of 4 units right and vertical translation of 9 units down . c) y-intercept 7 18. a) The original function is 4 units lower. b) Ул b) The original function is 2 units to the right. c) The original function is 3 units lower and 5 units left . 4- | f ( x ) = x2 g ( x) = 3f (x ) 2 h(x ) = = f( x) -4 -2 0 2 4 6 Χ d ) The original function is 4 units higher and 3 units right . 19. a ) The new graph will be translated 2 units right and 3 units down . b ) YA 2- £ 2 N -4-20 y = x3- x2 -N 2 L O - c o 8 X y = (x - 2 ) - ( x - 2 ) -3

h(x) = −x2 - 1 g( x): domain {x | x € R} , range { y y≥ 1 , y € R} h (x) : domain {x | x Є R } , range ( y y≤ -1 , ye R } 1 h ( x ) = X 1. a) X f ( x) = 2x + 1 g( x) = −f ( x ) h( x) = f(−x) -4 -7 7 9 -2 ལ -3 3 5 0 1 -1 1 20 k( x ) 2 5 -5 -3 4 9 -9 -7 4. a) b) y c) The y - coordinates of | g ( x) = − f ( x )\ 2 g(x ) have changed f ( x ) = 3x | h ( x ) = f ( x ) \ sign . The invariant point is ( -0.5, 0 ) . 2 4 -4 -26

| f( x) =2x + NO 2x The x-coordinates of h( x ) have changed -2- g (x ) = - 3x sign . The invariant point is (0 , 1) . b) yɅ d) The graph of g( x) is the reflection of the graph of f(x) in the x - axis , while the graph of h ( x) is the reflection of the graph of f (x) in the y - axis . 2 2. a) X f(x) = x2 g(x) = 3f(x) h(x) = = f(x ) 0 2 -6 36 108 12 c) -3 9 27 3 O 0 0 0 3 6 36 9 6 27 3 108 12 20 k (x ) = −1 h (x ): domain { x | x 0, x Є R} , range { yy0 , y € R} xk ( x ) : domain {x | x 0, x Є R } , range (yy0, y € R} g(x ) = -3x

4f(x ) b ) The graph of g (x) is a reflection in the x - axis of the graph of f (x ) . y = −f ( x ) c ) The graph of g ( x) is a horizontal stretch by a factor of of the graph of f(x ). y = f (3x) d) The graph of g( x ) is a reflection in the y - axis of the graph of f ( x ). y = f ( x ) 13. a) DA 80 1 D = 60 3052 40 201 0 20 40 60 8 80 S b) As the drag DA factor decreases, the length of the 80- skid mark D + 27 5 increases for the 60 same speed. 40 D = 16.552 20 8 . y= f ( 0.5x ) -10 -50 10 X D = 7.5 5 0 20 40 60

80 S 14. a) 9. a ) horizontally stretched by a factor of b) horizontally stretched by a factor of 4 c) vertically stretched by a factor of 1 1 X = −4, x = 3 c) x = -8 , x = 6 15. a ) I b) x = 4 , x = −3 d) x = -2 , x = 1.5 b) III c) IV d) IV 4 16. a ) yɅ 2 9 ( x ) d) vertically stretched by a factor of 4 e) horizontally stretched by a factor of and reflected in the y-axis f) vertically stretched by a factor of 3 and reflected in the x -axis f(x = x -20 2 4 6 - ∞ 8 X + x= 3 b ) У 2 10. a) b) y= | x | 2y = −3 | x || 2 4 Χ | y= |x | y = x f ( x) = |

(4,8) (2, 3) g(x) (5, 6) (-4, 8) Transformation reflection in the x- axis reflection in the y - axis (2,12 ) vertical stretch by a factor of 4 horizontal stretch by a factor of 글 (4, -12 ) (2 , −6) 1 and vertical stretch by a factor of 2 C4 У yɅ y= f ( x ) C5 a ) t = 4n - 14 n y = f3x ) 8 D b) t = -4n + 14 n c) They are reflections of each other in the x -axis . 1.3 Combining Transformations , pages 38 to 43 1. a ) y = f(x) or y = b) y= = -1 x2 f ( -4x ) or y = 4x2 2. The function f (x ) is transformed to the function g( x) by a horizontal stretch about the y - axis by a factor of 1. It is vertically stretched about the x -axis by

3 . a factor of 3. It is reflected in the x -axis , and then translated 4 units right and 10 units down. Reflections V er ti c al S tr et c h F a ct o r H or iz o nt al S tr et c h F a ct o r Vertical Translation Horizontal Translation 7. a) vertical stretch by a factor of 2 and translation of 3 units right and 4 units up ; (x, y ) → (x + 3, 2y + 4 ) b ) horizontal stretch by a factor of 1, reflection in the x -axis , and translation of 2 units down; (x, y) → (1- x , −y − 2) c) reflection in the y - axis , reflection in the x - axis , vertical stretch by a factor of 1 , and translation of 2 units left; ( x , y) → ( −x -2 , - 1) d) horizontal stretch by a factor of 1 , reflection in the x - axis , and translation of 2 units right and +3 ) 3 units up; (x , y) → ( x + 2 , −y + e) reflection in the y - axis , horizontal stretch by a factor of , reflection in the x- axis , and vertical stretch by a factor of 2; (x , y)

N -2- - ~ 2 4 8 C ח y = 2f (x) 41 y= f(x ) c ) S 2 4 -8 -6 -4 −2 0 y = 2 ( 3x) 2 4 6 Χ 2- b) УЛ y = f(x) y = -f( -x) y = -f(3 (x -2 )) + 1 0 2 4 6 - 1 0 X d ) yɅ y = 31 (3x) O N -6 -4 -2 0 2 4

6 -2+ -4- -~ - 1 0 8 Χ 12- 8- 3/1 y= ( x - 3 ) 4 4 6. a) ( -8 , 12) d ) ( 9 , -32 ) b) ( -4 , 72) c) (-6 , -32) e) ( -12 , -9) -16 -12 -8 -4 0 4 8 X Answers MHR 557 ⚫ e) yɅ 14. a) yɅ -12 -8 -4 0 Χ y = −x2 + 9 4- 4- -8+ 12+ -16- y = −3f( x + 4) −2 ▼ y b) f ) У А 글 와글 (x+2)-1 -8-4 0 4

2 1 | g ( x ) : 4 6 8x · ƒ (−2 (x +3 )) — 2 12. a ) A' ( - 11, −2 ) , B′( − 7 , 6) , C ’( − 3, 4 ) , D′( − 1 , 5) , E’( 3, −2 ) 4 b) y f (12(x+ = − f ( ( x + 3 ) ) + 13. a) The graphs are in two locations because the transformations performed to obtain Graph 2 do not match those in y = |2x - 6❘ + 2. Gil forgot to factor out the coefficient of the x-term , 2 , from - 6 . The horizontal translation should have been 3 units right , not 6 units. b) He should have rewritten the function as y = | 2 (x − 3) | + 2 . -16 -12 -8 −4 Q 12 ·(x + 6 ) ] +6 + 6) )* 15. a) ( -a , 0), (0, —b ) c ) + 6 X b) ( 2a, 0 ) , (0, 2b ) 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 vary with different base functions . 16. a) A = −2x3 + 18x c ) For (2,5), the area of the rectangle in part a) is 20 square units . A = −2x2 + 18x

A = −2 ( 2 )3 + 18 ( 2 ) A = 20 1 b) A = + 18x For (8, 5), the area of the rectangle in part b ) is 80 square units . A = -x3 + 18x A = −1 ( 8 ) 3 + 18( 8 ) A = 80 17. y = 36 ( x − 2 )2 + 6 (x − 2 ) — 2 18. Example: vertical stretches and horizontal stretches followed by reflections C1 Step 1 They are reflections in the axes. - 1: y = x + 3, 2: y = −x − 3 , 3 : y = x - 3 Step 2 They are vertical translations coupled with reflections . 1 : y = x2 + 1 , 2 : y = x2 – 1 , 3: y = −x2, 4 : y = -x2 - 1 C2 a ) The cost of making b + 12 bracelets , and it is a horizontal translation . b) The cost of making b bracelets plus 12 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 bracelets , and it is a = horizontal stretch . 2 C3 y 2( x - 3)2 + 1 ; a vertical stretch by a factor of 2 and a translation of 3 units right and 1 unit up C4 a) H is repeated ; J is transposed; K is repeated and transposed b) 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 1.4 Inverse of a Relation, pages 51 to 55 1. a) У y = f ( x ) , N b) YɅ 41 2