19. 1. y = -x - 1 b) $ y^{3} - x^{3} = 3xy $ c) $ (3/2, 3/2) $ d) $ (x + (3/2))^{3} + y^{3} = 3y(x + (3/2)) $

5. $ (x + (3/2))^{3} + (y + (3/2))^{3} = 3(x + (3/2))(y + (3/2)) $

20. 1. $ (0, 1/3) $ and $ (0, 1) $ b) $ y^{2}(1 - ) = ( + 2y - 1)^{2} $

3. $ y^{2}(1 - 4(x - )^{2}) = (4(x - )^{2} + 2y - 1)^{2} $ d) $ (1 - x^{2}) = (x^{2} - - 1)^{2} $

21. 1. the graph is symmetric about the x-axis. b) $ (x - x_{0})^{4} - 2(x - x_{0})^{3} + 4(y - y_{0})^{2} = 0 $

\[ \mathsf{c})\left(x-x_{0}\right)^{4}+2(x-x_{0})^{3}+4\big(y-y_{0}\big)^{2}=0 \]

22. 1. $ (0,0) $ , $ (0,) $ , $ (,0) $ b) Substitute $ y $ for y. New eqn: $ -x^{4}+10x{2}+y^{4}-y{2}=0 $ c) Substitute $ x $ for x. New equation: $ -x^{4}+25x{2}+y^{4}-y{2}=0 $
23. 1. $ (,0) $ , $ (,0) $ , and $ (0,0) $ b) The fish would become very thin, while maintaining its current length.

3. The fish would be twice as long and facing the opposite direction.

30. 1. v-stretch by a factor of 5 b) shift left by 2 units c) flip over x-axis d) shift up by 1 unit e) v-stretch by a factor of 7/9 f) h-stretch by a factor of 1/6 g) h-stretch by a factor of 3/2 h) shift up by 5 units i) v-stretch by 3 and flip over x-axis j) v-stretch by 3/5, and flip over x-axis. k) h-stretch by a factor of 1/4 and flip over y-axis l) shift down by 2 units.

31. a,b,c are false, for none of those equations are functions. d is true.

32. 1. sub. $ (x - 8) $ for x b) sub. $ (x + 8) $ for x c) sub. $ (y + 8) $ for y d) sub. $ (y - 8) $ for y e) mult. RHS by 6 f) mult. RHS by 1/3 g) sub. $ (x/10) $ for x h) sub. 10x for x i) mult. RHS by 3/16 j) mult. RHS by -1 k) sub. $ (-x) $ for x l) mult. RHS by -5 m) mult. RHS by -3/7.

33. (Check your graphs with Geogebra or another a computer program. I’ve listed key features below.)

1. \((0,-5)\), \((1,0)\) b) \((0,\sqrt{3}+1)\), endpoint: \((-3,1)\) c) \((0,3)\), \((\pm(3/2),0)\)

2. \((0,5/3)\), \(\left(-2\pm\sqrt{3/2},0\right)\), vertex: \((-2,-1)\) e) \((0,-2)\). Asymptotes: \(y=0\) and \(x=2\)

3. \((0,-1)\), \((1\pm\sqrt{2/3},0)\), vertex: \((-2,-1)\) g) \((3/7,0)\). Asymptotes: \(y=7\) and \(x=0\).

4. \((0,-2)\), \((1,0)\) i) \((0,3)\), \((\pm3\sqrt{3}/2,0)\). Endpoints: \((\pm3,-3)\)

5. \((0,-1)\), \((\pm\sqrt{3},0)\). Endpoints: \((\pm2,1)\) k) \((0,-1/2)\), \((-1,0)\)

34. 1. \(y = 2|x - 1|\) b) \(y = \frac{1}{2}|x| + 1\) c) \(y = -3|x - 2| + 1\)

35. Yes: multiply the function by m, substitute $ (x - x_{0}) $ for x, then add $ y_{0} $ to the function (v-stretch by a factor of m, h-shift by $ x_{0} $ units, v-shift by $ y_{0} $ units.)

36. a)(0,1), (4 ± √15,0), vertex: (-4, -15). b)(0,2), vertex: (-1,0) c)(±√3/5,0), vertex: (0,-3) d)(0,-7), (5 ± 3√2,0), vertex: (5,18) e)(0,5), vertex: (-2/3,11/3) f)(0,2), (6 ± √46/5,0), vertex: (-6/5,46/5) g)(0,0), (-4,0), vertex: (-2,-6) h)(0,1), (2 ± √46/3,0), vertex: (2/3,23/21)

37. No: for any k, the vertical line x = k crosses the graph of $ y = x^{2} $ [at $ (k, k^{2}) $ , in fact].

38. 1. Complete the square and follow the vertex during the various transformations. b) $ (-2/5, -9/5) $

39. No: consider a $ 2 $ rectangle and a $ 3 $ rectangle. 41. $ 25 $ feet

40. 1. $ (, ) $ b) about 3865 feet. c) 4000 feet d) about 3361 feet

41. Yes. Vertex: $ (0,0) $ , Focus: $ (0,1/8) $ , Directrix: y = -1/8