Translations

1. 🔥 ChallengingDescribe fully how y = f(x - 4) + 6 transforms y = f(x), and give the mapping notation.

2. 🔥 ChallengingThe graph of y = \sqrt{x} becomes y = \sqrt{x + 2} - 5. Describe the translation and state the new domain and range.

3. 🔥 ChallengingWrite y = x^{2} + 8x + 10 in vertex form and describe it as a translation of y = x^{2}.

4. 🔥 ChallengingThe point (3, -2) is on y = f(x). Find its image on y = f(x - 5) + 4.

5. 🔥 ChallengingThe point (-1, 6) is on y = f(x). Find its image on y = f(x + 2) - 9.

6. 🔥 ChallengingA translation maps (5, -3) to (2, 4). Find h and k and write the transformed equation of y = f(x).

7. 🔥 ChallengingUnder y = f(x - 6) + 2, which point of the original graph maps to (10, 10)?

8. 🔥 ChallengingThe parabola y = x^{2} is translated so its vertex is at (-7, -2). Find the image of the point (1, 1) (which lies on the original).

9. 🔥 ChallengingExpress y = x^{2} - 6x + 11 as a translation of y = x^{2} and give the vertex.

10. 🔥 ChallengingFind the x-intercepts of y = (x - 3)^{2} - 16.

11. 🔥 ChallengingFind both intercepts of y = (x + 2)^{2} - 9.

12. 🔥 ChallengingFor y = \sqrt{x - 4} + 1, state the starting point of the graph and its domain and range.

13. 🔥 ChallengingSolve for the x-intercepts of y = |x - 5| - 8.

14. 🔥 ChallengingFind both intercepts of y = \dfrac{1}{x + 1} - 2.

15. 🔥 ChallengingState the equations of the asymptotes of y = \dfrac{1}{x - 2} + 3.

16. 🔥 ChallengingFind the x-intercept of y = (x - 1)^{3} + 8.

17. 🔥 ChallengingGive the domain and range of y = \sqrt{x + 9} - 2.

18. 🔥 ChallengingGive the domain and range of y = \dfrac{1}{x - 6} + 5.

19. 🔥 ChallengingA function has domain [-2, 5] and range [1, 7]. State the domain and range of y = f(x - 4) - 3.

20. 🔥 ChallengingThe function y = f(x) has range [-4, 4]. Find the range of y = f(x + 1) + 10.

21. 🔥 ChallengingWrite y = x^{2} + 12x + 40 in vertex form and describe the translation of y = x^{2}.

22. 🔥 ChallengingThe graph of g(x) = (x - 2)^{2} + 5 is translated to h(x) = (x + 1)^{2} - 3. Describe the translation from g to h.

23. 🔥 ChallengingA function f has a maximum at (3, 10). Where is the maximum of g(x) = f(x + 5) - 4?

24. 🔥 ChallengingThe zeros of y = f(x) are x = -2 and x = 4. Find the zeros of y = f(x - 3).

25. 🔥 ChallengingThe point (0, 6) is the y-intercept of y = f(x). Find its image under y = f(x - 2) + 1, and explain why the new graph's y-intercept is generally not 6.

26. 🔥 ChallengingIf y = f(x - h) + k has exactly the same graph as y = f(x) for every x and f is not periodic, what are h and k?

27. 🔥 ChallengingWrite the equation of y = x^{3} after it is translated up 5 and then right 2.

28. 🔥 ChallengingThe point of inflection of y = x^{3} is (0,0). Find it for y = (x + 4)^{3} - 7.

29. 🔥 ChallengingA translation maps (0, 0) to (-4, 5). Write the mapping and the transformed equation of y = f(x).

30. 🔥 ChallengingFind the x-intercepts of y = (x + 5)^{2} - 3, in exact form.

31. 🔥 ChallengingThe domain of y = f(x) is [1, 9]. After the translation y = f(x + 3) + 2, state the new domain.

32. 🔥 ChallengingA projectile's height is y = -(x - 20)^{2} + 100. A second is launched from 15 m farther along with a summit 25 m higher. Write its equation.

33. 🔥 ChallengingTemperature over the year is T(d) = f(d). Because of a climate shift, temperatures are 1.5° higher and peak 10 days earlier. Write the new model.

34. 🔥 ChallengingThe cost of producing x items is C(x). Fixed costs rise by $500 and the break-even quantity increases by 20 units. Express the new cost function.

35. 🔥 ChallengingGiven f(x) = \sqrt{x}, find the value of k so that y = f(x - 4) + k passes through (8, 7).

36. 🔥 ChallengingGiven f(x) = |x|, find h so that y = f(x - h) + 1 passes through (9, 4) with h < 9.

37. 🔥 ChallengingThe graph of y = \dfrac{1}{x} is shifted to give asymptotes x = -3 and y = 8. Write the equation.

38. 🔥 ChallengingA circle x^{2} + y^{2} = 25 is translated so its centre moves to (4, -3). Write the new equation.

39. 🔥 ChallengingRewrite x^{2} + y^{2} - 6x + 8y + 9 = 0 by completing the square and describe it as a translated circle.

40. 🔥 ChallengingThe parabola y = f(x) has axis of symmetry x = 2. Find the axis of symmetry of y = f(x - 7) + 3.

41. 🔥 ChallengingThe graph of y = f(x) has a hole at (1, 4). Where is the hole on y = f(x + 3) - 2?

42. 🔥 ChallengingTwo absolute-value graphs y = |x - 1| + 2 and y = |x - 6| - 4 have the same shape. Describe the translation from the first to the second.

43. 🔥 ChallengingA function y = f(x) has a minimum value of -3. What is the minimum value of y = f(x - 8) - 5?

44. 🔥 ChallengingThe point (a, b) lies on y = f(x). Express, in terms of a and b, the point on y = f(x - 3) - 3.

45. 🔥 ChallengingFind the value(s) of h so that the graph of y = (x - h)^{2} - 4 passes through the origin.

46. 🔥 ChallengingA water level model L(t) peaks at t = 6\ \text{h}. A new dam raises the baseline 3 m and delays the whole cycle 2 h. Write the new model and its new peak time.

47. 🔥 ChallengingThe graph of y = f(x) passes through (-4, 0) and (6, 0). After y = f(x - 1) + 0, find the distance between the new x-intercepts.

48. 🔥 ChallengingIf g(x) = f(x - 2) and h(x) = g(x - 3) + 5, express h directly in terms of f.

49. 🔥 ChallengingThe maximum of y = f(x) is at (2, 9) and its zeros are x = -1 and x = 5. For the pure horizontal shift y = f(x - 4), give the new maximum point and new zeros.

50. 🔥 ChallengingA parabola with vertex (h, k) passes through (0, 0) and (4, 0) with a maximum value of 8. Find its equation in vertex form.