Translations

50 written problems across three levels — 20 foundational · 20 intermediate · 10 advanced. Work through them on paper, then reveal every answer to check. These are not graded and are counted separately from adaptive practice.

Foundational

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

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

3. FoundationalThe point $(-1, 6)$ is on $y = f(x)$ . Find its image on $y = f(x + 2) - 9$ .

4. FoundationalUnder $y = f(x - 6) + 2$ , which point of the original graph maps to $(10, 10)$ ?

5. FoundationalFind the x-intercepts of $y = (x - 3)^{2} - 16$ .

6. FoundationalFind both intercepts of $y = (x + 2)^{2} - 9$ .

7. FoundationalFor $y = \sqrt{x - 4} + 1$ , state the starting point of the graph and its domain and range.

8. FoundationalState the equations of the asymptotes of $y = \dfrac{1}{x - 2} + 3$ .

9. FoundationalGive the domain and range of $y = \sqrt{x + 9} - 2$ .

10. FoundationalGive the domain and range of $y = \dfrac{1}{x - 6} + 5$ .

11. FoundationalThe function $y = f(x)$ has range $[-4, 4]$ . Find the range of $y = f(x + 1) + 10$ .

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

13. FoundationalThe zeros of $y = f(x)$ are $x = -2$ and $x = 4$ . Find the zeros of $y = f(x - 3)$ .

14. FoundationalThe 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.

15. FoundationalWrite the equation of $y = x^{3}$ after it is translated up 5 and then right 2.

16. FoundationalThe point of inflection of $y = x^{3}$ is $(0,0)$ . Find it for $y = (x + 4)^{3} - 7$ .

17. FoundationalThe parabola $y = f(x)$ has axis of symmetry $x = 2$ . Find the axis of symmetry of $y = f(x - 7) + 3$ .

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

19. FoundationalA function $y = f(x)$ has a minimum value of $-3$ . What is the minimum value of $y = f(x - 8) - 5$ ?

20. FoundationalThe 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.

Intermediate

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

22. IntermediateWrite $y = x^{2} + 8x + 10$ in vertex form and describe it as a translation of $y = x^{2}$ .

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

24. IntermediateThe 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).

25. IntermediateExpress $y = x^{2} - 6x + 11$ as a translation of $y = x^{2}$ and give the vertex.

26. IntermediateSolve for the x-intercepts of $y = |x - 5| - 8$ .

27. IntermediateFind both intercepts of $y = \dfrac{1}{x + 1} - 2$ .

28. IntermediateFind the x-intercept of $y = (x - 1)^{3} + 8$ .

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

30. IntermediateWrite $y = x^{2} + 12x + 40$ in vertex form and describe the translation of $y = x^{2}$ .

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

32. IntermediateA translation maps $(0, 0)$ to $(-4, 5)$ . Write the mapping and the transformed equation of $y = f(x)$ .

33. IntermediateFind the x-intercepts of $y = (x + 5)^{2} - 3$ , in exact form.

34. IntermediateThe domain of $y = f(x)$ is $[1, 9]$ . After the translation $y = f(x + 3) + 2$ , state the new domain.

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

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

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

38. IntermediateTwo 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.

39. IntermediateThe point $(a, b)$ lies on $y = f(x)$ . Express, in terms of $a$ and $b$ , the point on $y = f(x - 3) - 3$ .

40. IntermediateFind the value(s) of $h$ so that the graph of $y = (x - h)^{2} - 4$ passes through the origin.

Advanced

41. AdvancedIf $y = f(x - h) + k$ has exactly the same graph as $y = f(x)$ for every $x$ , and $f$ is neither periodic nor constant, what are $h$ and $k$ ?

42. AdvancedA 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.

43. AdvancedTemperature 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.

44. AdvancedThe cost of producing $x$ items is $C(x)$ . New equipment raises fixed costs by $500, and a supplier now provides the first 20 units of material free, so for $x \ge 20$ the plant pays to produce only $x - 20$ units. Write the new cost function for $x \ge 20$ .

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

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

47. AdvancedA 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.

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

49. AdvancedThe 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. AdvancedA parabola with vertex $(h, k)$ passes through $(0, 0)$ and $(4, 0)$ with a maximum value of 8. Find its equation in vertex form.