## help_people one year ago The height of water shooting from a fountain is modeled by the function f(x) = −4x2 + 24x − 29 where x is the distance from the spout in feet. Complete the square to determine the maximum height of the path of the water. −4(x − 3)2 − 29; The maximum height of the water is 3 feet. −4(x − 3)2 − 29; The maximum height of the water is 29 feet. −4(x − 3)2 + 7; The maximum height of the water is 7 feet. −4(x − 3)2 + 7; The maximum height of the water is 3 feet.

1. help_people

@welshfella or @triciaal or someone :D

2. triciaal

|dw:1438815884398:dw|

3. zzr0ck3r

I am with @triciaal

4. help_people

ummm?

5. triciaal

|dw:1438816139795:dw|

6. help_people

i do not understand any of this i do but what she'll i do next :)

7. help_people

what is after that part @triciaal

8. triciaal

one approach compare your equation with the general equation identify your a, b, c and substitute

9. help_people

how she'll i do that or could you show me I'm confused if you cannot tell

10. triciaal

alternative |dw:1438816378738:dw|

11. triciaal

when you complete the square you are making a perfect square

12. triciaal

|dw:1438816857649:dw|

13. triciaal

does it make more sense now?

14. jdoe0001

@help_people have you covered what a "perfect square trinomial" is? sometimes just called "perfect square" if not, this may be the time to check your book for it then

15. help_people

i have @jdoe0001

16. jdoe0001

so.. this rings a bell $$\begin{array}{cccccllllll} {\color{brown}{ a}}^2& + &2{\color{brown}{ a}}{\color{blue}{ b}}&+&{\color{blue}{ b}}^2\\ \downarrow && &&\downarrow \\ {\color{brown}{ a}}&& &&{\color{blue}{ b}}\\ &\to &({\color{brown}{ a}} + {\color{blue}{ b}})^2&\leftarrow \end{array}\qquad % perfect square trinomial, negative middle term \begin{array}{cccccllllll} {\color{brown}{ a}}^2& - &2{\color{brown}{ a}}{\color{blue}{ b}}&+&{\color{blue}{ b}}^2\\ \downarrow && &&\downarrow \\ {\color{brown}{ a}}&& &&{\color{blue}{ b}}\\ &\to &({\color{brown}{ a}} - {\color{blue}{ b}})^2&\leftarrow \end{array}$$ right?

17. jdoe0001

let's see your equation, and let us do some grouping, as triciaal was indicating $$\bf f(x)=-4x^2+24x-29\implies y=-4x^2+24x-29 \\ \quad \\ y=(-4x^2+24x)-29\implies y=-4(x^2+6x)-29\impliedby \textit{common factor}$$ follow that?

18. jdoe0001

well... actually we took out -4, should be -6, one sec

19. jdoe0001

$$\bf f(x)=-4x^2+24x-29\implies y=-4x^2+24x-29 \\ \quad \\ y=(-4x^2+24x)-29\implies y=-4(x^2-6x)-29\impliedby \textit{common factor}$$

20. jdoe0001

21. help_people

yes

22. help_people

@jdoe0001

23. jdoe0001

ok so $$\bf y=-4(x^2-6x)-29\implies y=-4(x^2-6x+{\color{red}{ \square }}^2)-29$$ so... we have a missing number there, to make a "perfect square trinomial" what do you think that number is anyway?

24. jdoe0001

notice $$\begin{array}{cccccllllll} {\color{brown}{ a}}^2& + &2{\color{brown}{ a}}{\color{blue}{ b}}&+&{\color{blue}{ b}}^2\\ \downarrow && &&\downarrow \\ {\color{brown}{ a}}&& &&{\color{blue}{ b}}\\ &\to &({\color{brown}{ a}} + {\color{blue}{ b}})^2&\leftarrow \end{array}\qquad % perfect square trinomial, negative middle term \begin{array}{cccccllllll} {\color{brown}{ a}}^2& - &2{\color{brown}{ a}}{\color{blue}{ b}}&+&{\color{blue}{ b}}^2\\ \downarrow && &&\downarrow \\ {\color{brown}{ a}}&& &&{\color{blue}{ b}}\\ &\to &({\color{brown}{ a}} - {\color{blue}{ b}})^2&\leftarrow \end{array}$$ the middle term is just the other two terms without the "2" exponent times 2 and our last number is hidden in the middle term

25. help_people

-3? @jdoe0001

26. help_people

or 3 i meant to say @jdoe0001

27. jdoe0001

right.... 2 * x * 3 = 6x <--- middle term

28. jdoe0001

bear in mind two things one there's a -4 outside the group, that is multiplying the group and that what we're really doing is borrowing from our good fellow Mr Zero, 0 so, if we add anything, we also have to subtract it so, we'll add $$3^2$$ but is being multiplied by the -4 outside, that means that we're really adding $$3^2$$cdot -4 = -36

29. jdoe0001

and if we're adding -36, well, really subtracting, since it's negative we also have to add it so that -36 + 36 = 0 and our good felow Mr Zero gets back his figure

30. help_people

wait so the final answer is a ? @jdoe0001

31. jdoe0001

ahemm nope.. one sec

32. jdoe0001

$$\bf y=-4(x^2-6x)-29\implies y=-4(x^2-6x+{\color{red}{ 3 }}^2)+36-29 \\ \quad \\ y=-4(x-3)^2-7 \\ \quad \\ \quad \\ y=(x-{\color{brown}{ h}})^2+{\color{blue}{ k}} \qquad\qquad vertex\ ({\color{brown}{ h}},{\color{blue}{ k}})$$ so... what do you think is the vertex? or that y-coordinate for that matter

33. help_people

its either a or b 2 @jdoe0001

34. jdoe0001

hmmm actaully

35. jdoe0001

$$\bf yy=-4(x^2-6x)-29\implies y=-4(x^2-6x+{\color{red}{ 3 }}^2)+36-29 \\ \quad \\ y=-4(x-{\color{brown}{ 3}})^2+{\color{blue}{ 7}} \\ \quad \\ \quad \\ y=(x-{\color{brown}{ h}})^2+{\color{blue}{ k}} \qquad\qquad vertex\ ({\color{brown}{ h}},{\color{blue}{ k}})$$ can you see the vertex now?

36. jdoe0001

shoot even got 2 yy, =( $$\bf y=-4(x^2-6x)-29\implies y=-4(x^2-6x+{\color{red}{ 3 }}^2)+36-29 \\ \quad \\ y=-4(x-{\color{brown}{ 3}})^2+{\color{blue}{ 7}} \\ \quad \\ \quad \\ y=(x-{\color{brown}{ h}})^2+{\color{blue}{ k}} \qquad\qquad vertex\ ({\color{brown}{ h}},{\color{blue}{ k}})$$

37. jdoe0001

see the vertex now? see the y-coordinate? the y-coordinate of the verex, is the "maximum height" because the vertex is the U-turn point for the graph, it goes up, reaches a maximum, then back down

38. help_people

ok so i am trying to find the answer lol is it a or b i said it was a but you said i was wrong @jdoe0001

39. jdoe0001

well.... thata form of the quadratic is the so-called "vertex form" because it shows the vertex coordinates "x" is the horizontal distance "y" is the vertical height the equation has a leading term negative coefficient, thus is opening downwards thus the maximum point for it, is at the vertex

40. jdoe0001

and is obtained by "completing the square" so... the vertex should be pretty obvious there and so is the y-coordinate of the vertex and that y-coordinate, is the maximum height of it

41. help_people

so b? @jdoe0001

42. jdoe0001

dunno... what do you think is the vertex? notice to the equation in vertex form

43. help_people

vertex is 3 and 7 @jdoe0001

44. help_people

based on that i bleive its a @jdoe0001

45. jdoe0001

|dw:1438820047108:dw|

46. jdoe0001

equaiont in "a" doesn't even look like the obtained equation so it can't be that

47. help_people

the only other one i would think is c @jdoe0001

48. jdoe0001

$$\bf -4(x - {\color{brown}{ 3}})^2 {\color{blue}{ -29 }}$$ according to that, the vertex is 3, -29 which is not what the simplified version is

49. jdoe0001

yeap, the maximum height of the water is 7 units vertex at 3,7|dw:1438820313064:dw|

50. help_people

so it is c?

51. help_people

@triciaal please come back he just left and i am not sure what the answer is :/

52. help_people

i first thought it was a but then he told me no so now i believe it is c

53. help_people

never mind :)