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System Message

[00:00:00] *** Please note: All sessions are recorded for quality control. ***

Guest (Customer)

[00:00:08] Consider the parabolic paths of two soccer penalty kicks, represented in the graph at right. Kick #1 covers a horizontal distance of 20 yds and reaches a maximum height of 9yds. Kick #2 covers a horizontal distance of 24yds, but only reaches a maximum height of 6yds. Find an equation for the path of each kick? How far off the ground would they be when they collide? [ File > http://0-lhh.tutor.com.catalog.multcolib.org/SharedSessionFiles/501955c1-27d9-4ea3-8013-6947cc99f605_tn.jpg ]

Blythe L (Tutor)

[00:00:21] Welcome to Tutor.com !
Let's get started.

[00:00:23] hi

Guest (Customer)

[00:00:28] hello

Blythe L (Tutor)

[00:03:23] okay, here are the two parabolas

[00:03:33] does it make sense how I drew them so far?

Guest (Customer)

[00:03:44] by the info given

Blythe L (Tutor)

[00:04:09] so, now, we need to make the two equations for the parabolas

[00:04:34] and then we set them equal to each other and then solve

[00:05:08] so, let's use the vertex and another point to find the equation

[00:05:13] do you know how to do that?

Guest (Customer)

[00:05:24] no

Blythe L (Tutor)

[00:05:37] okay, I'll show you ...one sec...

[00:06:51] this is the vertex form of the equation

[00:07:44] the vertex is (h, k)

[00:08:22] so, I plug in (12, 6) for the vertex

Guest (Customer)

[00:08:33] ok

Blythe L (Tutor)

[00:08:49] and also, since (0,0) is another point on the parabola, then we can plug in that point for x,y and then we can find the "a"

[00:10:43] so, the "a" is -1/24

[00:11:04] and so the equation of the first one is y=-1/24 ( x-12)^2 +6

[00:12:03] okay, so that's the first one

[00:12:12] now, do you want to try to find the second equation

Guest (Customer)

[00:12:12] ok thanks

Blythe L (Tutor)

[00:12:26] using the (10,9) and (0,0) ?

[00:12:36] I'll tell you if there's a mistake

[00:13:31] wait, that's not how you do it

[00:13:52] do you see where the "h" and "k" is in the equation?

[00:14:03] the "h" is the x value of the vertex, 10

[00:14:12] the "k" is the y value of the vertex, 9

[00:14:35] and you will plug in (0,0) as the x,y in the equation

[00:17:23] so, that is the equation for the second one

[00:17:26] does it make sense?

Guest (Customer)

[00:17:34] kind of

Blythe L (Tutor)

[00:18:27] I plugged in (0,0) for (x,y) and I plugged in (10,9) for (h,k)....that was all just to find the "a" (the coefficient in front of the x)

[00:18:46] now, to find the point of intersection...

[00:18:58] you will need to set the equations equal to each other

[00:19:03] and solve for "x"

[00:19:16] do you understand what I'm talking about?

Guest (Customer)

[00:19:29] sort of

Blythe L (Tutor)

[00:20:45] set the two equations equal to each other and you are going to solve for "x"

[00:22:15] okay, now I have written them for you on the second whiteboard

[00:22:30] you will need to use this equation to solve for the x

[00:22:36] try to begin it

Guest (Customer)

[00:22:41] alright

Blythe L (Tutor)

[00:23:48] wait

[00:24:01] where did the 3 go?

[00:24:18] okay

[00:24:24] now remember the (x-12)^2

[00:24:35] is equal to (x-12)(x-12)

[00:24:39] you have to FOIL

[00:24:45] it's not the same as x-144

[00:24:58] that's not what it's equal to when you FOIL it out

[00:25:34] so, separate out the (x-12)^2 into (x-12)(x-12)

[00:25:39] and them multiply it out

[00:27:02] you need to multiply -12 times x and -12 times x

[00:27:04] in the middle

[00:27:13] it will be x^2 -12x-12x -144

Guest (Customer)

[00:29:20] is that correct so far

Blythe L (Tutor)

[00:29:39] it needs to be -10x -10x

[00:29:51] you left out one of them

Guest (Customer)

[00:30:28] i dont understand that part

Blythe L (Tutor)

[00:30:47] do you know what the FOIL method is?

Guest (Customer)

[00:31:11] kind of

Amanda S (Tutor)

[00:36:38] Hello,

Guest (Customer)

[00:36:52] hi

Amanda S (Tutor)

[00:36:58] Give me one moment to check over your progress so far :)

[00:38:43] Okay,

[00:39:15] So it seems at this point you're trying to foil (x-12)^2 is this correct?

Guest (Customer)

[00:39:27] yes

Amanda S (Tutor)

[00:39:57] Alright.
And you may have noticed you get
x times -12 twice when you do this,

[00:40:12] So you'd end up with x^2 -12x -12x - 144

[00:40:28] Does that make sense?

Guest (Customer)

[00:40:33] yes

Amanda S (Tutor)

[00:41:02] Alright

Guest (Customer)

[00:42:34] ok

Amanda S (Tutor)

[00:43:06] Sorry.
I'm just checking the original work to see if there's another method, just as this one seems ... a bit long?

[00:43:11] But,

[00:43:18] Continuing with this method,

[00:43:37] If you have x^2 -12x -12x

[00:43:44] and I'm sorry, I made a typo.

[00:43:54] The last term should be
plus...
+144

[00:44:04] -12 times -12 is positive, +144

Guest (Customer)

[00:44:27] alright

Amanda S (Tutor)

[00:44:40] I'm really sorry.
Would it be okay if I get a new board?
:(

[00:44:54] I can't see what I'm writing on this one

Guest (Customer)

[00:44:54] yes thats fine

Amanda S (Tutor)

[00:45:07] :-)
Thanks

Guest (Customer)

[00:45:22] welcome

Amanda S (Tutor)

[00:46:46] So at this point we were dealing with (x-12)^2 and I was mentioning we'd end up with

[00:46:54] x^2 -12x -12x + 144

Guest (Customer)

[00:47:09] yes

Amanda S (Tutor)

[00:47:34] -12x -12x would combine to give us -24x

Guest (Customer)

[00:48:10] yes

Amanda S (Tutor)

[00:48:26] At this point, I think I'd really want to just get rid of the fractions.

[00:48:54] The way to do that is to multiply everything by the least common denominator.

Guest (Customer)

[00:49:46] 4

[00:49:48] ?

Amanda S (Tutor)

[00:50:24] That would indeed be a common factor, but for a common denominator we want a common multiple

[00:50:59] :(
Does that make sense?
I know people usually don't stress the difference between those

Guest (Customer)

[00:51:12] yes

Amanda S (Tutor)

[00:51:20] Ok,

[00:51:28] For instance

[00:51:53] The method I sometimes use is a bit counter-intuitive,

[00:52:38] Both the terms have a 4

[00:52:55] But one lacks a six

[00:53:06] And the other lacks 25

Guest (Customer)

[00:53:23] 600

Amanda S (Tutor)

[00:53:28] Right!!

[00:53:45] So from this I get 600 as the lowest common denominator.

[00:53:54] :S
As I said, it's not a method most people use.

[00:54:10] But to cancel the fractions, we times each thing by 600.

[00:54:29] Does that reasoning work out for you?

Guest (Customer)

[00:54:37] yes

Amanda S (Tutor)

[00:54:45] Okay

[00:55:04] So that first part is times 600,

[00:55:24] +3 times 600

[00:55:31] and on the right, everything times 600.

[00:55:45] does that make sense?

Guest (Customer)

[00:55:57] yes

Amanda S (Tutor)

[00:56:06] Alright

[00:56:09] On the left side

[00:56:31] If you take 600 times .. well, all the stuff on the left.

[00:56:37] How would you work through tat?

[00:56:40] that*?

[00:57:20] Okay.

[00:57:23] Not quite,

[00:57:34] We'd really think of it as 600 over 1,

[00:58:07] So we'd really be canceling the 600 with the 100

[00:59:29] Okay.
For the most part you're getting the idea,

[00:59:41] Let me help with just one or two things here

[01:00:02] For instance you did cancel 600 and 100 out, but

[01:00:16] You still need to cross out the 00's there,

[01:00:26] So that's just 6

[01:00:58] Those parentheses in the first part are still important,

[01:01:18] And one last thing

[01:01:34] When you canceled 600 with -1/24

[01:01:58] So that's 600 times -1/24

[01:02:21] What's 600 divided by -24,
what would that give you?

Guest (Customer)

[01:02:47] .25

[01:02:52] -25

Amanda S (Tutor)

[01:02:58] Right,

[01:03:19] So in that whole part there, do you see where I was getting those numbers?

Guest (Customer)

[01:03:35] yes

Amanda S (Tutor)

[01:03:42] Okay.

[01:03:47] At this point

[01:04:05] We have 6(-9(x-10)^2) + 1800

[01:04:29] =

[01:04:41] -25(x^2-24x+144)

[01:05:23] Still a fairly involved problem.

[01:05:38] What do you think the next step would be at this point?

Guest (Customer)

[01:05:53] solve

Amanda S (Tutor)

[01:06:09] Right,

[01:06:27] You may consider simplifying first by taking 6 times -9 in the first part.

[01:07:00] Right,

[01:07:06] And the rest can be left the same for now,

[01:07:14] Unless you'd like to do two steps at once

Guest (Customer)

[01:07:16] ok

Amanda S (Tutor)

[01:07:24] And proceed by foiling (x-10)^2 ?

[01:08:54] Alright

[01:09:15] Though I think you might find you get x^2 -10x
and then another -19x

[01:09:19] *-10x
sorry

[01:09:30] :/

[01:09:40] x^2 -10x - 10x

[01:09:52] and then -10 times -10 would give you

[01:10:00] positive +100

[01:11:27] I'll get a new whiteboard,
:<
Sorry!
We ran out of room!!

[01:11:41] Do you have any questions so far?

Guest (Customer)

[01:11:40] alright

[01:11:55] no

Amanda S (Tutor)

[01:14:13] Okay, I believe that's what we have so far.

[01:14:16] I'm double checking.

[01:14:33] Could you double check as well to make sure that's the equation as we have it so far?

Guest (Customer)

[01:14:52] yes

Amanda S (Tutor)

[01:15:22] Alright; I believe everything so far should work out.

[01:15:34] What do you think the next step should be?

Guest (Customer)

[01:16:03] combine like terms

Amanda S (Tutor)

[01:16:11] Right,

[01:16:13] :)

[01:17:25] What would you do next?

Guest (Customer)

[01:17:44] get number to one ide

[01:17:47] side

Amanda S (Tutor)

[01:18:04] I can see what you mean, I think,

[01:18:20] But moving the +1800 around doesn't do much?
If that's what you mean?

Guest (Customer)

[01:18:44] i can subtract the 144

Amanda S (Tutor)

[01:18:57] Okay.
Actually you cannot,

[01:19:04] Because it's in parentheses.

[01:19:08] That's the only reason,

Guest (Customer)

[01:19:11] oh

Amanda S (Tutor)

[01:19:33] It's because it's actually times the -25 that's out front.

[01:20:02] Unless you take that -25 into account, you can't really move the -144, I don't know if that makes sense?

Guest (Customer)

[01:20:19] ok

[01:20:48] so would i have to time the -25 to the whole equation to get rid of it

Amanda S (Tutor)

[01:20:56] Yes;

[01:20:58] In fact,

[01:21:13] At this point it would be great if we could just get rid of any parentheses.

[01:21:18] This means

[01:21:32] On the right we take -25 times each thing it's next to in parentheses,

[01:21:48] And on the left side as well

[01:22:11] Continue,

[01:22:47] Right,

[01:23:20] So on the left side?

[01:24:16] Good!

[01:24:21] One thing on signs

[01:24:39] Negative 54, negative 20

[01:25:07] Okay!

[01:25:24] At this point we have

[01:25:48] -54x^2+1080x -5400

[01:25:49] =

[01:26:19] -25x^2-600x-3600

[01:26:54] *sorry missed the +1800

[01:28:05] :( I'm sorry this problem is taking so long,

[01:28:14] Does this correspond with what you have so far?

Guest (Customer)

[01:28:43] yes

Amanda S (Tutor)

[01:28:51] Okay, great!

[01:28:58] What would you do at this point?

Guest (Customer)

[01:29:39] subtract numebrs from sides

[01:29:45] or add numbers

Amanda S (Tutor)

[01:29:57] Okay,

[01:30:20] Looking at the problem at this point.
I'd agree, one one condition.

[01:30:27] We want to get everything on the left side.

[01:30:46] So we may start with -5400+1800 on the right

[01:31:04] Right

[01:31:16] Then looking on the right side, what do you see?

Guest (Customer)

[01:31:37] they cancel out

Amanda S (Tutor)

[01:31:41] :D
Right

[01:31:48] :)

[01:32:12] How would you work with the problem now?

Guest (Customer)

[01:32:46] i don't know i am stuck

Amanda S (Tutor)

[01:33:04] Well let's continue with trying to get things on the left side.

[01:33:14] You have -25x^2 on the right

Guest (Customer)

[01:33:35] add 600x

Amanda S (Tutor)

[01:33:48] Sure, that's a good start!

[01:36:07] Okay.
So, at this point you have 1680 = 29x^2

[01:36:11] Yes?

[01:36:27] Wait, hang on

Guest (Customer)

[01:36:26] square root

Amanda S (Tutor)

[01:36:40] Yes but I'm missing something.

[01:36:51] Oh !
wait,

[01:37:00] That's still 1680 c

[01:37:05] 1680x*

[01:37:17] Right?

Guest (Customer)

[01:37:26] oh yes

Amanda S (Tutor)

[01:37:49] So we have... 1680x = 29x^2

[01:37:54] Right?

Guest (Customer)

[01:37:59] yes

Amanda S (Tutor)

[01:38:06] Ok

[01:38:12] I'm going to get a new board,

Guest (Customer)

[01:38:45] ok

Amanda S (Tutor)

[01:39:07] One thing I'm thinking at this piont,

[01:39:09] point*

[01:39:55] And just see if this maybe makes sense

[01:40:01] We move everything to one side,

Guest (Customer)

[01:40:10] kind of

Amanda S (Tutor)

[01:40:24] Now both of these has an 'x' term in it

[01:40:38] So for factoring,

[01:40:55] We can take one x out

[01:41:25] Does that kind of make sense?

Guest (Customer)

[01:41:31] yes

Amanda S (Tutor)

[01:41:55] So one x-term out of 29x^2 is what?

Guest (Customer)

[01:42:02] 29x

Amanda S (Tutor)

[01:42:10] RighT!

[01:42:30] How about an x-term out of -1680x ?

Guest (Customer)

[01:42:46] 1680

Amanda S (Tutor)

[01:42:53] Right,

[01:43:07] and so last,

[01:43:12] what do you think we'd do?

Guest (Customer)

[01:43:26] check it

Amanda S (Tutor)

[01:43:44] Well right,

[01:43:47] But to get solutions

[01:43:51] we'd sed

[01:43:55] set*

[01:43:57] x=0

[01:44:07] and 29x-1680 =0

[01:44:48] Does that sound familiar?

[01:45:26] :)
Right.

[01:45:32] So that should be your answer,

[01:45:56] and I know that was a super, mega number of steps.

Guest (Customer)

[01:45:56] ok thanks

Amanda S (Tutor)

[01:46:12] Any questions on what we've worked through?

Guest (Customer)

[01:46:17] yeah and no

Amanda S (Tutor)

[01:46:40] Wow, okay!
:)
Well thanks for holding on there!
You did a super job.

[01:46:55] Thanks for using Tutor.com !
Please fill out the survey at the end, if you have time. :) Your feedback really helps!

Guest (Customer)

[01:46:59] thank you for helping me finish and understand

Amanda S (Tutor)

[01:47:10] :)
You're very welcome.

[01:47:12] Bye!!