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Course: Algebra 1 (Eureka Math/EngageNY) > Unit 4
Lesson 8: Topic A: Lessons 8-10: Parabolas introQuadratic word problems (factored form)
Solving a problem where a quadratic function (given in factored form) models the height of a launched rocket.
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I can solve these equations but I don't understand what each expression means.
x+2=0, x=-2.
x-18 =0, x=18.
(-2+18)/2 =8.
-4(8+2)(8-18)=400
What does -4 mean in this circumstance? and what are (x+2) (x-18) mean?(16 votes)
If you start from - 4(x+2)(x-18) and expand it to f(x) = - 4(x^2 - 16x - 36) or f(x) = -4x^2 + 64x + 144, the -4 has to do with the force of gravity trying to pull the rocket back to the ground, the 64 has to do with the initial velocity that the rocket was launched at, and the 144 has to do with the initial height of the rocket. The equation often uses t instead of x because t would stand for time and f(t) is height above ground. The -2 and the 18 are the solutions to the quadratic function, which in this case means that this will be either a real (18) or hypothetical (-2) time when the rocket is on ground level. the -2 is hypothetical because it is behind where the rocket is launched, so it could be shown as a dashed line from time -2 to 0 which fits the function, but not the situation.(42 votes)
At2:57isn't it supposed to be 20 seconds as the launch starts at x=-2 and goes to 18 which makes the total time in the air 20 seconds and make the rocket land after 20 seconds?(9 votes)
It would be 20 seconds if the rocket was launched from the ground. In this case, the rocket was launched from a raised platform. The x=-2 is basically denoting that 2 seconds of flight were saved by launching from the platform rather than the ground.
Hope this helps.(19 votes)
so if the equation is h(x) = -4(x - 18), how would you find the first x value?(10 votes)
if y=0 then either -4 or (x-18) equals zero. -4 cannot equal zero, so (x-18) has to. So x=18 and there is no second x.(10 votes)
Umm... Guys, what if we were asked to give the model?(6 votes)- Like graphing it? It'll look something like a curve with the 0's as -2 and 18.(9 votes)
At3:05, In the question it asked "how many seconds after being launched will the rocket reach its maximum height?" So what i thought of it was that, since the total air time of the rocket was 18 seconds so it would mean that at 9 seconds it will be at its maximum point so shouldn't 9 seconds be the correct answer?(5 votes)
Ok. I will solve it my way, and hope you can get something from it.
1)What is the height of the rocket at the time of launch ? Well, the time at launch is x=0, so h(0)=4(0+2)(0-18) , which equals -4(2)(-18)=(144), so the height at the time of launch is 144 assuming x=0 is the ground.
2)How many seconds after the launch will the rocket hit the ground ? The rocket will reach the ground when h(x)=0, which will hapen in two points. we can apply Bhaskara´s formula in order to find zeroes or just take a look at the function in this case.
h(x) = -4(x+2)(x-18), so h(x)=0 when (x+2)=0 or when (x-18)=0 .So h(x)=0 when x=-2 or when x=18. Lets check for extraneous solutions by plugging the results back in the function. -2 is not a valid solution, because the rocket cannot fall to the ground before it has taken of, so that leaves us with x=18 as the only valid answer.I checked all my answers with a graphic calculator.(9 votes)
My thinking is that the midpoint X should be 10. It's 20 units between them where X=seconds. If you follow the graph, that would mean that the rocket after 0 seconds launch would just appear so many meters above the ground. The rocket's path must start at 0 meters and 0 seconds. It would proceed with the same curve, and maximum height, but it cant launch 2 seconds in the past. Unless you actually think of it like this: two seconds after launch it reached Y meters. Am I making any sense?(3 votes)- While I like your thinking process, you make several incorrect assumptions. In the problem, it clearly states that it is launched from a platform, so your assumption that it must start at 0 meters and 0 seconds is just incorrect. If it were a function to be graphed independent of the word problem, both sides of the parabola would extend to negative infinity. However, within the context of the word problem, the domain would be from 0 to 18 seconds when it hit the ground. So the word problem would cut off all of the parabola less than 0 (including the part that is reflected from 16-18 seconds at -2-0) as well as the part greater than 18. Being launched from a platform that is 144 m about the ground gives a starting point at (0,144) not (0,0), and the symmetrical point would be at (16,144) which again gives the same midpoint at 8. The part between 16 and 18 just does not have a part that is reflected across the line of symmetry. While the -2 point is not within the domain of the word problem, it is still within the graph of the function and is the imaginary reflection of (18.0) which still gives midpoint at 8. It not that the midpoint is 10 units from the y intercept, it is 10 units from the endpoint 18 and 18-10 = 8. So what you really are attempting to do is shift the whole thing 2 units to the left so that you will begin at (0,0) because you do not like the fact that it was launched from a platform.(13 votes)
I've come to notice that when you are trying to find the y coordinate for the vertex that the expressions within the parentheses are evaluated as a number and the negative of the number. For example in this problem the evaluation of the parentheses was 10 and -10. Is it coincidental and if not what is the reasoning behind this remarkable observation?(6 votes)- It is what should be expected because you found the midpoint between -2 and 18. The distance between these two is 20, so the midpoint was at 8 which should be 10 units away from the two points. Thus, one of the points mush be + 10 away, and the other must be - 10 away. Even if you change points such as 6 and 18, you would find the midpoint as (6 + 18)/2 = 12. Since 18-6 = 12, they are twelve units away from each other, so dividing by two gives + 6 and -6 from the midpoint. There is no coincidence about it.(7 votes)
- Ive watched it many times. but i still dont understand how to solve for x.(6 votes)
That is a great question! The two Xs in the parentheses are going to be opposite of what they are being added/subtracted to since the first X in the problem is 0. If you are asking to answer how many seconds after the rocket was launched it hit the ground then here you go as well. Since after the rocket we are going forward in time, between the two answers of x=-2 and x=+18 the only one reasonable enough would be x=18 then that would be the answer. Not sure if I confused you more or not but that is all I have, I also had looked back at the video, amazing question.(3 votes)
The rocket launch in -2 sec, and it falls in 18 sec, I think the answer should be 20 and it's not 18.
Is my thinking correct?(3 votes)- What you are trying to answer is a question where the rocket is launched from ground level. The rocket was launched at t=0, not at t=-2.(5 votes)
I'm still very confused, how do you know how to answer each question? Since Sal has a different way of answering each problem...(5 votes)- Each question is essentially asking you to find one of the basic parts of the parabola. The first question asks you for the y-intercept. The second question asks you for the second x-intercept. The third question asks for the vertex of the parabola, and the last question asks for the y-value of the vertex. The question is disguising these queries as questions about the rocket, but in reality, they're asking for these values.(2 votes)
2:57Video transcript
- [Instructor] We are
told a rocket is launched from a platform. Its height in meters, x seconds
after the launch is modeled by h of x is equal to
negative four times x plus two times x minus 18. Now, the first thing they
ask us is what is the height of the rocket at the time of launch? Pause the video and see if
you can figure that out. Well, what is x at the time of launch? Well, x is the number of
seconds after the launch, so at the time of launch, x
would be equal to zero. The height of the rocket when
x is equal to zero, they're essentially saying,
"Well, what is h of zero?" To figure out h of zero, we
just have to go back to this expression and replace
all the xs with zeros. H of zero is going to be equal
to negative four, negative four times zero plus two,
which is just going to be two times zero minus 18,
which is just going to be negative 18. Let's see, this is going
to be negative eight times negative 18, negative
eight times negative 18, which is the same thing as negative eight times negative nine times two. This is going to be positive 72 times two, which is 144, so 144 meters. Did I do that right? Let's see. We're going to have ... Yep that sounds right. That's right. How many seconds after
launch will the rocket hit the ground? Pause this video again and
see if you can answer that. Well, what does it mean for
the rocket to hit the ground? That means that the height is
equal to zero, so if you want to figure out how many
seconds after launch, how many seconds that's x, so we want
to figure out the x when our height is equal to zero. We can set up an equation. Let's make our height h of
x equal to zero, so zero is equal to negative four times x plus two times x minus 18. Well, if you have the product
of three different things being equal to zero, the way
you get this to be equal to zero is if at least one of these three
things is equal to zero. Well, negative four can't be
equal to zero, so we could say x plus two equals zero. I got that from right over here. If x plus two were equaling
to zero, then this equation would be satisfied. That would be the situation
when x is equal to negative two, but remember x is the
number of seconds after the launch, so a negative
x would mean be going before the launch. We can rule that one out. Then, we could also think
about, "Well, x minus 18 if "that's equal to zero,
then this entire expression "could be equal to zero." X minus 18 equals zero,
you add 18 to both sides. You get x is equal to 18, so 18 seconds after launch, well we're going forward in time. 18 seconds after launch, we see that our height is zero, we have hit the ground. Next question, how many seconds
after being launched will the rocket reach its maximum height? Pause the video again and see
if you can figure that out. Well, the key realization here is if you have a curve, if you have a parabola in
particular and it's going to look something like this, if you're
gonna have a parabola that looks something like this,
you're going to hit your maximum height right over here
between your two zeros or between the two times
that your height is zero. If you figure out this x
value and this x value, the average of the two will give
you your x value the time after launch, when you're
at your maximum height. Well, we already figured out
what this x-value is and what this x-value is. We know that h of x is equal
to zero when x is either equal to 18, so that is x is equal to 18 or x is equal to negative two, so that is x is equal to negative two. To answer this question, we
just have to go halfway between negative two and 18, so let's do that. Negative two plus 18 divided by two gets us what? That's going to be 16 over
two, which is going to be equal to eight. This is right over here. This is x equals eight
seconds the rocket is at its maximum height. Last question, what is the
maximum height that the rocket will reach? Once again, pause the video
and try to answer that. Well, we already know from
the previous question that we reach our maximum height
when x is equal to eight, eight seconds after launch. To figure out the height
then, we just have to evaluate what h of eight is, h of eight. Remember that's what this
function does, you give me any x value, any elapsed time after
launch and it will give me the height, so eight seconds
after launch, I know I have maximum height. To figure out that height,
I just input it into the function, so h of eight is
going to be equal to negative four times eight plus two times eight minus 18. Eight plus two is 10. Eight minus 18 is negative 10. You have negative four times negative 100, so that's
going to be positive 400. H is given in meters, so
that's its maximum height, 400 meters.