Module 4: Find the Root of a Quadratic or Polynomial FunctionDue No Due DatePointsSubmitting extemal todlAvallable until Oct 11 at 11 59pmQuestion The height of a...

An object is thrown upward from the ground with an initial velocity of 64 feet per second. The height $h(t)$ in feet of the object after $t$ seconds is given by the polynomial function $$ h(t)=-16 t^{2}+64 t $$ a. Write an equivalent factored expression for the function $h(t)$ by factoring $-16 t^{2}+64 t$ b. Find $h(1)$ by using $$ h(t)=-16 t^{2}+64 t $$ and then by using the factored form of $h(t)$ c. Explain why the values found in part (b) are the same.

Problem. 30. An object is thrown upwards, so it's height and feet is why so wise and feet and then x seconds after being thrown. So we know X isn't seconds described by the function y equals negative. 16 X squared was 48 x was 64 on dso apart a breast to evaluate what X equals two and explain what it means. So we plug in X equals two into our function. We'll get negative 16 times four plus 96 plus 64. Sure, you cool negative 64 plus 96 plus 64. We see that these 60 fours can cancel negative in a positive. So we get 96. So we know that it will be 96 ft when it's two seconds. Part B was the height of the object three seconds after a stone. So we plug in three to our function for X, just do some algebra. We get negative. 16. 3 squared is nine plus 48 times 3, 144 was 64 when he got 16 times nine. It's always okay to do your work over on the side. Just 1 44 is well So we get negative 1 44 plus 1, 44 plus 64 and these cancel. So we get it's 64 this is a feat. It's 64 ft after three seconds. Finally, part C, whereas to evaluate it when X equals four. So we get negative. 16. I'm plugging. Four. Endure algebra. Negative 16 times four squared, which is 16 plus 48. It was four, which is 1 92 plus 64. And then we get negative 16 times 16, which is 2 56 plus 1 92 plus 64 which is zero. So does 0 ft. It is on the ground four seconds after.

So let's look at the equation. H equals negative 16 t squared plus VT. Now we can solve this equation fourty, but because we have an extra variable V, our answer will be in terms of me. So we're gonna start by putting zero equals native, 16 t squared class B T. And then we can go ahead. And remember the quota reform that this X equals minus B plus or minus the square root B squared rest for a C all over two. A. So is the coefficient of T squared, which is negative. 16 based coefficient of tea, which is V and C is the constant which, since we don't have one, will default to zero. It's good substitute into the form that we're gonna have minus V classical. Remind us the square root of the squared minus four times a time. See, sometimes negative 16 times zero all over two times ice to two times NATO 16. So it's go ahead. It's a Vitus have actually close minus b plus or minus the square root. No, because we have a zero here four times, 16 time zero. This is all just gonna cancel to zero. So we're gonna be left with just the B squared over a negative 32. First we're down a little bit and we can continue to simplify. So we've X equals negative B plus or minus the square root of V squared the square and a square root council and rich left with B. I'll be overnight of 32 so we can go ahead and continue to write are two answers. So we're gonna have a negative V plus V over NATO 32 which will be zero over negative 32 which will give us an answer of zero are other value of X will be negative b my A c overnight of 32 which is negative to the over negative 32 which would be the over 16 in those terms. And those will be our two solutions in terms of B

Here we have to find a quadratic function. Why is it goes to ex care plus off BX plus off? See who's good off passes Throw the points through The points Given is one and 200 24. The second point is three and 176. Well, the third point is given by 1400 four No, to find the equation XK plus BX plus e, that's just substituted these points into the given general equation. So we will have be bless off B plus off see is equals to 200 24 when we will have X is it goes to one and why is it goes to 224 So we can name this equation as a question one? No. For the second case, we will have 90 plus off three B plus off see is equals to 176 hair we have taken X is equals to three on by physical stew. 1 76 week unnamed. This equation ist equation Don't now considering the third case. So we will have the graciousness 16. A bless off four B plus off See is equals 204 hair. We have used X physicals to four, and we have used by is equals 204. We can name this question is a question three. Now they're going to sold these three linear equations to find developed off A B and C. So for that, let's apply the operation a question toe minus off equation one. So we will have that would be bless off Will be Is it goes to minus off 48. Well, now we will use the question to wear any question three on we will subtract aggression toe from a question three. So we will have three minus off to So we will have the question as 70 bless off B is equal to minus off 72 so we can name these equations as Equation four and Aggression five. No, we will solve these 14 50 question so we can write these questions as a day plus off. Toby is equal to minus off or did. And so money plus off b is equals to minus off 70. So we're going to use the operation two multiplied by question five minus off equation for So this will be calls to. We will have 14. A bless off you will be is it cools to minus off 1 40 Ford and we will have The second question is, which is fourth equation a day with the negative sane minus off will be Is it going strong? 40 eight. So cornered in this we will have six e is it goes toe minus self 96. What? We will have a physical stoop minus off 16 Now we will use this really off is equal to minus 16. Put it is a close to minus off 16 in equation five. So from this we will have seven and don't minus off 16 plus off. B is equals to minus off 70 too. So this will be close to minus off 112 plus off B is equals to minus off 72. Or we will have be physical stupor 40. No, we have the value off B and we have the value off. See? So Louisville substitute the value off the value off be and see in equation first. So from there we will have minus off 16 plus off 40. Bless off. See? Is it cool? sto minus off sounded or 224 224. So from here we will have C is equals to 200. Now we have the value off A, B and C so substituting these values in duty givens and early questions so we will have the question is why is equals to minus off 60 next year? Bless off 40 x plus off 200. No considered in the but be so here. Here we have to find developed off by when we have X physical stoop. Five. So this will be close to Why is it goes to minus off 16 into five. Scare Bless off 40 Endo five plus off 200 and this will be close to right is equal to minus off 400 plus off 200 plus off 200 Vigen plays that viable because to Zito So we have developed off way attacks Physical store five. Now we are asked that what does this result means? So this result means that means that the bowl, the ball but it is the rich is the ground after five seconds, so the ball will reach the ground after five seconds

So if we're looking at 2.5 seconds, which is the X value? So let's go and put 2.5 into the X here to both exes and let's solve this. Let's go to simplify evaluated. So 2.5 squared. So your 2.5 squared is going to be 6.25 And let's multiply that by that negative 16 as well. This is gonna give us negative a negative 100. Now it's mostly that 60 times that 2.5, which is gonna give us 150 and then that plus 80. So let's add all these together. And when we do that, negative 100 plus 1 50 is 50 and then add another 80 is 100 30. So what this means is filling in the blanks. It would be if if 2.5 seconds have, um, have gone by. We're talking about an object that is 130 feet in the air