in-9-14-a-sketch-the-graph-of-each-function-b-from-the-graph-estimate-the-roots-of-the-function-to-5

Question

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}-2 x-1
$$

Foster Wisusik · Accepted Answer

Okay, This question asks us about the roots to this quadratic equation. So the first thing it wants us to do is try and find these routes graphically, which I have. The graph over here ends in black. Those points are where f of X equals zero. So these air, where we have solutions to the equation f of X equals zero so we can approximate the X values that will give us zero. So we get X values of approximately negative zero point for and 2.4 when we do this. So now we want to solve it. Exactly. So we sent our equation f of X equals zero, and then we&#x27;re going to complete the square. So completing the square turns a quadratic X squared plus B x plus C into a new quadratic a Times X plus d squared plus e where de is equal to be over to a and E is equal to b squared, see minus B squared over for a So now let&#x27;s find D and E for our quadratic. So de is equal to be over to a which is negative two divided by two or negative one and then e is equal to C minus. D squared over for a and that&#x27;s just equal to negative one minus or sorry. This should be a B squared one minus B squared, which is four divided by for a and that is just equal to one minus 4/4 or one minus one for negative one minus one, which is negative. Two. So now are transformed. Equation is X minus one quantity squared minus two is equal to zero. So now we can add to to both sides to get X minus. One squared is equal to two. And then if we square root both sides, we get X minus. One is equal to plus or minus the square root of two. And now we can split up our solutions. We have X minus one is equal to the positive squared of to and X minus one is equal to the negative squared too. So in the first case, we have X equals one plus the squared of to giving us our positive solution. And then we have X equals one minus squared of two, which gives us our negative solution. And that&#x27;s all right

Aditya Balasubramanian · Answer

with a function X squared minus two X minus two and you want to be able to draft this in part? A. So if we start dissecting the ocean Comptel that the coefficient of the term in the standard form is a positive one, which means the graphs should be opening up. You can also start by finding me pretexts. So we find the Vertex, which means the exit value in standard for the negative be over two. A second term is be so would be to over two times the pressure. Just another two. It&#x27;s a naturist one. And in order to find the vibe value, why value of the vortex of are probably here. We have to plug in the extra right so it could be a tough one. Was one squared minus two minus two inches. Just negative three. So it&#x27;s Write that down and go over when we have a 1 to 1 scale honor axes right here. So just to go over one and go down the re So if you start graphing on either side of the Vertex will notice our problem&#x27;s gonna be symmetric. So let&#x27;s start doing that right. Go over one sinner. I just be two because one plus one is two. We end up with two squared minus two minus two, which is just a negative to do the same thing on the other side, from zero equals zero squared, minus zero minus two. Negative. Too true. Those children that. Then we do that again. Right? Three. I was nine. My six minutes to just run. Same thing. Another side of negative one one plus two minus two blood. So you born those two coordinates and then finally, it&#x27;s good to have two or three and then decide. I&#x27;m gonna go for the reciever connected on the graph. But generally, as long as you have two or three and other side of the Vertex, you&#x27;ll be good. So it&#x27;s Shrine bidding. The four in the negative to 16 minus eight minutes to schools six for plus four months goes under six. So here, around years, mobile incident will be ableto point the proble truth in that direction. Um, I&#x27;m gonna try and sketch this right here. Forgive me from sketching, but he&#x27;s get this that direction. Same thing on our side. No, you fix that, actually try and draw differently. there we go. And in parts be we have to estimate the roots of the function to the nearest 10. So if you look at where the function is equal to zero, we can try and estimate that to the 10th. So since there is 1234 four lines in between each number here, we can show that is 2040 60 80. So, for example, in this case, on the right side, it will be two point around two point, uh seven. Because it goes in between the six and a mommy Got a point there. It will be 2.7 zero this in between 2.62 point eight, even though there&#x27;s no distinct line there. So that&#x27;s one the other side. It should be around negative point eight. We&#x27;ll call that a negative 48 comma zero, so the routes will be around 2.7 and negative 0.8. Throwing the X equals there. So in part C, now we actually have to find those roots. In order to do that, we actually got to get the standard form into a quadratic, so it&#x27;s used a quartet formulary. So say X equals negative B plus or minus the route B squared minus for a C all over to A which moves hold on extension on Ranger right there. Which means that it equals two plus or minus the route off or minus four times one times negative too. So plus eight all over two. Which means it should the to plus or minus +23 over two, which he waas one closer, minus root three and what is it?

Foster Wisusik · Answer

Okay, This question would like us to find the solutions to this quadratic equation. First, it wants us to find a graphical estimate for these solutions. So I grabbed this in Dismas here, and we&#x27;re looking for the points where F of X is equal to zero. And that&#x27;s just where it&#x27;s crossing the X axis. So our solutions air here and here and zooming in the approximate X values of these solutions Negative 0.6 for ah, one more to the right and then negative 3.4 for the left. But now we want to complete the square to find the exact values of thes roots. So we want to set our quadratic equation equals zero. And now, to solve this, we&#x27;re gonna use the complete the square method where we convert X squared plus B X plus C into the form a times X plus d squared plus e and de is a constant that&#x27;s be over to a and E is a constant that c minus b squared over four a. So now we just need to find the DND values for the specific quadratic so d is equal to be over to a which is four over to purchased two and then e is equal to C minus. B squared over for a which is tu minus. Be square to 16 over two A. So we get to minus eight, which is negative. Six. So now if we do this you or on my bed this should be a four on the bottom, which is negative. Two. So now we can rewrite this in our completed square form of X plus d quantity squared plus e, which is minus two. And now this is much easier to solve. So now we have X plus two quantity squared is equal to two when we add to double sites. So taking the square root we get X Plus two is equal to plus or minus the square root of two or splitting our solutions we get X Plus two equals square root of two and X plus two equals minus square root of two and then solving we get X equals negative two plus route to and X equals negative too minus root too. And these are the exact forms of our approximations we found from the graph

Aditya Balasubramanian · Answer

All right, so here we have the function X squared minus 10 X plus 18 and we are shown base a graph. We have to sketch the career here now, So let&#x27;s start up a And if we analyze this coefficient of the eight terms or one, you know it&#x27;s gonna open up and our Vertex to start off with will be. But again, that X equals negative b over to a formula, which is 10 over to just wait or sorry, five ages 10 minus two. So, yeah. Um, no. I have to find the white corn it at five. Peoples 25 minus 50 plus 18 which is Meggett to 25 plus a team, which is just a negative seven. So the Vertex just five negative seven and I doubled up the y scale here to account for the larger numbers here. So be a little easier for us to do that. And let&#x27;s also thinks our ex killer ate here because Arbor, Texas, five cents will be able to do numbers on the right side. Most we fix it, so let&#x27;s make are right side over here at 10. In her left side. Uh, zero. So we go back to what we get. Five negative. Seven of the pretext. It&#x27;s a negative. Seven will be around right over here. But the critics was started by the numbers around it. F of six equals 36 minus 60 plus 18. That&#x27;s six, if of four equals 16 minus 40. Native 24. That is closely team once again. Negative six. What about the ***? Right there. Um, put it in. I&#x27;m not inside. Let&#x27;s move on. F of seven equals 49 minus 70. Negative 21 plus 18 which is in negative three. Same thing for the other. Side nine minus 31 Because it in committee three. So right here will be negative. Three. We still have to get above that on XX, sis, so we can actually do part B. So we have to keep chugging along with these coordinates plotting. So let&#x27;s try out eight and two, right? 64. Mine&#x27;s a *** 16 plus 18. And we finally got to the positive side for minus 40. All right. Minus 20. That is you, um, 16 plus 18 which is to So I got a two on either side right here, and let&#x27;s plot. And after we stay, catch this out. We have two points here. These points are gonna end up being useful for part because now we have to estimate the roots of the graph. And when we estimate we aren&#x27;t actually finding the exact routes that is, we&#x27;re trying to find something that&#x27;s within the range of I would say it, like two or three tents would be good enough, since parting is inexact science. So let&#x27;s see what we end up with we get. This is probably around. I so remember that we have, ah five in the middle, so we have seven points. Call that 7.8 with reasonable reasons. A 7.8 zero on the right side and then or three So should be a two point to North Side would be also pretty reasonable. So roots X equals to put two and seven point eats part. See, we actually have to find those roots and we go back to the quadratic formula to find the exact radical for it. So X equals negative B plus or minus the root of B squared minus for a c oh, over to A and we end up with 10 plus and minus the route of 100 minus four times 18 four times 18 72. So it&#x27;s 100 minus 72 all over. Two times lunches, too. She So that&#x27;s 10 plus or minus the roots of 28 over to we can simplify that radical. Route 28 is the same thing as true. Seven not equals five plus or minus. Oh, my clear five plus and minus Route seven and that is it.

Foster Wisusik · Answer

Okay, This question asks us about this quadratic function so first and wants us to graph the quadratic and estimate its roots to the nearest tent. So I went ahead and drafted in Dismas here. So this is our function after vax, and we want to know where f of X is equal to zero. So let&#x27;s just where it crosses the X axis. So we see two points one here on one here and we see that are approximate X values free to these routes is for the first route. That&#x27;s around 0.8 and the other route is around 5.2. So that&#x27;s our estimates for the solutions. But we can actually find the solution exactly by completing the square. So we start with X squared minus six. X plus four equals zero, and we want to solve this. So we&#x27;re going to complete the square by turning X squared plus B X plus c into are complete the square for him. Where d is equal to be over to a and E is equal to see minus B squared over four a. So let&#x27;s find D and E S. O. D. Is equal to be over to a which is negative. Six over to or just negative three on then E is equal to C minus. B squared over four A, which is four minus. B squared is 36 divided by four A just four. So we have four minus nine, which is negative. Five. So that means we can transform our equation into X minus three squared minus five equals zero. So if we want to solve this equation now, this is something that&#x27;s much more doable for us. I&#x27;m sorry. Minus five equals zero. So we just acted five. We just add five to both sides here and get X minus three is equal to five X minus two squared. So then we just take a square root and get X minus. Three is equal to plus or minus the square of five. So that means we have two options. We have X minus three equals positive square 25 and X minus three equals negative squirt of five. So we have X equals three plus Route five, and we have X equals three in minus root five as our solutions. And if we punch these into our calculator, we&#x27;ll see that these are

Aditya Balasubramanian · Answer

case are right here we have the function Experiment six Experts six. Let&#x27;s start by sketching the graph of this, and you&#x27;ll notice that if you take a look at the standard form, you&#x27;ll see that the coefficient of the terms of positive one. You know that the graph is there if we&#x27;re gonna have to open up, so we&#x27;ll start by freeing the Vertex. So the Vertex should be X equals negative B over to A, which is the second term times negative one, which is six over two times the first term, which is just a one 76 over to, and that&#x27;s a positive three. Let&#x27;s play that in to find the white term of the vortex and after three equals nine minus 18 plus six, which is another 32 This time it&#x27;s negative, and we have what seems to be a 1 to 1 scale right here. So we graft that. We end up three. Make it two three. So let&#x27;s start by finding some of the points on either side. Let&#x27;s try a four so recalls. One is for right, so I&#x27;ll be 16 minus 24 plus six. Just, um, negative eight plus six inches and negative, too. Um, two equals four minus 12 plus six. Another negative, too. So skipping two in there f of five equals 25 minus 30 plus six. So be one same thing on the other side. Except it&#x27;s one this time one minus six negative five plus six, which should be just a one again. So one in either said. I think two points and other side&#x27;s good enough structure because I don&#x27;t have any more room on the right side of the graph. Cells just sketched this right now. We fixed that. So it goes through that first point right there. Student, other side as well. First, it struck the area. Now let&#x27;s move to the other side. And now, if we estimate where this function is zero as private, fastest by the roots, we can do that. And if you&#x27;ll notice that since we have 2040 60 80 in terms of the 10th 2468 in between each of the whole numbers on this on this scale, we can estimate so on the left side it will be around one 0.2 euro and on the right side, this will be a little harder because of the arrow, but it should be around a nine over there. So since seems to be in between E. T and 100 should should be a 34.9 your own that right side. So roots X equals 1.2 and 4.9. We make that look less like currencies, then part See, we actually have to find the roots. In order to do that, we have to find our radical form, and we have to move this into apologetic form. So we have to use our handed any quadratic formula, which states X equals negative B plus or minus the route B squared minus or a C all over two A. That means it&#x27;s negative one times negative. Six. Just six. What&#x27;s the minus? The roots 36 minus four times one times six minus 24 All over two. The chief was 6% minus 23 over. To which equals three plus or minus. Recruited me the vial those numbers to to simplify. And that is our answer

Problem

In $9-14 :$ a Sketch the graph of each function. …

Problem 13 Medium Difficulty

In $9-14 :$ a Sketch the graph of each function. b. From the graph, estimate the roots of the function to the nearest tenth. c. Find the exact irrational roots in simplest radical form.
$$
f(x)=x^{2}-2 x-1
$$

Answer

a) graph
b) The roots of $f$ ROUNDED TO THE NEAREST TENTH are the $x=-0.4$ and $x=2.4$ as shown in the pic.
c) $x=1+\sqrt{2}$ and $x=1-\sqrt{2}$

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Ann Xavier Gantert

Algebra 2 and Trigonometry

Alayna H.

Kayleah T.

Caleb E.

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Absolute Value - Example 1

Absolute Value - Example 2

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a) graphb) The roots of $f$ ROUNDED TO THE NEAREST TENTH are the $x=-0.4$ and $x=2.4$ as shown in the pic.c) $x=1+\sqrt{2}$ and $x=1-\sqrt{2}$

Algebra 2 and Trigonometry

Alayna H.

Kayleah T.

Caleb E.

Maria G.

Absolute Value - Example 1

Absolute Value - Example 2

Ann Xavier GantertAlgebra 2 and Trigonometry

Alayna H.

Kayleah T.

Caleb E.

Maria G.

a) graph
b) The roots of $f$ ROUNDED TO THE NEAREST TENTH are the $x=-0.4$ and $x=2.4$ as shown in the pic.
c) $x=1+\sqrt{2}$ and $x=1-\sqrt{2}$

Ann Xavier Gantert

Algebra 2 and Trigonometry