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Channel: Школа Пифагора ЕГЭ по математике

математика

трушин

егэ

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today we will analyze a problem about which

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many schoolchildren have already asked me, this is a

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new problem in the demo version of 2022 and the

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profile is everything as you like the problem is

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simple and many understand that it is

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simple, but I want to not only be able to

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solve it, solve it, not waste it there’s

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too much time on it, so today’s project

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we’ll talk about how to solve it exactly even if

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you know almost nothing and how to

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solve it if you have a good feel for

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mathematics without spending even five minutes, well, it’s

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complete, well, you’re at most minutes, okay,

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the problem is like this, there’s a graph of a parabola,

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that is, it’s known that it was time that it was

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even known that the coefficient is integer,

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although this is not really needed for the solution, and

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they ask, having this graph, how to

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understand what value our function has at the

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point minus 12, that is, what f is equal to

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minus 12, how to solve this can be solved in

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very different ways

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One of the ways is this: if

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you turn off your brain and don’t think about anything, then

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the task comes down to this: you need to

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look carefully to find

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some 3 integer points,

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well, for example, this is it, that is, the point at

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which both x and y are integers. Here .

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Well, I don’t know. Yes, it’s usually

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drawn in such a way that these points are clearly visible and

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what we see is that f from minus 4

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to grows 0 minus 1 minus 2 minus 3

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minus 4 f from minus 4 is equal to minus 3, that

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is, if you substitute minus 4 instead of X, you

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should get minus 3

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16 a

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minus 4 b + c should be minus 3,

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that is, if we take

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minus 4 as xo, we should end up in -3, so here’s the

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next one. with x equal to minus three,

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it should be -2, and

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minus three b plus c is equal to -2, and

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here is the third. with x equal to -2, the result should

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be one 4 and minus 2b plus c

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equals 1, in fact, essentially at this

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moment the problem is solved because we

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received a system of three linear

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equations with three unknowns and many

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of you, having become fifth-sixth grade, have

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already solved this well and all we need is

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to find a pc if you found a pc then you

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know what your function is and knowing the

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function, substitute x equals minus 12

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you got the answer, well, then whoever

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knows how to solve quickly solves the system

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there in a minute and a half who

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solves slowly spend about five minutes, but overall

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it’s clear that essentially the problem has been solved, well,

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let’s discuss how to do it, for example, you

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can take the last line and subtract it

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from the first two, then c will go away, but if

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you subtract 3 from the first line, then it will be

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12 a

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minus 4b minus minus 2b, that is plus 2 b

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it turns out minus 2b c it took equals

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-4 so if you subtract this line from this line

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you get 5 and

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minus b c it took equal to -3 that’s

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the system, well, let’s cut it by two here,

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yes, this is the same thing as

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6 a minus b is

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equal to minus 2, well, then look at

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these two lines, if you subtract this from this one, you

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get a equals 1, if a equals 1,

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substitute either here or here where

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you prefer, well, let’s

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get 5 minus b equals minus 3,

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then 8 equals b husband b equals 8,

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so, knowing a and b, we substitute them

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here, for example, and find c, it turns out 4

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minus

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16 plus c equals one, that is, minus 12

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plus c equals one, well, that means we’ll

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write c equals

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13, it didn’t take long to find all the values of a

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b c, which means ours function f from x is equal to x

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squared plus 8x plus 13 means f from

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minus 12 is equal to

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144

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minus 8 by 12 is 80 and 16 minus 96 + 13

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well, let's do the math here it turns out

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48 plus 13 is 61 that's

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all, that's the answer if it's nothing don’t

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think, then this is quite a solution,

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especially if you don’t have goals of getting

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100 points on the Unified State Exam, if

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7580 points are enough for you, you can easily

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spend 5 10 minutes on this problem there,

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no big deal, you have a lot of

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time, so you can do it like this it’s

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great to solve, but in fact, if

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you think a little, you can greatly

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shorten the reasoning, even in this

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solution you can shorten the reasoning,

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firstly, look what you can understand if

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you look closely at this parabola, and here’s how

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it’s structured near the top, so draw the

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coordinate axes here for yourself just imagine that

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the parabola is here and you can immediately

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see what kind of parabolas they are when you

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approach one, you get

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one until you move away by two, you get

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four, that is, this is an ordinary parabola y

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equals x square which

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has been moved a little, but if you understand that when

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the parabola moves along the coordinate

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plane your leading coefficient does not change,

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then you immediately know that

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the coefficient of the original parabola is the same

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as this one, and this one is obviously

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equal to one to 1 in one, the value of

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one means y is equal to x squared, that is,

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you know

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that a is equal to one, and knowing this is enough

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level 2, yes, for example, you take

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some two points there, this one and

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this one, you also know that and equal to one

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you get a system of two equations with

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two unknowns

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and you don’t have to look for the third point, but

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let’s not do that, but it’s clear

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that knowing that a is equal to 1 is already much simpler; the

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second step, which can also simplify

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if you know that the vertex of the parabola

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is at the point with the coordinate minus b

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on 2a and already knowing that a is equal to one, you

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get that the vertex is at minus

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b in half, but the vertex of our parabola

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is located in minus 4 and this means that

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minus b is equal to -4 in half and that means b is

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equal to 8 we knew that b is equal to 8 and is equal to

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one to find c one equation is enough,

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for example this if you

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know that a is equal to 1 and b is equal to 8 and it

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works out right away 16 minus 32, that is,

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minus 16 plus c equals -3, which means c equals

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13, that’s all, yes, that is, you know a little

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about it was time that the leading coefficient

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can be determined by how it

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behaves near the top before and there the coefficient

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b can be determined by where is

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the vertex all a and b you know, all that remains is to find

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the goals, just substitute just

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one point that you

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like best, this is the solution

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for those who like to solve systems either from

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three equations or from 2, well, or from

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1, but in fact if you

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understand a little more about the parabola, life

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can be even easier, watch it for those who

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understand well how parabolas are structured,

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especially for those who watched my old

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video about how to

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build a parabola using the equation of a parabola,

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understand the following, what

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was this pair it turned out from the original and how

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it turned out it turned out 2 gom by 4 to the

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left and three down and we know what

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happens from the parabola to with the equation it

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was time to move it like that,

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look if you had and rick is equal to x

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square to move it four to the left

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you need

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to do it like this, so that when x is equal to

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minus 4, you get 0 to if you moved the parabola

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four to the left, that’s what you

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did with it, and if you also moved it

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down three, then you need to subtract 3, that is, in

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fact, if you understand how it

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was time to arrange it, then you immediately say

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that the equation of this parabola is like this,

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and if you understand this, then

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just substitute minus 12 and you

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get that minus 12 plus four

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squared minus 3, that is, minus 8

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squared 64 minus 361, that is, if a little -you just

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understand that the new pair was just an

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ordinary parabola y equals x square

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which was first

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shifted but 4 to the left and then lowered

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three down, then you immediately know that it has

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this equation, which means you can

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answer the question what value will be

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if x is equal minus 12 but for those who have a

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good imagination, you can

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do without any formulas at all, look

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if you understand that this pair was, yes,

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this is an ordinary pair, it was shifted and

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look here, what can you say, you

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see, here we have drawn this

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new local system coordinates, but

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in this new local coordinate system

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we can easily understand how the

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value works, if x is equal to 1, well, the new

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x is equal to 1, then y is equal to 1, x is equal to 2y, equal to

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4, there in minus 11 in -2 is also four, and so

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on and so on Now let’s understand that they

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are asking us what the value of -12 is, it’s

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somewhere here, far away, you can’t draw it, it

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was time to run very far away, but they

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’re asking us where -12 is here,

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asking what the value is, and

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let’s look at this new

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system of ours coordinates, what’s happening here

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at -12 due to the fact that they are shifted by 4

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here, this is minus 8, in these new

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local coordinates at minus 8, we know

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what the value is because it’s just a

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parabola y is equal to x squared at minus 8,

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our parabola has a value 64 but our pair

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was lowered down by 3 relative to

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this axis, which means in the old coordinates

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its value is three less, that is,

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61, it’s clear, but what if along this

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brand new axis it has a value of 64, then

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the pastor of the y axis has a value three less that

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out of 61, that is, if you understand a little about

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the geometry of the parabola, then you immediately understand

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that the value of -12 of the old parabolas

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is practically the same as the value

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of minus 8, this one is the simplest, it

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was time to just subtract 3 and not

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necessarily even in this form It’s up to

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you to write it, it’s easier for the majority, it’s

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probably easier to see the equation,

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especially if you’re used to writing these

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equations and substituting -12, but who

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understands that this is the most ordinary

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pair that was moved a little, then

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you can understand without explicitly looking at the equation

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that if x is equal to - 12 then y is equal to 61

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this is the problem, that is, the good problem is that

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it is solved very differently

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depending on what set of knowledge

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you have, if you all forgot about

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parabolas, then just look at the

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three points and see what they are

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values, write a system and solve this way,

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you can solve any problem,

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no matter what function you have here,

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you will have a cubic parabola, yes, but you

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will need to take four points and look and

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solve a system of 4 linear equations with

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four unknowns, it’s understandable and in

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general you won’t have to use the properties

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because that you don’t know any special properties for the

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cubic pore, but when

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it is a well-known

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quadratic function, you know how

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parabolas work, if you know how

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parabolas work, then at least

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you can quickly find the coefficients

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a and b if you know any properties

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are connected with the vertices, and if

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you think a little, immediately say the

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answer if you understand how

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parabolas work, I hope that after this video

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you will run out of questions about how to

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solve this new strange problem from games,

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and that’s all, likes, subscribe, all the way to

00:13:08

new ones bye bye guys

00:13:10

I want to say thank you to all those who

00:13:12

help and financially support

00:13:15

the channel and if you want to

00:13:17

join them then

00:13:18

click here and the third fact f from

00:13:22

minus 2 is equal to one

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