<p>Explain why: No matter what the value of x, x²-8x+22 is never less then 6.</p>
<p>Find the vertex of the graph. Use x = -b/2a to find the x-coordinate of the vertex. You will get 4. Plug that into the equation, and you should get 6 as y. The vertex of the parabola will be (4,6).</p>
<p>Since "a," the leading coefficient of x^2, is positive, it tells us that the graph will have a minimum at the vertex (4,6). In other words, y will satisfy the interval [6,inf] for all x's.</p>
<p>I dont know if im explaining well but let's first plug in some sample numbers. Try -10, -1, 0, 1, 5, 10. All the numbers are over 6.<br>
Why so? Well notice the x^2 always makes the first term positive. Then that positive number has 22 added to it. If x is negative, we are simply adding more to the x^2 and the 22 when we subtract 8x because a negative and a negative is a positive. If that x is positive, we are taking away but the number we are taking away can not make the x^2 and 22 less than 6 because no matter what number x is, there is no way taking 8 times the number away from the number squared and adding 22 is less than 6. sorry if thats confusing.</p>
<p>EDIT::Disregard my nonsense, the above poster gave a better response. i forgot that -b/2a thingy</p>
<p>Because that's what my calculator tells me.</p>
<p>On a more serious note, to find the minimum value of a function we can use various methods to obtain it. The one I prefer to use is the derivative, basically finding where the slope of the line is = to zero. For example, in this case, the derivative of the given function is 2x - 8, which, when set to 0, gives x = 4. When x = 4, y = 6, the minimum value of the function.</p>
<p>Does that answer your question, or are you really asking why a minimum value exists in your function?</p>
<p>if x is a negative number, the result is sure to be bigger than 22, so cross out all the negatives.
if x is 0, you try it, no, so cross out zero.
if you have a positive number, x^2 will be always bigger than or equal to 8x until x=8. so corss out 8~infinity. now you have 1~7 to try out.
1: 15
2: 10
3: 6
4: 6
5: 6
6: 10
7: 15</p>
<p>i know, you asked 'why', but i had to try this myself. haha..its pretty cool. its a pattern! 15,10,6,6,6,10,15...COOL :)</p>
<p>Your way works fine too. My way was more of a graphical method, while yours was more of an algebraic/intuitive method.</p>
<p>if you try graphing it probably on the graphing calculator the graph would show that the range is y is greater or equal to 6</p>
<p>mines more algebraic/intuitive because my level of math intelligence does not exceed algebra2. shame on me :(</p>
<p>i still didnt explain WHY....</p>
<p>wow that was alot of responds thanx guys</p>
<p>oh but, it sort of does. if you think about parabolas (pair-a-bowl-as according to my teacher), that patten of 6s you found is simply where the bottom/minimum is. your numbers look like a parabola in nature.</p>
<p>I'll go with lil_killer129's solution, that's the best way...</p>
<p>This is another way:
You take the derivative of it, which is 2x-8. Find the local min, which is at x = 4, y= 6.
Therefore, the graph can never be less than 6 ^_^</p>
<p>the pattern is because its a parabola...</p>
<p>y=eq</p>
<p>2nd calc, min</p>
<p>trace and click, trace and click</p>
<p>enter</p>
<p>4,6</p>
<p>6 is y</p>
<p>there you go</p>
<p>guys if hes asking about basic functions, why are u using calc</p>
<p>btw, dont be intimdated by derivatives an whatnot, for a polynomial you simply multiply each term by its exponent, then knock exoponent down 1 degree.</p>
<p>and that is your slope...were the variable is your x value</p>
<p>thats why y=mx+b has slope m always</p>
<p>dy/dx (thats der notation)=(m<em>1)(x^(1-1)) + (b</em>0)=m</p>
<p>use it for anything really, but later you will have to learn derivatives of special things, sinx ln[x] etc</p>
<p>I still think derivative is the best way to do it, unless you can graph the function by hand faster than taking the derivative. Sorry but I just dont like calculator language. ^_^</p>