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For the question $f(X) = X^2 + 4KX + (3+11K)$, where $K$ is a constant. given that $K = 1$, Sketch the graph $y=f(X)$, showing the coordinates of any point at which the graph crosses a coordinate axis. )" I get

... $(x+2)^2 - 4^2 + (3 + 11)$ which is equivalent to $(x+2)^2 + 10$ and i get $(-2, 10)$ for the minimum coordinates, however i do not get how the $-2$ value was achieved for the first coordinate value.

Display Name
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Dan
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  • It is the x coordinate for the vertex (in your case a minimum indeed) which can be achieved by performing $-b/(2a)$, a well known formula to find the axis of symmetry of a parabola of the form $y=ax^2+bx+c$ – imranfat Sep 17 '16 at 16:27
  • How did you get $(-2,10)$ for the minimum coordinates? What part of that method did you not trust? – David K Sep 17 '16 at 16:57
  • If you know that the minimum value of $f(x) = (x+2)^2 + 10$ is 10, then what input value for $x$ would give you an output of 10? – Display Name Sep 17 '16 at 17:01
  • Sorry i don't understand. Could you please explain further ? thanks. – Dan Sep 17 '16 at 20:27
  • @DavidK well i know the coordinates are (a,b) where b = +10 but i dont understand how (x+2)^2 = -2 for a. I done this through a walkthrough tutorial on youtube thats how i got the first answer. – Dan Sep 17 '16 at 20:29
  • It sounds like you found a method that works but the explanation of why it works either was missing or did not satisfy you. That's a reasonable thing to ask about. It might help clarify the question (by showing what you did not have trouble with) if you edit the question to show the steps you followed (based on the youtube tutorial). You might also look at the answers to http://math.stackexchange.com/questions/325164/help-in-understanding-quadratic-equation -- do they help? – David K Sep 18 '16 at 00:21
  • @DavidK so would the answer be (-2,10) because -16 + 14 = -2 , and (3 + 11) which is 14 then use -4^2 to get 14 - 4 = 10 ? Is this correct ? And if so, why was this done for the second coordinate 10 ? Thanks. – Dan Sep 18 '16 at 04:26
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    It looks like you accidentally squared $2$ twice to get $(x+2)^2 - 4^2 + (3 + 11)$. It should have been $(x+2)^2 - 4 + (3 + 11)$, in which $4 + (3 + 11) = 10$, so when you wrote "equivalent to $(x+2)^2 + 10$" you had a correct formula. It looks like Mick explained the $-2$. – David K Sep 18 '16 at 12:55

2 Answers2

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The reason why we perform the completing square trick is to change the original to the form $y = (x+h)^2 + k$ so that we can deduce the minimum of y.

Since the squared term is always positive, the minimum value of it is 0. This occurs when x = -h. In your case, h = 2.

Answering part

Performing completing square, we have $y = x^2 + 4x + \4 - \4 +(3+11) = (x+2)^2 - 4 + (3+11)= (x+2)^2 + 10$.

The -2 comes directly from setting $x + 2 = 0$.

Mick
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  • so would the answer be (-2,10) because -16 + 14 = -2 , and (3 + 11) which is 14 then use -4^2 to get 14 - 4 = 10 ? Is this correct ? And if so, why was this done for the second coordinate 10 ? Could you please show your working too so I understand this much better? Thanks. – Dan Sep 18 '16 at 04:35
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    @Dan See the answering part. – Mick Sep 18 '16 at 04:51
  • So where would the + 10 come from ? Thanks. – Dan Sep 18 '16 at 05:10
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    @Dan When x = -2, y attains its minimum and $y_{min} = (0)^2 - 4 + 3 + 11 = 10$. – Mick Sep 18 '16 at 05:12
  • Thank you so much I understand it now. – Dan Sep 18 '16 at 05:18
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Observe that the minimum value of $(x+2)^2$ is 0 at $x=-2$, since squares are non negative.

Then the minimum value of $(x+2)^2 + 10$ is 10, achieved also at $x=-2$.

yoyostein
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