Sketching a graph of q(x)=x^2-8x+19 and finding minimum

[ReneO923]

ok so i had an exam with a question I was completely stuck on for about twenty minutes. I totally blanked out, I would like to know the steps i should have followed because i didn't even know where to start.

QUESTION

sketch the graph of q(x)=x^2-8x+19 and label its minimum. what formula do i use to even get started?

 

[ksdhart2]

You don't really need a "formula" as such to solve this problem; all of the instructions are in the question. Most of the time, when sketching a function, it's helpful to first make a table of values. My teachers in high school called it a "T-table." Yours may have called it something different. Either way, the process is easy. If we let q(x)=y, the table will look something like this:

  x  |  y
-----------
 ... |
 -2  |
 -1  |
  0  |
  1  |
  2  |
 ... |

So, if x=0, what is the value of y? If x=1? Etc. You're asked to find a minimum value, so a good first strategy might be to plug in successively smaller numbers and see what happens. Where does that lead? Eventually, with enough data points, you should begin to see a pattern and thus can deduce the answer to your question.

 

[pka]

QUESTION sketch the graph of q(x)=x^2-8x+19 and label its minimum. what formula do i use to even get started?

Complete the square, then \(\displaystyle q(x)=(x-4)^2+3\). Explain why \(\displaystyle (4,3)\) is the minimum.

 

[stapel]

Q: Sketch the graph of q(x)=x^2-8x+19 and label its minimum.

What formula do I use to even get started?

You could memorize the vertex formula, or use the completing-the-square process they taught you. (here) To refresh, check in your book's index (in the back), under something like "completing the square", "quadratics", or "vertex, finding".

 

[Ishuda]

ok so i had an exam with a question I was completely stuck on for about twenty minutes. I totally blanked out, I would like to know the steps i should have followed because i didn't even know where to start.

QUESTION

sketch the graph of q(x)=x^2-8x+19 and label its minimum. what formula do i use to even get started?

Just to add my 2￠. Something you should remember: Given the function
f(x) = a x² + b x + c
with a not equal to zero, let
v = \(\displaystyle -\dfrac{b}{2a}\)
Now, if a is positive f has a minimum and if a is negative f has a maximum. That minimum/maximum occurs at v.

Something else I have found extremely nice to remember is the f(x) above can now be written as
f(x) = a [(x-v)² - d]
where
d = \(\displaystyle \dfrac{b^2 - 4 a c}{4 a^2}\)
This is completing the square.

We can now graph our function by just graphing x², then translate (possibly twice) and stretch/shrink that graph. Actually though I (sort of) use ksdhart2's method now except I put down my x's in the table as positive distances from v realizing I will get the same answer as for negative distances from v.

Your question is made easier in that it is slightly simplified, i.e. a = 1 and b is 'doubly even' [divisible by 4]. Thus you would be working with integers for v and d and your graph is simply a graph of x² shifted to the left or right and then shifted up or down..