MYTUTOR SUBJECT ANSWERS

466 views

How do I find the minimum or maximum of a quadratic function?

              Take a general function f(x) = ax²+bx+c

   First thing to do with every problem is sit back and think a bit: there are usually several ways to solve a problem but one will likely be easier and shorter.

 In this case we don’t want to start differentiating the function and looking for points where f’(x)=0 . This method does work, but it can involve quite a bit more fiddling around than other ways.

 Since this is a quadratic this problem can be solved by simply ‘completing the square’ - we need to get our function into the form a(x+d)²+e . With this done we can simply read off the minimum or maximum e reached when x+d=0 (x=-d).

 To get the quadratic into the right form we just need to follow a few simple steps :

1) Factor out a -> ax²+bx+c = a(x²+b/ax+c/a)

2) Add and subtract by a number d to such that the form ‘x² + 2dx + d²’ is present:

= a(x²+b/ax+(b/2a)²-(b/2a)²+c/a)

3) Rearrange the ‘x² + 2dx + d²’ form into ‘(x+d)²’

= a((x+b/2a)²-(b/2a)²+c/a)

4) Multiply out by a to get the final expression

= a(x+b/2a)²-a(b/2a)²+a(c/a)

= a(x+b/2a)²+(-b²/4a+c)

  So here the minimum or maximum is given by (-b²/4a+c). Whether it is a maximum or minimum will depend on the sign of a.

  As you can see the final expression gives us d= b/2a and e= (-b²/4a+c). These values are very quick and easy to derive so do not need to be learned by heart.

Max V. A Level French tutor, A Level Chemistry tutor, A Level Geograp...

1 year ago

Answered by Max, who has applied to tutor A Level Maths with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

295 SUBJECT SPECIALISTS

£20 /hr

Jigar P.

Degree: Mathematics (Bachelors) - Bath University

Subjects offered: Maths, Further Mathematics + 2 more

Maths
Further Mathematics
Economics
-Personal Statements-

“Hi, I'm currently a second year student at the University of Bath, studying Mathematics. I am looking forward to meeting you :)!”

£22 /hr

Greg C.

Degree: Economics with Econometrics (Bachelors) - Exeter University

Subjects offered: Maths, Further Mathematics + 2 more

Maths
Further Mathematics
English
-Personal Statements-

“I consider myself a very extrinsic, hardworking person. I like to get involved in as much as I can and this is another opportunity to try something new! At the University of Exeter I’ve involved myself in a vast range of societies inc...”

MyTutor guarantee

£20 /hr

Wesley M.

Degree: Mathematics (Masters) - Bristol University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Chemistry

“I am a current Mathematics student at The University of Bristol. Maths can be a love-hate sort of subject for many people, but with my absolute love and enthusiasm for both maths itself and the idea of inspiring others, I hope to be a...”

About the author

£20 /hr

Max V.

Degree: Geology (Bsc) Hons (Bachelors) - Edinburgh University

Subjects offered: Maths, Physics+ 6 more

Maths
Physics
Geology
Geography
Further Mathematics
French
Chemistry
Biology

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

how do integrate an equation with a surd or a fraction?

Why is it that sin^2(x) + cos^2(x) = 1?

How do you find the area between two lines?

Why does ln(x) differentiate to 1/x ?

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok