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October 22nd, 2016, 06:10 AM  #1 
Newbie Joined: Oct 2016 From: South Africa Posts: 3 Thanks: 0  Nature of Roots for Quadratic and Cubic Functions
Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that  the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to determine the values of the shift to obtain the required roots. I attach 3 example questions... They are 8.1.4, 7.4 and 4.5 Please help me find Youtube Videos/Websites or any resource that might help me understand how to approach these questions. Thanks Last edited by GardeeZak; October 22nd, 2016 at 06:13 AM. 
October 22nd, 2016, 06:57 AM  #2 
Senior Member Joined: Dec 2012 From: Hong Kong Posts: 781 Thanks: 284 Math Focus: Linear algebra, linear statistical models 
You could use the discriminant. For example, for the leftmost question, you can find the discriminant of f(x)  g(x)  k, set it as > 0 and solve the inequality.

October 22nd, 2016, 07:45 AM  #3  
Newbie Joined: Oct 2016 From: South Africa Posts: 3 Thanks: 0  Quote:
At our level we have only been taught how to use the discriminant for quadratic graphs  the discriminant for a cubic graph is not examinable  
October 22nd, 2016, 09:57 AM  #4 
Senior Member Joined: May 2016 From: USA Posts: 530 Thanks: 233 
We have no idea what math you already know. Do you know derivatives? Also, these thumbnails are very difficult to read. I have no idea what two of them say at all and must guess for some parts of the first problem. So I think the first problem starts with $f(x) = x^3 + px^2 + qx  12.$ Is that correct? And it has a local maximum at ( 4, 36). Is that correct? If all that is correct $f'(x) = 3x^2 + 2px + q.$ Furthermore, $f(\ 4) = 36 = (\ 4)^3 + p(\ 4)^2 + q(\ 4)  12 \implies 36 = 16p  4q  64  12 \implies$ $16p = 112 + 4q \implies 4p = 28 + q.$ And finally $f'(\ 4) = 0 = 3(\ 4)^2 + 2p(\ 4) + q \implies 0 = 48  8p + q \implies 8p = 48 + q.$ $\therefore 8p  4p = 48 + q  (28 + q) \implies 4p = 20 \implies p = 5.$ $\therefore q = 8 * 5  48 = \ 8.$ $Or\ f(x) = x^3 + 5x^2  8x  12.$ This involves nothing more than using the function, its derivative, and the value of both at x =  4 to solve for the unknown parameters. The next part of the problem involves nothing more than the fundamental theorem of algebra, the zero product property, and the quadratic formula. $f(\ 1) = \ 1 + 5 * 1  8(\ 1)  12 = \ 1 + 5 + 8  12 = 13  13 = 0.$ That  1 is a zero of the function was given to you. $\therefore x^3 + 5x^2  8x  12 = \{x  (\ 1)\}(x^2 + mx + n) \implies$ $\dfrac{x^3 + 5x^2  8x  12}{x + 1} = x^2 + mx + n \implies x^2 + 4x  12 = x^2 + mx + n.$ So the other zeroes are: $\dfrac{\ 4 \pm \sqrt{16  4(1)(\ 12}}{2} = \dfrac{\ 4 \pm \sqrt{16 + 48}}{2} = \dfrac{\ 4 \pm \sqrt{64}}{2} = \ 6\ or\ 2.$ Nothing after this is legible to me, but so far, none of this involves doing much more than using what you know about polynomials and their derivatives. 
October 23rd, 2016, 11:18 AM  #5 
Newbie Joined: Oct 2016 From: South Africa Posts: 3 Thanks: 0 
Hi  I'm so sorry about the unclear pics; here they are in clearer form: http://bit.ly/2eyoBdz I have done calculus with derivatives  I have not done integration at school level. I am not sure how your explanation helps me to understand the concepts I don't get to answer 8.1.4, 7.4 and 4.5 Please assist me, and thanks. Last edited by skipjack; January 30th, 2017 at 12:21 PM. 
January 30th, 2017, 12:22 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 16,605 Thanks: 1201 
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cubic, functions, nature, quadratic, roots 
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