Help with understanding Nature of Roots for Quadratic and Cu

[Zakariyya]

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

 

[Simon Bridge]

OK the questions do not ask for exact values for roots so there is not much calculation needed: it's a way of testing your understanding.
Going: $g(x)\to g(x)+k$ ... and asking about the roots, is the same as asking for the intercept between curve $y=g(x)$ and the line $y=-k$ ... for example, asking what k must be to give the g(x)+k two positive real roots, you are just looking for a horizontal line that cuts g(x) twice... or the set of such lines, and so on depending on the wording.

ie. $g(x)=x^2$ has 1 real root at x=0, you see why that is right?
A horizontal line drawn through y>0 will cut g(x) twice - do you see why?
Therefore, g(x)+k will have two real roots for k<0.

ie. $g(x)=(x-1)x^2$ has 2 roots, x=0 and x=1 (where are the turning points?)
A horizontal line drawn for any y<0 will cut g(x) once only, do you see why?
So k>0 makes g(x)+k have one root, and the root is positive.
It has three roots for a small range of values - can you see which?