Nature of Roots for Quadratic and Cubic Functions

[GardeeZak]

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

 

[Ishuda]

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

You might want to read through
http://www.purplemath.com/modules/fcntrans.htm

The basic concepts are
(1) Moving the graph up (down) is adding (subtracting) 'outside the function itself'. That is, if
g(x) = f(x) + 5
[5 is added outside the function f], the graph of g is the same as the graph moved up 5 units.

(2)Moving the graph left (right) is adding to (subtracting from) the x value 'inside the function'. That is, if
g(x) = f(x+5)
[5 added to the x], the graph of g is the same as the graph of f moved 5 units to the right.

(3) Rotating about the x axis is 'negating f'. That is, if
g(x) = -f(x)
[g is the negative of f], the graph of g is the same as the graph of f rotated about the x axis.

(4) Rotating about the y axis is 'negating x'. That is, if
g(x) = f(-x)
[the x for g is the negative of the x for f (and vice versa)], the graph of g is the same as the graph of f rotated about the y axis.

Look at graphs for (1) and (3): x in the function doesn't change so it must be something is happening to y. For (1) y gets moved up or down. For (3) y gets negated, i.e. positive y turns into negative y and negative y turns into positive y.

Look at the graphs for (2) and (4): x is the thing in the function which is changing and y is remaining the same. For 2 x is moved left or right. For (4) x gets negated, i.e. positive x turns into negative x and negative x turns into positive x.

Finally, these can be combined, i.e.
g(x) = f(x+5) + 3
That is move the graph of f left 5 units and then up 3 units [or up then left]

Good Luck