1. ## Graph transformations...

Hello! I am a little confused about graph transformations and I have some questions that I don't really know how to solve...Please help! Thank you!

1)
The transformations A,B and C are as follows:
A: A translation of 1 unit in the negative y direction
B: A stretching parallel to the y-axis (with x-axis invariant) with a scale factor of 2
C: A reflection about the x-axis

A curve undergoes in succession, the above transformations A,B and C and the equation of the resulting curve is y = -2(x^2) -6. Determine the equation of the curve before the transformations were effected. [Is it possible to view the equation of the resulting curve as y= -2 (x^2 -3)? But I think this will give a different original equation from if I use the given resulting equation??]

2)
The equation of a graph is y = (ax)/(bx-y), and y= c is the horizontal asymptote while x = d is the vertical asymtote of this graph. Given that c = d = 1/2, what are the values of a and b? [If I am not wrong, b should be 2, but I don't know how to find a....Is a = 1?]

3)
Consider the curve (5x + 10)^2 - (y-3)^2 = 25. Prove, using an algebraic method, that x cannot lie between two certain values (which are to be determined). [eartboth has solved this here but I don't understand the method...Can someone explain?]

4)
The curve of y = x^2 + (3/x) undergoes a reflection about the line y=4 and the point (2, 5.5) is mapped onto (2, 2.5). Hence find the equation of the new curve in the form y = f(x).

Thank you very very much for helping me!

2. Originally Posted by Tangera
...
3)
Consider the curve (5x + 10)^2 - (y-3)^2 = 25. Prove, using an algebraic method, that x cannot lie between two certain values (which are to be determined). ...
...

$\displaystyle \frac{(x+2)^2}{1^2} - \frac{(y-3)^2}{5^2} = 1$

which is the equation of a hyperbola with the center (-2, 3) and the semi-axes a = 1 and b = 5.

Then I calculated the coordinates of the vertices and got the intervall in which x cannot lie.

I've attached a drawing of the curve to show where you can find this interval.

3. Originally Posted by Tangera
Hello! I am a little confused about graph transformations and I have some questions that I don't really know how to solve...Please help! Thank you!

1)
The transformations A,B and C are as follows:
A: A translation of 1 unit in the negative y direction
B: A stretching parallel to the y-axis (with x-axis invariant) with a scale factor of 2
C: A reflection about the x-axis

A curve undergoes in succession, the above transformations A,B and C and the equation of the resulting curve is y = -2(x^2) -6. Determine the equation of the curve before the transformations were effected. [Is it possible to view the equation of the resulting curve as y= -2 (x^2 -3)? But I think this will give a different original equation from if I use the given resulting equation??]

...
You have:

$\displaystyle y = -2x^2-6 = -2(x^2+3)$

Reflection about the x-axis will yield: $\displaystyle y = 2(x^2+3)$

The stretching parallel to the y-axis (with x-axis invariant) with a scale factor of $\displaystyle \frac12$ will yield: $\displaystyle y = x^2+3$

The translation of 1 unit in the positive y direction will yield: $\displaystyle \boxed{y = x^2+4}$

I've attached a sketch of the complete process. You start with the blue graph and you end with the red one.