Thread: Graph

I need hints on the following questions:


1. Sketch the curve (y2-2)2+(x2-2)2=2. What does it look like?

I got 4 circles which kind of merge together where they meet.


2. kx^4=x^3-x Find the real roots when k=0. Sketch the graph when k is small and then when k is large, and find approximations of the real roots in both cases. When else does x have 3 real roots?

I don't understand why the graph has 3 roots, even when k is not 0?


3. sketch x^x

The graph is a bit complicated, and broken down in the negative x values. I'm not sure how to draw it when x = 0, on the calculator it doesn't show it as an asymptote or anything, is it just supposed to end there or what? Also, I dont understand how to work out the fluctuations between positive and negative values of y for the negative values of x.


4. Sketch f(x) = (x - R(x))2, where R(x) is x rounded up or down in the usual way. then sketch g(x) = f(1/x)

For f(x) I got like multiple humps between the integer, with maximum at half way between 2 integers of 1/4, and then 0 at the integers. However, I have no idea for g(x), it doesn't seem to follow any pattern.

Hello Aquafina

Originally Posted by Aquafina

I need hints on the following questions:


1. Sketch the curve (y2-2)2+(x2-2)2=2. What does it look like?

I got 4 circles which kind of merge together where they meet.

Note that we have just even powers of and , so the graph is symmetrical about both axes. If we re-arrange:



Note that whenever is real, so is real.

Then is real whenever

i.e. or .

These give approximate values of and .

By the symmetry of the equation, has the same ranges of values.

So the graph consists of four non-overlapping loops as in the attached sketch.

Grandad

3. y = x^x. Details are not included, these are the progressive graphs, try to fill in the details