Can a single node graph be considered a (strongly) connected component? I’m confused because I was reading about cut vertex which by definition is a vertex that if eliminated increases the number of connected components. So if we were to remove vertex number 2, we would get 2 connected components? Thanks!

A function $f(x)$ is continuous in the interval $[0,2].$ It is known that $f(0)=f(2)=−1$ and $f(1)=1.$ Which one of the following statements must be true$?$ Options are $:$ There exists a $y$ in the interval $(0,1)$ such that $f(y)=f(y+1)$ For every $y$ in the interval $(0,1),f(y) = f(2−y)$ The maximum value of the function in […]

I tried to get a equation that can represent a spiral like spring: $$z=\frac{height}{2\pi }\cdot \arctan \left ( \sqrt{\frac{radius^{2}-x^{2}}{radius^{2}-y^{2}}} \right )$$ I am not sure about the equation I got, so I am confused about… whether this equation really represent a spiral like spring? whether an equation can represent a curve or a line in […]

I found this and I just couldn’t verify it. How come it is true? The maximum number of nodes that a balanced binary tree with depth $d$ is a complete binary tree with $2^d-1$ nodes. Let say I have the following tree 1 / \ 2 3 The tree is balanced as well as a […]

I am solving a question, where I need to find the expected value of the absolute value of $X-Y$ where $X$ and $Y$ are two independent uniformly distributed random variable. So I was looking for the graph of the absolute value of $X – Y$.

I was wondering, can we have an equation for any graph even the most zigzagged one?

Say I have a cos function, $y = 1.2cos(0.503x) + 5$ And say I want to replicate it using sine, so expressing the above function in sine so that it gives “the same wave”. How would I do this? I know I have to find a phase shift, but I don’t get the steps on […]

I have an equation like $$ (x – a)^2 + (y – b)^2 = r^2 $$ that represents a circle. I need to “plot” it very basically with a programming language. Computer graphics coordinate generally use the “lower-right” quadrant of a Cartesian plane (0, 0 being top left). I want my circle to have its […]

I’m trying to figure out this question: in the Cartesian coordinate system, how many grid points (x,y) satisfy $(|x|-1)^2 + (|y|-1)^2 < 2$? When I plugged it in to a graphing calculator, it didn’t let me use the < sign, so I changed it to the = sign, and got a graph that looked like […]

This may be an obvious question, but I can’t seem to figure it out. I found an algorithm that receives an equation, and in order to plot its graph, for each pixel in the screen, it will replace the variables for the coordinates of the pixel and evaluate both sides of the equation keeping track […]

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