In this section we’re going to discuss the most important properties of parabola, also we’ll show how to find its vertex, roots and intercepts. This information is needed to sketch the graph and also allows to analyze the behavior of given parabola. We’ve already shown how to sketch graphs of quadratic function, and now we’re going through most essential details. Let’s start with considering the equation of parabola. (more…)
Have you ever seen a lotus? If yes, you’ve probably noticed that its flowers are not only incredibly beautiful, but somehow stay clean even when the water around is muddy and dirty. Leaves and flowers are not wetted by water, so the droplets roll off them like balls of mercury, washing away all the dirt. Do you know why the leaves of this amazing plant possess self-cleaning property? This is called lotus effect and we’re going to discuss how it works.
What is a parabola? How to sketch its graph? These are the questions for this section. Sketching parabolas is commonly asked in algebra homework tasks, so some tips will be helpful. We’ll discuss the basic concepts and show you on simple examples how to deal with parabolas.
Let’s get started. In simple words, parabola is a U-shaped curve with some specific properties. However, note that not any U-shaped curve can be called parabola. It’s a common fact that math likes to describe everything by formulas. No surprises here, we also can write a formula for parabola. Basically, any parabola is described by some quadratic equation. (more…)
In previous sections we’ve talked about Gauss elimination method and considered some examples. Today we have one more systems of linear equations to solve, but this time we’ll apply matrix representation for systems of linear equations:
\left\{\begin{aligned}2x_1+4x_2-3x_3+2x_4=0\\x_1-2x_2+5x_3+2x_4=2\\2x_2-4x_3+5x_4=-1\\-2x_3+x_4=-3 \end{aligned}\right. (more…)
If you need to solve a system of linear equations, Gaussian elimination is often the best choice. We’ve considered this topic in one of the previous sections. For better understanding we prepared several examples covering this topic similar to those in linear algebra homework. Now let’s solve the system of 3 linear equations with 3 unknowns x,y,z (more…)
Have you ever tried to comb a hairy ball? Really, it is a very important and responsible work. It roots back to old 1912. A Holland mathematician Luitzen Brouwer, who worked in a field of topology and set theory, had shown that hairy ball cannot be brushed in such a way to be smooth and without intersections of hairsprings. At least one hairspring should stick up. This result is now called a hairy ball theorem. (more…)
What do you know about solving systems of linear equations? This type of tasks is often offered to students in linear algebra homework. In one of our videos we discussed that a system of linear equations can be solved using Cramer’s rule which involves finding determinants and it might get tricky when dealing with systems of 4, 5 or more equations. Luckily, there is another method which allows to avoid massive calculations. And it is called Gaussian elimination (or row reduction). (more…)
