Popular Articles (30d)
From the Circle to the Astroid: How Powers, Pythagoras and Parametric Equations Are Connected
A simple equation such as x² + y² = 1 looks familiar. It is the equation of a circle. But why? Why does the power 2 produce a circle? What happens if the right-hand side changes from 1 to 2 or 3? What happens if the powers change from 2 to 3? And why does…
Why Does x² + y² = 1 Give a Circle?
At first glance, the equation x² + y² = 1 may look like just another algebraic equation. But it hides something much deeper. The reason it produces a circle is not simply that x and y represent two-dimensional coordinates. The real reason is that x² + y² is directly connected to the Pythagorean theorem and…
From Diamond to Circle to Square: How Changing an Equation Changes the Shape
One of the most interesting things about coordinate geometry is that a very small change in an equation can produce a completely different geometric shape. Consider the family of equations 
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How Powers of x and y Shape Curves
When studying equations in two variables, it is fascinating to see how simply changing the powers of x and y can dramatically change the shape of a curve. A familiar example is: x² + y² = 1 which produces a perfect circle. But what happens if the powers are changed? What if we use fourth…
Tips for Choosing a Parameter in Parametric Equations
Choosing the parameter is often the hardest part of setting up a parametric equation. Fortunately, there are some simple rules that make the process much easier. The main idea is: Choose a parameter that makes x(t) and y(t) as simple as possible and describes the required part of the curve without ambiguity. Here are some…
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Understanding Parameters in Parametric Equations: Where Does x(t) Come From?
When a curve is first introduced in the familiar form y = f(x) it is natural to wonder how it can later be written using parametric equations: x = x(t)y = y(t) A particularly important question is: If y = f(x) is already known, where does x(t) come from? The answer reveals an important idea…
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