91Ó°ÊÓ

Cartesian equations describe a curve in the coordinate plane using only the variables \(x\) and \(y\). These equations are widely used because they allow us to visualize and analyze relationships between coordinates directly on a graph. In the given exercise, it starts with parametric equations defined by \(x = s\) and \(y = \frac{1}{s}\). To convert these into a Cartesian equation, we express both \(x\) and \(y\) terms without the parameter \(s\).
By setting \(x = s\), \(s\) gets removed from the equations, giving us the replacement \(y = \frac{1}{x}\). This shows the relationship directly between \(x\) and \(y\), eliminating the need for the parameter \(s\). As a result, the Cartesian equation is \(y = \frac{1}{x}\). This form is particularly helpful when graphing the curve or analyzing its properties.

Eliminating the parameter involves expressing one variable in terms of another without relying on a third, independent parameter. In parametric equations, a parameter often simplifies calculations or defines specific sections of the graph more clearly when \(x\) and \(y\) are complex or periodic. However, finding the Cartesian form, which does not involve a parameter, often simplifies understanding and working with the curve.
In the exercise, the parameter \(s\) relates \(x\) and \(y\) through \(x=s\) and \(y = \frac{1}{s}\). To eliminate \(s\), observe that \(s = x\), meaning we can substitute \(x\) for \(s\) in the second equation. This gives us \(y = \frac{1}{x}\), creating a direct relationship between \(x\) and \(y\). This step converts the problem from parametric to a more traditional Cartesian form.

Graphing curves involves plotting points that satisfy an equation, helping to visually understand the behavior and properties of a curve. With the Cartesian equation \(y = \frac{1}{x}\), the curve forms a specific path on a graph.
The equation \(y = \frac{1}{x}\) produces a hyperbolic shape. Since \(x\) cannot be zero (as this would make \(y\) undefined), the curve exists where \(x\) is greater than zero. For the defined domain \(1 \leq x \leq 10\), the curve begins at \((1,1)\) and stretches towards \((10, 0.1)\) as \(x\) increases. This represents the right branch of the hyperbola in the positive quadrant of the Cartesian plane, decreasing slowly as \(x\) increases.

The domain of a function describes all possible input values \(x\) for which the function is defined. Determining the domain ensures we understand the constraints and scope of the function.
Given the original parameter range \(1 \leq s \leq 10\), and knowing \(x = s\), we can define the domain for \(x\) in the Cartesian form. Since \(x = s\), the domain for \(x\) is \(1 \leq x \leq 10\). This restriction tells us the set of \(x\) values we can input into the function without causing it to become undefined. Inside this domain, \(y = \frac{1}{x}\) is valid and provides meaningful values. Hence, for every \(x\) between 1 and 10, there is a corresponding \(y\) value on the curve.