Problem 37 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=t\\\ &y=\sqrt{t} \end{aligned}$$

Step-by-Step Solution

Answer

After substituting \(x = t\) into the equation for \(y = \sqrt{t}\), we get \(y = \sqrt{x}\). The graph of this equation forms a curve in the first quadrant, starting from the origin (0,0) and increasing in y value as x increases. For the exact plot, a graphing utility should be used.

Step 1: Understanding the Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as 'parameters'. Here, there are two parametric equations presented: \(x = t\) and \(y = \sqrt{t}\). These equations show that the variables x and y are both expressed as a function of the parameter \(t\).

Step 2: Substitute x into the equation for y

Since \(x = t\), we can substitute x into the equation for y to give us \(y = \sqrt{x}\). This gives us a single function that defines y in terms of x.

Step 3: Graphing the Equation

Now that we have the equation \(y = \sqrt{x}\), we can use this to graph the curve. The graph of this equation forms a curve in the first quadrant, starting from the origin (0,0) and increasing in y value as x increases. Please use a graphing utility to generate the exact graph.

Key Concepts

Parametric Equations

Parametric equations provide a powerful way to describe a curve by linking multiple equations through a parameter. This parameter, often denoted as \( t \), determines the position of a point on the curve.
In our example, the parametric equations given are \( x = t \) and \( y = \sqrt{t} \). Here, the parameter \( t \) is the independent variable for both \( x \) and \( y \). As \( t \) changes, so do the values of \( x \) and \( y \), tracing out a path on the coordinate plane.

These equations give us a systematic way to see how \( y \) relates to \( x \), often revealing more complex behaviors than simple Cartesian form equations. By eliminating the parameter, we can derive the Cartesian equation \( y = \sqrt{x} \), which simplifies graphing and understanding the curve.

Graphing Utilities

Graphing utilities are indispensable tools for visualizing mathematical equations and functions. They can range from simple software calculators to advanced graphing applications that allow for detailed exploration of functions and their behaviors.

Using a graphing utility makes graphing parametric equations much easier. You can input the parametric equations \( x = t \) and \( y = \sqrt{t} \) directly and immediately see the resulting graph. This instant visualization helps in understanding the shape and direction of the curve on the coordinate system, confirming theoretical analysis.

• Input both parametric equations into the graphing tool simultaneously.
• Adjust the range of \( t \) to ensure it covers the regions of interest.
• Observe how the graph appears, typically as a curve in the first quadrant due to \( y = \sqrt{x} \).

This process provides a clear and interactive means to study complex mathematical relationships without manual plotting.

Square Root Function

The square root function, expressed as \( y = \sqrt{x} \), is a fundamental mathematical function that showcases a distinct curve. It is defined for non-negative values of \( x \) only, as the square root of a negative number is not a real number in standard arithmetic.

In the graph of \( y = \sqrt{x} \), you'll see it starts at the origin \((0,0)\) and increases as \( x \) increases.

• The graph only appears in the first quadrant as both \( x \) and \( y \) are non-negative.
• The slope of the curve decreases as \( x \) increases, showing a vertical line tangent at the origin and becoming less steep farther along the x-axis.
• This creates a smooth, ever-rising curve that is continuous and well-known for its gentle curvature.

Understanding this behavior is crucial in various fields of math and science, as the square root function often appears in applied mathematics and natural phenomena.