Question

Consider the parametric equations \(x=+\sqrt{t}, y=t\), for \(0 \leq t \leq 10\) (a) Draw up a table of values of \(t, x\) and \(y\) for values of \(t\) between 0 and 10 (b) Plot a graph of this function. (c) Obtain an explicit equation for \(y\) in terms of \(x\).

Short answer

Question: Given the parametric equations \(x = \sqrt{t}\) and \(y = t\), find the explicit equation for y in terms of x. Answer: Using the process of substitution and elimination, we obtained the explicit equation for y in terms of x as \(y = x^2\).

Step-by-step solution

Create a table of values for t, x, and y

To create the table of values, we need to plug in the given range of t values into the parametric equations and find the corresponding x and y values: \(x = \sqrt{t}\) \(y = t\) We will do this for t values from 0 to 10.

Plot the graph of this function using the table of values

Using the table of values from Step 1, we can plot the points on a coordinate grid to get an idea of the function's graph. The points will be (x, y) coordinates for each corresponding value of t.

Obtain the explicit equation for y in terms of x

To find the explicit equation for y in terms of x, we need to eliminate t from the given parametric equations: 1. \(x = \sqrt{t}\) 2. \(y = t\) First, let's solve equation (1) for t: 1.' \(t = x^2\) Now, substitute the expression from equation 1.' into equation 2: 2.' \(y = x^2\) So the explicit equation for y in terms of x is given by: \(y = x^2\)

Key concepts

graphing functions

Graphing functions helps us visualize mathematical relationships. Let's learn how to graph functions using parametric equations. Parametric equations are equations where one or more variables are expressed in terms of another variable, often 't', called a parameter.

Consider the parametric equations:

• \(x = \sqrt{t}\)
• \(y = t\)

Here 't' changes from 0 to 10. We find values of \(x\) and \(y\) by substituting values of \(t\). For example, when \(t=1\), \(x=\sqrt{1}=1\) and \(y=1\).

Creating a table helps, where you write down values of \(t\), \(x\), and \(y\) for various \(t\). You then plot these (x, y) pairs on a coordinate grid to see the graph. This visual helps us understand how x and y change as t changes, and see the overall shape of the function.

explicit equation

An explicit equation expresses one variable directly in terms of another. In this context, it means finding a formula for \(y\) in terms of \(x\), with no \(t\) involved.

Given our parametric equations where \(x = \sqrt{t}\) and \(y = t\), our goal is to combine these into just one equation. To achieve this, solve \(x = \sqrt{t}\) for \(t\). You get \(t = x^2\). Now substitute this \(t = x^2\) back into \(y = t\), resulting in \(y = x^2\).

This is our explicit equation, \(y = x^2\). It shows how \(y\) changes directly with \(x\) without having to consider \(t\). This form is often easier to work with, especially when graphing.

coordinate system

A coordinate system allows us to visually position points on a plane. The most common is the Cartesian coordinate system, based on perpendicular axes: usually the x-axis (horizontal) and y-axis (vertical). Each point on this plane is specified by an (x, y) coordinate.

When graphing the function from our parametric equations, each calculated pair \((x, y)\) for each specific value of \(t\) is a point on this coordinate plane. Starting from \(t = 0\) to \(t = 10\), and plotting each corresponding \((x, y)\), you trace the path or curve of the function.

The coordinate grid is critical for visual tools like plotting, which transforms abstract numerical relationships into clear, interpretable shapes. These visual shapes represent the behavior of mathematical relationships over the defined parameter range.