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Chapter 6: Problem 17

Eliminate the parameter and graph the equation. $$x=\sqrt{t+1}, y=t, \text { for } t \geq-1$$

Short Answer

Expert verified
The equation after eliminating the parameter is \(x = \sqrt{y+1}\), valid for \(y \geq -1\). The graph is a portion of the simple square root curve shifted one unit to the left and starting at \(y = -1\).

Step by step solution

01

Express parameter in terms of y

From the parametric equation \(y=t\), express \(t\) as \(t = y\).
02

Substitute \(t\) in terms of \(y\) into \(x\) equation

Next, substitute \(y\) for \(t\) in the first equation, to get \(x = \sqrt{y+1}\). Thus, we鈥檝e eliminated the parameter \(t\) by expressing \(x\) solely in terms of \(y\).
03

Graph the equation

To graph the function, make a table for some values of \(y\) and find the corresponding \(x\) values using the equation \(x = \sqrt{y+1}\). Start with \(y = -1\) since \(t \geq -1\). Substituting \(y=-1\), \(y=0\), \(y=1\), and \(y=2\), we get four points \((-1,0), (0,1), (1,1.41), (2,1.73)\). Plot these points on a graph and connect them with a curve, keeping in mind that the graph is valid for \(y \geq -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parametric Equations
Parametric equations are a versatile way to represent a curve using parameters. Instead of describing a relationship between two variables directly (like in Cartesian equations), parametric equations use one or more parameters. These parameters act as intermediaries that help us express each variable separately.
For example, in the given exercise, we have:
  • For the x-coordinate: \( x = \sqrt{t+1} \)
  • For the y-coordinate: \( y = t \)
Together, these give a set of coordinates \((x, y)\) that form a curve as the parameter \( t \) changes. Parametric equations are helpful in describing trajectories and paths as they clearly assign a direction defined by the parameter's range.
Graphing Equations
Graphing equations involves plotting points that satisfy the equation on a coordinate plane. When dealing with parametric equations, the goal is to find a range of parameter values, plot the corresponding points, and then connect them smoothly.
In our case:
  • Use the given x and y equations and substitute parameter values.
  • Create a table listing values of \( t \), the parameter variable.
  • Convert these to x and y coordinates.
  • Plot the points from your table against each other on a graph.
This method helps visualize the path described by the parametric equations. Make sure to consider any constraints on the parameter, like \( t \geq -1 \) in this case, which affects the graph's starting point and domain of the function.
Eliminating Parameters
Eliminating a parameter means rewriting the equations to remove the intermediary variable. This will convert parametric equations into a standard Cartesian form where one variable is directly related to the other.
To eliminate parameters:
  • Express one of the parameter equations in terms of the parameter (for instance, \( y = t \)), and then turn it around to express \( t \) in terms of \( y \).
  • Substitute this back into the other equation to solve for x in terms of y.
This yields a simpler equation, in this case, \( x = \sqrt{y+1} \). Now, you have a Cartesian equation, which can be graphed more intuitively while still reflecting the original path defined by the parametric equations, starting from the domain constraint provided by the parameter \( t \).

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