Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=2^{t}, y=\sqrt{3 t-1},\) for \(t\) in \(\left[\frac{1}{3}, 4\right]\)window: \([-2,30]\) by \([-2,10]\)

Step-by-Step Solution

Answer

The rectangular equation is \(y=\sqrt{3 \log_2 x - 1}\) with domain \(x \geq 2^{1/3}\).

Step 1: Graph the Parametric Equations

Using a graphing calculator, input the parametric equations \(x=2^t\) and \(y=\sqrt{3t-1}\). Set the parameter \(t\) in the interval \([\frac{1}{3}, 4]\). Adjust the viewing window to \([-2, 30]\) for the x-axis and \([-2, 10]\) for the y-axis. This should show the curve generated by the parametric equations over the specified interval.

Step 2: Isolate 't' in terms of 'x'

To find a rectangular equation, first solve for \(t\) in terms of \(x\). From \(x=2^t\), take the logarithm base 2 on both sides: \(t=\log_2 x\).

Step 3: Substitute 't' in terms of 'x' into 'y'

Substitute \(t = \log_2 x\) into the equation for \(y\): \(y = \sqrt{3(\log_2 x) - 1}\). This yields the rectangular equation of the curve: \(y=\sqrt{3 \log_2 x - 1}\).

Step 4: Determine the Domain of the Rectangular Equation

The domain of this new equation can be found by ensuring the expression under the square root is non-negative: \(3\log_2 x - 1 \geq 0\). Solving this gives \(\log_2 x \geq \frac{1}{3}\), hence \(x \geq 2^{\frac{1}{3}}\).

Key Concepts

Graphing Calculator

A graphing calculator is an essential tool for visualizing complex equations and functions. When working with parametric equations like these, a graphing calculator can help generate the curve that represents the relationship between the parameters. To do this:

• Input the equations: For the problem you're solving, input the equations \(x=2^t\) and \(y=\sqrt{3t-1}\) into your calculator.
• Set the parameter range: Ensure that the parameter \(t\) is set within the interval \(\left[\frac{1}{3}, 4\right]\).
• Adjust the viewing window: Adjust the x-axis to \([-2, 30]\) and the y-axis to \([-2, 10]\). This setting ensures you see all relevant parts of the curve.

Using a graphing calculator in this way allows you to verify your theoretical work and better understand the shape and position of the curve in a visual format.

Rectangular Equation

The goal of converting a parametric equation to a rectangular equation is to express the relationship between \(x\) and \(y\) without the parameter \(t\). This often simplifies the analysis of the curve.

To find the rectangular equation here, follow these steps:

• Isolate \(t\): Starting with \(x=2^t\), solve for \(t\) by taking the logarithm base 2 of both sides to find \(t=\log_2 x\).
• Substitute \(t\) into \(y\): Replace \(t\) in the equation for \(y\) with \(\log_2 x\), yielding \(y=\sqrt{3(\log_2 x) - 1}\).

This equation \(y=\sqrt{3\log_2 x - 1}\) is your rectangular form, directly relating \(x\) and \(y\) without involving \(t\) anymore. This form can be more familiar and simpler to work with for further analysis.

Domain

Determining the domain of a function is crucial as it highlights the valid inputs (\(x\) values) that work within the equation.

For the equation \(y=\sqrt{3\log_2 x - 1}\), we must ensure that the expression inside the square root is non-negative, as taking the square root of a negative number results in an undefined real number. To find the domain:

• Set the condition: Ensure \(3\log_2 x - 1 \geq 0\).
• Solve for \(x\): This simplifies to \(\log_2 x \geq \frac{1}{3}\).
• Convert the inequality: In exponential form, this gives \(x \geq 2^{1/3}\).

Hence, the domain is \(x \geq 2^{1/3}\), meaning that only \(x\) values greater than or equal to \(2^{1/3}\) will keep the function defined.

Interval

Intervals are used to define the range of values a parameter, like \(t\), can take. A specified interval helps in limiting the calculation domain and visualization on tools like graphing calculators.

For the parametric equations \(x=2^t\) and \(y=\sqrt{3t-1}\), the interval given is \([\frac{1}{3}, 4]\). This interval specifies that:

• \(t\) starts at \(\frac{1}{3}\) and ends at \(4\).
• It constrains the parametric equations to produce results only for values of \(t\) within this range, affecting both the shape and position of the resulting graph.

When interpreting graphs or solving problems, always pay attention to intervals to understand which part of a function's output is being focused on.