Problem 23 - Algebra and Trigonometry

Question

Eliminate the parameter t. Then use the rectangular equation to sketch the plane curve represented by the given parametric equations. Use arrows to show the orientation of the curve corresponding to increasing values of t. (If an interval for t is not specified, assume that \(-\infty< t <\infty.\)) \(x=2 t-4, y=4 t^{2}\)

Step-by-Step Solution

Answer

The rectangular equation that corresponds to the given parametric equations is \(y = (x+4)^{2}\), denoting a parabola opening upwards. The orientation of the curve for increasing t-values points upwards.

Step 1: Isolate the parameter

To convert parametric equations into a rectangular form, we first need to isolate the parameter t from one of the equations. From the first equation \(x=2 t-4\), we can find t in terms of x as follows: \(t = \frac{x + 4}{2}\)

Step 2: Replace t in the second equation

Now we will replace t in the second equation \(y=4 t^{2}\) with \(\frac{x + 4}{2}\). This leads to: \(y = 4 (\frac{x + 4}{2})^{2}\)

Step 3: Simplify the resulting equation

The equation from Step 2 will simplify to \(y = (x+4)^{2}\). This equation is the rectangular form of the initial parametric equations.

Step 4: Sketch the curve

This step is about sketching the curve \(y = (x+4)^{2}\), which is a parabola opening upwards. Since the orientation of the curve corresponds to increasing t-values, arrows pointing upwards would denote an increase in t.

Key Concepts

Eliminate the Parameter

Understanding how to eliminate the parameter is crucial when working with parametric equations. Parametric equations represent curves by expressing the coordinates of the points on the curve as functions of a parameter, usually denoted as 't'. To analyze these curves in a more traditional sense, you sometimes need to express the relationship as a rectangular equation, that is, in terms of 'x' and 'y' only.

The process involves isolating the parameter 't' from one of the parametric equations and then substituting this expression into the other equation. For instance, given the parametric equations \(x=2t-4, y=4t^2\), you start by solving the first equation for 't', resulting in \(t = \frac{x + 4}{2}\). Next, you substitute this expression for 't' into the second equation, replace \(t\) with \(\frac{x + 4}{2}\), and then simplify the equation to remove the parameter 't', leaving you with a relationship solely in terms of 'x' and 'y'.

Rectangular Equation

The rectangular equation is the result of eliminating the parameter to express the relationship between 'x' and 'y' directly. This equation is named for its use in Cartesian, or 'rectangular', coordinates. For an equation like \(y = 4t^2\), after eliminating the parameter as described previously, the rectangular equation would be \(y = (x+4)^2\).

Rectangular equations are beneficial because they are often easier to analyze and graph than their parametric counterparts. They can be used to find intercepts, analyze symmetry, determine the location of critical points, and understand the general shape of the curve—all important aspects when sketching graphs.

Plane Curve Sketching

The art of plane curve sketching becomes much more straightforward once you have the rectangular equation. With the equation \(y = (x+4)^2\), you can identify it as a parabola, a specific type of curve. Recognizing this allows you to use your knowledge of parabolas to sketch the graph.

Knowing where the vertex is (in this case at \(x=-4\), \(y=0\)), and the direction in which the parabola opens (upwards because the coefficient of \(x^2\) is positive), you can sketch the general shape. Additionally, you should consider the orientation of the curve in relation to the parameter 't'. For increasing 't', you'll need to include arrows on the curve to show this directionality.

Parabola Graphing

When it comes to parabola graphing, recognising that the equation \(y = (x+4)^2\) represents a parabola is just the start. The vertex form of a parabola's equation is \(y = a(x-h)^2 + k\), where \(h\) and \(k\) are the coordinates of the vertex, and 'a' determines the width and the orientation of the parabola.

In this case, the vertex is at \(x=-4\), \(y=0\), and since 'a' (the coefficient of \(x^2\)) is positive, the parabola opens upwards. The larger the value of 'a', the narrower the parabola. Now, when you graph this parabola, you plot the vertex and then create a symmetrical 'U' shape around this point. Adding arrows at the end of these branches that point upwards indicates the direction of increasing 't', completing the sketch.