Problem 11 - Precalculus with Limits

Question

In Exercises 7-26, (a) sketch the curve represented by the parametric equations (indicate the orientation of the curve) and (b) eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Adjust the domain of the resulting rectangular equation if necessary. \(x=t+2\) \(y=t^2\)

Step-by-Step Solution

Answer

The rectangular equation whose graph represents the curve is \(y=x^2-4x+4\). The orientation of the curve lies to the right.

Step 1 - Express Parameter t in terms of x

From the first equation \(x=t+2\), we can express the parameter 't' in terms of 'x' by subtracting 2 from both sides which gives us \(t=x-2\).

Step 2 - Substitute t in Second Equation

Substitute 't' from the first equation into the second equation, replacing 't' with '(x-2)', to eliminate the parameter. This results in \(y=(x-2)^2\).

Step 3 - Simplify the Resulting Equation

Expand the equation \(y=(x-2)^2\) to its basic form by expanding the square term. This yields \(y=x^2-4x+4\).

Step 4 - Sketching the Curve

The given parametric equations suggest the graph will have a parabolic shape since 'y' is equal to 't' squared. As 't' increases, the value of 'x', which is 't + 2', also increases, indicating that the orientation of the parabola opens towards the right.

Key Concepts

Rectangular Equation

In mathematics, a rectangular equation is an algebraic expression that defines a relationship between two variables, often labeled as 'x' and 'y'. This type of equation presents the curve or surface in terms of these coordinates.

For instance, when working with parametric equations such as \( x = t + 2 \) and \( y = t^2 \), we aim to convert these into a single rectangular equation by eliminating the parameter 't'. By expressing 't' in terms of 'x' (from \( x = t + 2 \), we get \( t = x - 2 \)) and substituting it into the equation for 'y', we derive a rectangular equation. This results in \( y = (x - 2)^2 \), which simplifies to \( y = x^2 - 4x + 4 \). This equation describes a parabolic curve on a Cartesian plane, characterized by its quadratic form.

Eliminating the Parameter

Eliminating the parameter is a crucial step in transforming parametric equations into a rectangular form. Parametric equations involve a third variable, known as a parameter (often denoted as 't'), which defines the x and y coordinates of points on a curve.

To eliminate the parameter, you must express 't' in terms of one of the variables, usually 'x'. Using the example \( x = t + 2 \), we can solve for 't' by isolating it, giving us \( t = x - 2 \).

We then substitute this expression into the second parametric equation \( y = t^2 \), which results in \( y = (x - 2)^2 \). Once simplified, this generates the rectangular equation \( y = x^2 - 4x + 4 \). By eliminating the parameter, the relationship between 'x' and 'y' becomes direct, allowing for easy plotting and analysis of the curve.

Parabolic Curve

A parabolic curve is a symmetrical, U-shaped path defined by a quadratic equation of the form \( y = ax^2 + bx + c \). Parabolas are ubiquitous in mathematics and physics due to their elegant properties and simple geometric shape.

In our example, the rectangular equation \( y = x^2 - 4x + 4 \) generates a parabolic curve. The coefficient '1' in front of \( x^2 \) indicates that the parabola opens upwards.

The vertex of the parabola, which is its highest or lowest point depending on the orientation, can be found using the vertex formula \( x = -\frac{b}{2a} \). For our equation, \( a = 1 \), \( b = -4 \), which gives a vertex at \( x = 2 \). Substituting \( x = 2 \) into the equation gives \( y = 0 \), so the vertex is at the point (2, 0).

Recognizing and sketching such curves are foundational skills in understanding their applications across various fields.