Problem 8 - Algebra and Trigonometry

Question

\(3-24=A\) pair of parametric equations is given. (a) Sketch the curve represented by the parametric equations. (b) Find a rectangular coordinate equation for the curve by eliminating the parameter. $$ x=t^{2}, \quad y=t^{4}+1 $$

Step-by-Step Solution

Answer

Sketch: Parabola opening upwards. Rectangular equation: \( y = x^2 + 1 \).

Step 1: Identify the Parametric Equations

The given parametric equations are \( x = t^2 \) and \( y = t^4 + 1 \). These equations describe how \( x \) and \( y \) change with the parameter \( t \).

Step 2: Sketch the Curve

To sketch the curve represented by these parametric equations, calculate some values: For \( t = -2, -1, 0, 1, 2 \), compute the respective \( (x, y) \) pairs: \((4, 17), (1, 2), (0, 1), (1, 2), (4, 17)\). These points show that the curve expands and rises as \( t \) increases or decreases from zero.

Step 3: Eliminate the Parameter

To eliminate the parameter \( t \), start with \( x = t^2 \). Solving for \( t \), we get \( t = \pm \sqrt{x} \). Substitute into \( y = t^4 + 1 \): \( y = (\sqrt{x})^4 + 1 = x^2 + 1 \). This gives the rectangular coordinate equation \( y = x^2 + 1 \).

Step 4: Conclusion

The rectangular equation \( y = x^2 + 1 \) describes a parabola shifted upwards by 1 unit. This corresponds to the curve previously sketched with the parametric equations.

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of specifying points on a plane using two perpendicular lines called axes. These two axes are typically the x-axis (horizontal) and the y-axis (vertical). Each point in this system is defined by a pair of numbers

• The first number is the x-coordinate, which indicates the horizontal position of the point.
• The second number is the y-coordinate, representing the vertical position of the point.

Using rectangular coordinates helps in visualizing mathematical equations as curves or shapes on a graph. In our example, we have parametric equations given, which we'll convert into a rectangular form to better understand the shape of the curve. With parametric equations, points are defined based on a parameter, usually denoted as \( t \).
This leads us to explore how each value of \( t \) gives us specific \( (x, y) \) pairs, which we then plot to sketch the curve.

Eliminating Parameters

Eliminating the parameter in a set of parametric equations means rewriting the equations in terms of \( x \) and \( y \) only, getting rid of the parameter \( t \). This process is essential as it helps convert parametric equations into a more familiar form, often making it easier to identify the type of curve described.
Here's how we can eliminate the parameter:

• Start with the parametric equations. For example, we have \( x = t^2 \) and \( y = t^4 + 1 \).
• Express \( t \) in terms of \( x \) using the equation \( x = t^2 \). Solving for \( t \) gives \( t = \pm \sqrt{x} \).
• Substitute \( t = \sqrt{x} \) into the equation for \( y \), resulting in \( y = (\sqrt{x})^4 + 1 \), which simplifies to \( y = x^2 + 1 \).

By following these steps, we eliminate the parameter and express the relationship between \( x \) and \( y \) directly. Now, we end up with the equation \( y = x^2 + 1 \), which is a familiar quadratic equation representing a parabola. This technique is a critical tool in calculus and algebra for simplifying and analyzing curves.

Parabolas

Parabolas are a type of curve that appears frequently in mathematics, particularly in quadratic equations. They have a characteristic "U" shape and can open upwards or downwards depending on the equation's form.
In the rectangular coordinate system, the most general equation for a parabola is \( y = ax^2 + bx + c \). However, in many simple cases, like our current exercise, the parabola is in the form \( y = ax^2 + c \).

• The coefficient \( a \) determines the width and direction of the parabola.
• If \( a \) is positive, the parabola opens upwards; if negative, it opens downward.
• The term \( c \) indicates a vertical shift of the parabola along the y-axis.

In our example, the obtained rectangular equation is \( y = x^2 + 1 \). This means:

• Our parabola is shaped by \( a = 1 \), leading it to open upwards with a relatively standard width.
• The \( + 1 \) shifts the parabola one unit up along the y-axis.

Understanding these properties allows us to sketch and interpret parabolas easily. These curves not only have applications in geometry and algebra but also in physics, engineering, and various real-world scenarios.