Problem 20 - A Graphical Approach to Precalculus with Limits

Question

For each plane curve, (a) graph the curve, and (b) find a rectangular equation for the curve. $$x=2 t-1, y=t^{2}+2 ; \text { for } t \text { in }(-\infty, \infty)$$

Step-by-Step Solution

Answer

Rectangular equation: \( y = \frac{x^2 + 2x + 9}{4} \). Graph a parabola opening upwards.

Step 1: Substitute Parametric Equations

To find the rectangular equation for the curve, we first look at the given parametric equations: \( x = 2t - 1 \) and \( y = t^2 + 2 \). Our goal is to eliminate the parameter \( t \) from these equations.

Step 2: Solve for t in terms of x

We start by solving the equation for \( x = 2t - 1 \) to express \( t \) in terms of \( x \). \[ x = 2t - 1 \] Adding 1 to both sides gives \[ x + 1 = 2t \] Now, divide by 2 to solve for \( t \): \[ t = \frac{x + 1}{2} \]

Step 3: Substitute expression for t into y equation

Now that we have an expression for \( t \), substitute \( t = \frac{x + 1}{2} \) into the equation for \( y \): \[ y = \left(\frac{x + 1}{2}\right)^2 + 2 \]

Step 4: Simplify the expression

Expand the square in the equation \( y = \left(\frac{x + 1}{2}\right)^2 + 2 \): \[ y = \left(\frac{x^2 + 2x + 1}{4}\right) + 2 \] Simplify further by multiplying out: \[ y = \frac{x^2 + 2x + 1}{4} + \frac{8}{4} \] Simplify to get the rectangular equation: \[ y = \frac{x^2 + 2x + 9}{4} \]

Step 5: Graph the Curve

Graph the curve by plotting the rectangular equation \( y = \frac{x^2 + 2x + 9}{4} \). This represents a parabola opening upwards, shifted vertically and adjusted in width by the factor of \( \frac{1}{4} \).

Key Concepts

Rectangular Equations

Rectangular equations involve expressing a relationship in terms of just the two variables, usually denoted as 𝑥 and 𝑦. These are classic equations that are straightforward to graph using a Cartesian coordinate system. The key here is to eliminate the parameter, in this case, the parameter \( t \).

• We start with parametric equations like \( x = 2t - 1 \) and \( y = t^2 + 2 \).
• Our goal is to convert these into a single equation involving only \( x \) and \( y \).
• This process typically involves solving one parametric equation for \( t \) and substituting it into the other.

By doing this, you translate the situation from one of motion or change, represented by \( t \), into a static relationship between \( x \) and \( y \). This provides a clear geometric explanation of the relationship - like turning abstract direction instructions into a map.

Graphing Curves

Graphing curves requires plotting the relationship defined by the rectangular equation on a coordinate plane. This visual representation helps understand complex algebraic relationships intuitively.When graphing the rectangular equation \( y = \frac{x^2 + 2x + 9}{4} \), we can follow these steps:

• Identify the type of curve: From the given equation, we deduce it is a parabola because the equation follows the format \( y = ax^2 + bx + c \).
• Determine the curve's direction: The positive coefficient of \( x^2 \) implies it opens upwards.
• Plot points by plugging in various \( x \) values to find corresponding \( y \) values. This provides precise points through which the curve passes.
• Sketch the curve, ensuring it aligns with the deduced direction and shape.

Graphing curves gives a dynamic representation of algebraic relations and helps in understanding their properties visually. This method reveals the nature and extent of how one variable dimensionally interacts with another.

Parabolas

Parabolas are U-shaped curves that are symmetric and can open up or down in the coordinate plane. They are a common subject of study in both algebra and geometry because of their applicability and interesting properties.

• The equation \( y = \frac{x^2 + 2x + 9}{4} \) is characteristic of a parabola.
• Its symmetry is around its vertex, which, in this case, occurs where the first derivative (i.e., slope) is zero.
• The coefficient \( \frac{1}{4} \) multiplies the entire expression, indicating how "wide" or "narrow" the parabola is.
• This particular parabola opens upwards because the quadratic term (\( x^2 \)) has a positive coefficient.

Understanding parabolas is crucial. From designing satellite dishes to understanding projectile motion, parabolas are prevalent in multiple practical applications. Identifying their vertex, focus, and axis of symmetry is essential in recognizing and applying these curves effectively.