Problem 9 - Essential Calculus

Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$

Step-by-Step Solution

Answer

The rectangular equation is a piecewise-defined function: for \(x \geq 4\), \(y = x/2 - 2\), and for \(x < 4\), \(y = 2 - x/2\). The curve is oriented upwards for \(x \geq 4\) and to the left for \(x < 4\).

Step 1: Express the parameter \(t\) in terms of \(x\)

From the first parametric equation, \(x = 2t\), we can express \(t\) in terms of \(x\) by dividing both sides by 2. We get \(t = x/2\). This expression will be useful for eliminating the parameter in the second parametric equation.

Step 2: Substitute \(t\) into the second parametric equation

Substitute \(t = x/2\) from Step 1 into the second equation, \(y=|t-2|\), to get \(y = |x/2 - 2|\). This equation is in rectangular form, but the absolute value symbols need to be addressed.

Step 3: Simplify the absolute value

The expression inside the absolute value can be positive or negative. For \(y = |x/2 - 2|, y = x/2 - 2\) when \(x/2 - 2 \geq 0\) or in other words when \(x \geq 4\), and \(y = -(x/2 - 2)\) or \(y = 2 - x/2\) when \(x/2 - 2 < 0\) or \(x < 4\). This gives us the piecewise-defined function for \(y\) in terms of \(x\).

Step 4: Sketch the curve

Using the piecewise-defined function from Step 3, the curve can be graphed in the rectangular coordinate system. For \(x \geq 4\), \(y = x/2 - 2\), and for \(x < 4\), \(y = 2 - x/2\). The curve shifts at the point (4, 0). The orientation depends on the direction of \(t\); in this case, for larger values of \(t\), \(x\) and \(y\) both increase, so the orientation is upwards for \(x \geq 4\) and to the left for \(x < 4\).

Key Concepts

Rectangular Equations

Rectangular equations, also known as Cartesian equations, are expressions that describe a curve by relating the coordinates in a plane, usually as an equation in terms of \( x \) and \( y \). These types of equations do not involve any parameters, making them straightforward for direct graphing on a Cartesian plane.

To obtain a rectangular equation from parametric equations, our main goal is to eliminate the parameter. In our exercise, we started with the parametric equations \( x = 2t \) and \( y = |t - 2| \). By expressing \( t \) in terms of \( x \) using the first equation, we found \( t = x/2 \).

Next, we replaced \( t \) in the second equation, resulting in \( y = |x/2 - 2| \). This transformation simplifies our parametric equations into a single expression linking \( x \) and \( y \), forming the basis of our rectangular equation. This transformation is particularly useful as it allows us to understand the shape and behavior of curves without having to consider the parameter, making it easier for plotting and further analysis.

Sketching Curves

Sketching curves requires understanding of the behavior of equations over different intervals. It involves translating a mathematical equation into a visual representation on the coordinate plane.

In our problem, we end up with a piecewise function after eliminating the parameter. This function consists of two linear equations: \( y = x/2 - 2 \) for \( x \geq 4 \) and \( y = 2 - x/2 \) for \( x < 4 \).

To sketch the curve, note that at \( x = 4 \), both equations yield \( y = 0 \). Thus, the line segments meet at point (4,0). When visualizing this piecewise function:

• For \( x \geq 4 \): The line \( y = x/2 - 2 \) represents a line with positive slope, indicating that as \( x \) increases, \( y \) also increases.
• For \( x < 4 \): The line \( y = 2 - x/2 \) has a negative slope, showing that \( y \) decreases as \( x \) increases to 4.

By considering these two segments, you draw two intersecting line segments forming a V-shape. The direction or orientation of the curve will also depend on the parameter: here, it moves upwards and to the left based on \( t \) values.

Piecewise Functions

Piecewise functions are defined by different expressions over particular intervals. They allow us to express equations that behave differently over different ranges of the input variable. This is especially useful for functions involving absolute values, like in our exercise.

After eliminating the parameter, we obtained the equation \( y = |x/2 - 2| \). The nature of absolute values necessitates breaking the equation into parts where the expression inside the absolute value is positive (or zero) and where it's negative.

For our example:

• When \( x \geq 4 \), we have \( y = x/2 - 2 \). Here, the inside of the absolute value is positive or zero.
• For \( x < 4 \), it becomes \( y = 2 - x/2 \). Here, the expression inside the absolute value is negative.

These distinct expressions for \( y \) over specified domains create a piecewise-defined function, which seamlessly transitions at \( x = 4 \). Knowing how to work with piecewise functions is crucial for plotting and interpreting complex real-world applications that change behavior at certain thresholds.