Problem 3 - Precalculus with Calculus Previews

Question

Sketch the curve that has the given set of parametric equations. $$ x=t-1, y=2 t-1,-1 \leq t \leq 5 $$

Step-by-Step Solution

Answer

The sketch forms a line, \( y = 2x + 1 \), from \( (-2,-3) \) to \( (4,9) \).

Step 1: Understand the Parametric Equations

The parametric equations given are \( x = t - 1 \) and \( y = 2t - 1 \). These equations express both \( x \) and \( y \) in terms of the parameter \( t \).

Step 2: Eliminate the Parameter

To eliminate the parameter \( t \), solve the equation \( x = t - 1 \) for \( t \), giving \( t = x + 1 \). Substitute this into the equation for \( y \) to get \( y = 2(x + 1) - 1 \). Thus, \( y = 2x + 2 - 1 \), simplifying to \( y = 2x + 1 \). This is the equation of a line.

Step 3: Determine the Range

The parameter \( t \) is given by the interval \(-1 \leq t \leq 5\). Based on this, calculate the range for \( x \): when \( t = -1 \), \( x = -1 - 1 = -2 \); when \( t = 5 \), \( x = 5 - 1 = 4 \). Thus, \( -2 \leq x \leq 4 \).

Step 4: Plot the Graph Using Points

Use several values of \( t \) to find specific points: For \( t = -1, x = -2, y = -3 \), point is (-2,-3); for \( t = 0, x = -1, y = -1 \), point is (-1,-1); for \( t = 2, x = 1, y = 3 \), point is (1,3); for \( t = 5, x = 4, y = 9 \), point is (4,9).

Step 5: Sketch the Line

Draw a line through the points (-2,-3), (-1,-1), (1,3), and (4,9). This line is \( y = 2x + 1 \) with \( x \) ranging from -2 to 4.

Key Concepts

Eliminating Parameters

Eliminating parameters is a technique used to convert parametric equations into a single equation in terms of just one variable, usually expressed as "y" in terms of "x". In our exercise, we start with parametric equations, where both the x and y coordinates of points on a curve are expressed in terms of a third variable "t", called the parameter.

To eliminate "t", we solve one of the equations for "t" and substitute it into the other. Consider our example:

• Given: \( x = t - 1 \).
• Solving for "t": \( t = x + 1 \).
• Substituting into \( y = 2t - 1 \) gives \( y = 2(x + 1) - 1 \).
• This simplifies to the linear equation: \( y = 2x + 1 \).

By eliminating the parameter, we've shown that the original parametric curve represents a straight line, providing a clearer understanding of the graph's nature. This technique is particularly useful when we seek to understand the relationship between two variables without the intermediary role of the parameter.

Sketching Graphs

Sketching graphs from parametric equations involves finding critical points and understanding the curve's direction and range. Once we convert parametric equations into a familiar form such as "y = mx + b", our job of sketching becomes straightforward. Let's break this down:

• The line equation from our exercise is \( y = 2x + 1 \) which demonstrates a straight line.
• We need to focus on the given range of "t", which in turn gives us the range of "x": \(-2 \leq x \leq 4\).

By substituting distinct values within the parameter's range, we calculate the corresponding x and y values:

• For \( t = -1 \): \( (x, y) = (-2, -3) \)
• For \( t = 0 \): \( (x, y) = (-1, -1) \)
• For \( t = 2 \): \( (x, y) = (1, 3) \)
• For \( t = 5 \): \( (x, y) = (4, 9) \)

These points help us sketch the line segment, remembering the parameter limits restrict our line between certain points. It's important to observe both extremes of the range for a complete understanding of the curve's behavior.

Linear Equations

Linear equations represent some of the simplest forms of relationships between variables, typically in the format \( y = mx + b \), where "m" is the slope and "b" is the y-intercept. They are fundamental in graph theory for depicting direct relationships.

In our exercise, after eliminating the parameter, we derived the linear equation \( y = 2x + 1 \). This equation has some key properties:

• Slope (m): In this case, 2, which tells us the line rises by 2 units for every 1 unit it moves horizontally.
• Y-intercept (b): The line crosses the y-axis at \( y = 1 \), meaning when x is zero, y is one.

The discovery that our parametric equations translated into a linear form reassures us of the predictable nature of such equations, confirming they represent straight lines. Being able to identify this helps articulate the overall behavior and positioning of the line in the graph, aiding in both understanding and visualization.