Problem 2 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) whose graph contains the points on the curve \(C\). Sketch the graph of \(C\), and indicate the orientation. $$x=1-2 t, \quad y=1+t, \quad-1 \leq t \leq 4$$

Step-by-Step Solution

Answer

The Cartesian equation is \(x + 2y = 3\). The graph is a line with orientation from (-7, 5) to (3, 0).

Step 1: Analyze the Parametric Equations

The given parametric equations are \( x = 1 - 2t \) and \( y = 1 + t \) with the parameter \( t \) ranging from -1 to 4.

Step 2: Solve for t in terms of x

To convert the parametric equations to a Cartesian equation, we first solve one of the equations for \( t \). From \( x = 1 - 2t \), we can rearrange to find \( t = \frac{1-x}{2} \).

Step 3: Substitute t in the y equation

Substitute \( t = \frac{1-x}{2} \) into the equation for \( y \). Thus, \( y = 1 + \frac{1-x}{2} \).

Step 4: Simplify the Cartesian Equation

Simplify the expression for \( y \):\[ y = 1 + \frac{1-x}{2} = 1 + \frac{1}{2} - \frac{x}{2} = \frac{3}{2} - \frac{x}{2} \].Thus, the Cartesian equation is \( 2y = 3 - x \) or equivalently, \( x + 2y = 3 \).

Step 5: Sketch the Graph and Indicate Orientation

The graph of the equation \( x + 2y = 3 \) is a straight line. To find points on this line, use the parametric range:When \( t = -1 \), \( (x, y) = (3, 0) \).When \( t = 4 \), \( (x, y) = (-7, 5) \).Plot these points and draw a line through them.Orientation follows the direction of increasing \( t \) from \( (-7, 5) \) towards \( (3, 0) \).

Key Concepts

Cartesian Equation

In mathematics, a Cartesian equation involves variables like \(x\) and \(y\) without parameters. This makes them easy to graph on a 2D plane. Here, our goal is to convert the given parametric equations into a single Cartesian equation.

The parametric equations given are \( x = 1 - 2t \) and \( y = 1 + t \). To eliminate the parameter \(t\), we first express \(t\) in terms of \(x\). From the equation \(x = 1 - 2t\), rearrange to get \(t = \frac{1-x}{2}\).

Next, substitute \(t = \frac{1-x}{2}\) into the equation for \(y\):

• Begin with the parametric form: \( y = 1 + t \).
• Replace \(t\) with \(\frac{1-x}{2}\) to get: \( y = 1 + \frac{1-x}{2} \).
• Simplify: \( y = \frac{3}{2} - \frac{x}{2} \).

This simplifies to the linear Cartesian equation \(x + 2y = 3\). With these steps, the parametric form is successfully converted into a Cartesian form, making graphing and analyzing the relationships of variables straightforward.

Graph Orientation

Graph orientation refers to the direction in which we're supposed to move along the graph, often indicated in relation to parameters like \(t\) in parametric equations. It tells us how a graph "flows" over its entire range.

For the exercise at hand, we begin with given parametric equations \(x = 1 - 2t\) and \(y = 1 + t\). The parameter \(t\) ranges from \(-1\) to \(4\). These parametric bounds help us determine the starting and ending points which guide the orientation.

- When \(t = -1\), we calculate that \((x, y) = (3, 0)\), which represents one endpoint.- At \(t = 4\), the coordinates become \((-7, 5)\), marking the other endpoint.
This tells us the graph starts at the point \( (3, 0) \) and ends at \( (-7, 5) \). By drawing a straight line through these points, orientation is shown by direction from the starting point towards the ending point. With increasing \(t\), the graph flows from right to left.

Linear Equations

Linear equations are among the simplest forms of equations, known for their straight-line graphs. In a Cartesian coordinate system, any two linear variables \(x\) and \(y\) relate in a manner that can be expressed with a polynomial of the first degree.

In this exercise, our converted Cartesian equation \(x + 2y = 3\) is a clear example of a linear equation. Its general form can typically be written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.

• The coefficients \(A\) and \(B\) (here 1 and 2) determine the slope of the line.
• The constant \(C\) (here 3) determines where the line intersects the axes.

To sketch this line, identify key points derived from the equation. For example, set \(x = 0\) to find the \(y\)-intercept or \(y = 0\) to find the \(x\)-intercept. This method of analysis illustrates the simplicity and usefulness of linear equations in depicting relationships between variables.