Question

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=3-t, y=2 t-3,1.5 \leq t \leq 3 $$

Short answer

The curve is the line segment \( y = -2x + 3 \) from \( (1.5, 0) \) to \( (0, 3) \), oriented from right to left.

Step-by-step solution

Solve for the Parameter

We begin with the parameter equations: \( x = 3 - t \) and \( y = 2t - 3 \). To combine these, we first solve one equation for \( t \). Let's rearrange the first equation for \( t \):\[ t = 3 - x \]

Substitute Parameter

Substitute the expression for \( t \) from Step 1 into the second equation: \( y = 2(3 - x) - 3 \).

Simplify the Equation

Simplify the equation in Step 2:\[ y = 6 - 2x - 3\]Further simplify to obtain:\[y = -2x + 3\]

Determine the Range

Check the range of \( t \) which is \( 1.5 \leq t \leq 3 \). Use these bounds with \( t = 3 - x \) to find the corresponding values of \( x \). For \( t = 1.5 \), \( x = 1.5 \). For \( t = 3 \), \( x = 0 \). This gives us the range \( 0 \leq x \leq 1.5 \).

Determine the Orientation

The orientation indicates the direction of increasing \( t \). As \( t \) increases from 1.5 to 3, \( x \) decreases from 1.5 to 0. Therefore, the curve runs from the right to the left, along the line \( y = -2x + 3 \), starting at \( (1.5, 0) \) and ending at \( (0, 3) \).

Sketch the Curve

Sketch the line \( y = -2x + 3 \) over the range \( 0 \leq x \leq 1.5 \) and indicate the orientation with an arrow pointing from \( (1.5, 0) \) to \( (0, 3) \).

Key concepts

Curve Sketching

When sketching curves, especially those defined by parametric equations, we often start by eliminating the parameter to express the curve in a more familiar Cartesian form. In our example, we have the parametric equations:

• \( x=3-t \)
• \( y=2t-3 \)

To sketch the curve, we need to find a relation directly between \( x \) and \( y \). This involves substituting the parameter \( t \) with a relationship that connects \( x \) and \( y \). After substituting and simplifying, we find: \[ y = -2x + 3 \] Sketching this line on the Cartesian plane involves plotting this equation over the specified range for \( x \). The curve takes the shape of a straight line. Begin by plotting points using the bounds of \( x \): where \( x = 0 \), \( y = 3 \), and \( x = 1.5 \), \( y = 0 \). Draw the line through these points to visualize the curve.

Equation Simplification

Simplifying equations in mathematics helps in making complicated equations easier to work with by reducing them to simpler forms. With parametric equations, the goal is often to transform them into a simpler form or a standard equation, such as the line equation we derived earlier. Starting with:

• \( x=3-t \)
• \( y=2t-3 \)

We solved the first equation for \( t \) resulting in \( t=3-x \), and substituted this into the second equation. This way, we effectively eliminate the parameter \( t \), allowing us to focus solely on \( x \) and \( y \). The substitution results in:\[ y = 6 - 2x - 3 \]Which simplifies stepwise to:\[ y = -2x + 3 \]This transformation highlights the linearity of the relationship between \( x \) and \( y \). Simplifying equations not only unveils these relationships but also makes it easier to sketch and analyze the behavior of the curve.

Range and Orientation

In parametric equations, understanding the range and orientation is essential for accurately sketching the curve. The range provides the span of \( x \) values that need to be plotted, while the orientation indicates the direction in which the curve is drawn. Given:

• \( 1.5 \leq t \leq 3 \)

This translates the parameter \( t \) into limits for \( x \) using the equation \( t = 3 - x \). Compute the bounds for \( x \): when \( t = 1.5 \), \( x = 1.5 \); when \( t = 3 \), \( x = 0 \). This verifies that the range of \( x \) is from 0 to 1.5. The orientation requires evaluating the behavior of \( x \) as \( t \) increases. For this problem, as \( t \) goes from 1.5 to 3, \( x \) decreases from 1.5 to 0, giving the curve a leftward orientation. Thus, the curve on the line \( y = -2x + 3 \) is sketched from right to left, from \((1.5, 0)\) to \((0, 3)\). Understanding these attributes ensures accurate representation of the curve.