Question

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=t^{2}+2 t, y=t+1 $$

Short answer

The curve is a parabola \( x = y^2 - 1 \), oriented from left to right as \( t \) increases.

Step-by-step solution

Express x in terms of t

Given the parametric equations \( x = t^2 + 2t \) and \( y = t + 1 \), start by expressing \( t \) in terms of \( y \). From the second equation, solve for \( t \): \( t = y - 1 \).

Substitute t in x equation

Substitute \( t = y - 1 \) into the equation for \( x \). This becomes \( x = (y - 1)^2 + 2(y - 1) \).

Simplify the Expression

Expand and simplify the expression from Step 2. This gives \( x = y^2 - 2y + 1 + 2y - 2 \). Simplifying further, we obtain \( x = y^2 - 1 \).

Sketch the Curve

Now we have the equation \( x = y^2 - 1 \), which is a parabola opening to the right (horizontal parabola). The vertex of this parabola is at \((-1, 0)\).

Determine the Orientation

As \( t \) increases from negative to positive, \( y = t + 1 \) also increases. The parametric form gives a direction: the curve is traced from left to right as \( t \) increases.

Key concepts

Curve Sketching

Curve sketching is a method used to draw a rough image of a curve defined by mathematical equations without plotting numerous points. It helps to understand the overall shape, size, orientation, and key features of the curve.

For curves defined parametrically, like in our original exercise, sketching involves understanding how the parameters control the coordinates. These kinds of curves can often represent complex shapes that aren't easily graphed with standard functions.

Key steps in curve sketching include:

• Finding the Cartesian equation from the parametric equations.
• Identifying critical points such as vertices, intercepts, and points of inflection.
• Determining the orientation and direction of the curve based on how the parameter changes.

In our example, we converted the parametric functions into a single Cartesian equation, which helped to visualize and understand the form and position of the curve in the Cartesian plane.

Parabolas

Parabolas are u-shaped graphs that are easy to understand once their basic properties are known. In the plane, they can open up, down, left, or right depending on their equations. Parabolas are common in many real-world situations, from the path of thrown objects to the design of satellite dishes.

A standard form of a parabola is usually either centered vertically (opening up or down) or horizontally (opening left or right).

In our problem, the simplified equation was \( x = y^2 - 1 \) indicating a horizontal parabola:

• The vertex is the highest or lowest point, pivotal in defining the curve's overall position. Here, it's at (-1, 0).
• Opening direction, horizontal in this case, is determined by squaring \(y\).
• Additional features include the axis of symmetry, a line that reflects the parabola onto itself.

This understanding allows us to predict the curve's behavior and sketch it accurately.

Eliminating Parameters

Eliminating parameters is a crucial technique in mathematics to convert parametric equations into standard Cartesian equations. This simplification removes the parameter, leaving a relation purely between the x and y coordinates.

In our exercise, we started with the parametric equations:

• \( x = t^2 + 2t \)
• \( y = t + 1 \)

The goal was to express \( x \) in terms of \( y \) by eliminating the parameter \( t \). We first solved for \( t \) using the second equation, giving \( t = y - 1 \). This result was substituted into the \( x \) equation, resulting in a step-by-step transformation: \( x = (y - 1)^2 + 2(y - 1) \), which simplifies to a familiar quadratic equation \( x = y^2 - 1 \).

Eliminating parameters often makes mathematical relationships clearer, eases graphing, and improves our understanding of the relationship between the variables.