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Parametric equations are a way to express the coordinates of the points that make up a curve. Instead of describing the curve with a single equation in terms of x and y, parametric equations express both the x and y coordinates as functions of a third variable, usually denoted as t, known as the parameter.
In the vector-valued function \( \mathbf{r}(t) = \left\langle 6t, 6t - t^2 \right\rangle \), the parametric equations are:

• \( x(t) = 6t \)
• \( y(t) = 6t - t^2 \)

These equations tell us that for any given value of the parameter \(t\), there is a specific point \((x, y)\) on the curve. For instance, when \(t = 1\), \(x(1) = 6\) and \(y(1) = 5\), giving the point \((6, 5)\) on the curve.
The advantage of parametric equations is that they can easily represent complex curves and motions which might be difficult to express as a single equation relating x and y. They also show the trajectory and position over time, making them especially useful in physics and engineering applications.

Curve sketching involves plotting a curve by determining the key characteristics and points along the curve. It helps in visualizing how a curve behaves and changes. For our example, the simplified equation obtained from eliminating the parameter \(t\) was \(y = x - \frac{x^2}{36}\).

Key Points to Analyze:

• Identify the vertex: The vertex of the parabola is at \((0, 0)\). This means the point where the curve changes its direction.
• Determine direction: The negative coefficient of \(x^2\) indicates that the parabola opens downward.
• Intercepts: When \(x = 0\), \(y = 0\). This is the intersection point at the y-axis.
• Additional Points: By substituting values like \(x = 6\), \(x = 12\) and \(x = -6\), we derive more points to plot.

Plotting these points on a graph provides a framework for smoothly sketching the curve. Visualization like this can solidify understanding of the behavior of the function and how the equation translates to a geometric shape.

A quadratic equation is any equation that can be written in the form \(y = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. These equations create parabolas on a graph. Depending on the sign of \(a\), the parabola will open upwards (\(a > 0\)) or downwards (\(a < 0\)). In our case, the quadratic equation derived was \(y = x - \frac{x^2}{36}\).
Because the term \(- \frac{x^2}{36}\) carries a negative coefficient, the parabola opens downwards.

Analyzing a Quadratic Equation:

• Vertex Form: The equation already suggested that the vertex is at \((0, 0)\), as the standard vertex form \(y = a(x-h)^2 + k\) points towards \(h = 0\) and \(k = 0\).


• Axis of Symmetry: This is the vertical line \(x = h\), here \(x = 0\), meaning the curve is symmetric about the y-axis.


• End Behavior: As \(x\) moves further from 0 in either direction, \(y\) becomes more negative, clearly indicating the downward opening.

Understanding quadratic equations allows for predicting the shape and position of parabolas without always having to plot several points.