GRAPHS

• Five parametrizations of the parabola \(y = x^2\):
  • Graph 1
  • Graph 2
  • Graph 3
  • Graph 4
  • Graph 5
• Graph 6 — Three parametrizations of a circle.
• Friends of Circles: An ellipse and a helix.

EXERCISE 1 — The Return of Vector Equations of Lines

Background & Notes

A vector function \(\vec{r}(t)\) is a function whose input is a scalar — the parameter — and whose output is a vector evaluated on the parameter.

In 3-space:

\[\vec{r}(t) = \big< x(t), y(t), z(t) \big>\]

• \(t\) — the parameter
• \(x(t), y(t), z(t)\) — the parametric functions (or coordinate functions).

The terminal points of a vector function are a 1-dimensional curve.

Vector functions are often used to represent the position of a moving object as a function of time.

Recall from the topic on lines and planes: A line can be parametrized using the vector function:

\[\begin{align*}
\vec{r}(t) &= t \vec{v} + \vec{r}_0\\
&= \big< at + x_0, bt + y_0, ct + z_0 \big>
\end{align*}\]

Assumptions

Let \(L\) be the line:

\[y = -x + 3\]

And let \(\vec{r}_A(t), \vec{r}_B(t), \vec{r}_C(t)\) be the position functions for three objects, all traveling along the same line:

\[\begin{align*}
\vec{r}_A(t) &= \big< 3t - 1, - 3t + 4 \big>\\
\vec{r}_B(t) &= \big< -2t + 1, 2t + 2 \big>\\
\vec{r}_C(t) &= \big< -5t + 5, 5t - 2 \big>
\end{align*}\]

Instructions

1. Evaluate & Graph

In your notebook, sketch the \(xy\)-coordinate axes with the line \(y = -x + 3\) three times.

Then, evaluate each of the three vector functions at time \(t = 0\).

\[\begin{align*}
\vec{r}_A(0) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>\\
\vec{r}_B(0) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>\\
\vec{r}_C(0) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>
\end{align*}\]

Add each of these three vectors to your sketch — label them so you can identify them later!

Repeat for time \(t = 1\): Evaluate the three vector functions at time \(t = 1\), and add them to your sketch (label them!).

\[\begin{align*}
\vec{r}_A(1) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>\\
\vec{r}_B(1) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>\\
\vec{r}_C(1) &= \big< \, \underline{\hspace{1cm}} \, , \, \underline{\hspace{1cm}} \, \big>
\end{align*}\]

2. Analyze

Answer the following questions about the three position functions:

• Do these three position functions point to the same point when \(t = 0\)?
• Do these three position functions point to the same point when \(t = 1\)?
• Are all three objects traveling in the same direction?
• Are all three objects traveling at the same speed?

3. Challenge — Construct Examples

Find a vector function \(\vec{r}(t)\) that describes the position of an object traveling along the line \(y = -x + 3\) so that:

• It’s located on the \(y\)-axis — at the point \((0,3)\) — when \(t = 0\).
• It’s located on the \(x\)-axis — at the point \((3,0)\) — when \(t = 1\).

Find a vector function \(\vec{r}(t)\) that describes the position of an object traveling parallel to the \(z\)-axis so that:

• It’s located on the \(xy\)-plane at the point \(P_0 (2,3,0)\) when \(t = 0\).
• It’s located at the point \(P_1(2,3,5)\) when \(t = 1\).

EXERCISE 2 — Parametrizing Graphs of Functions

Background & Notes

Graphs of functions can be parametrized using the relationship between the coordinate functions.

If \(y = f(x)\), then one way to parametrize its graph is:

\[\vec{r}(t) = \big< t, f(t) \big>\]

or, more generally:

\[\vec{r}(t) = \big< x(t), f\big(x(t)\big) \big>\]

Assumptions

Let \(\vec{r}_1, \vec{r}_2, \vec{r}_3, \vec{r}_4, \vec{r}_5\) be vector functions that parametrize the graph \(y = x^2\):

Instructions

1. Graph & Analyze

Visit the graphs corresponding to the five vector functions.

For each, locate the t-slider at the top of the command window, and use it to move the position vector \(\vec{r}_i(t)\).

Observe the motion of the position vector (blue) over the graph of the parabola \(y = x^2\) as the \(t\)-values increase over the parameter interval.

(On these graphs, “\(\infty\)” is represented by "-10$.)

Based on your observations, answer these questions:

• Which of the five vector functions trace out the entire parabola? Which ones trace out only half of the parabola? Which ones trace out some other part of the parabola?
• Compare the graphs for \(\vec{r}_1, \vec{r}_2, \vec{r}_3\). What effect does changing the endpoints of the parameter interval have on the amount of the graph \(y = x^2\) traced out by the vector function?
• Which of the five vector functions move strictly from left to right on the graph as \(t\) increases? Which move from strictly right to left as \(t\) increases? Which move in some other way?

2. Construct Examples & Graph

Construct vector functions for each of the three graphs:

\[\begin{align*}
y &= x^3\\
y &= e^x\\
x &= \cos y
\end{align*}\]

Identify the coordinate functions \(x(t), y(t)\) of your examples.

Then, go to the graph of the corresponding function:

• Graph of \(y = x^3\)
• Graph of \(y = e^x\)
• Graph of \(x = \cos y\)

From the Add to graph: Select ... drop-down menu, select Space Curve: r(t).

In the new curve, set \(x(t)\) and \(y(t)\) equal to the coordinate functions of your example, and set \(z(t) = 0\).

If needed, change the endpoints of the parameter interval to -10 ≤ t ≤ 10.

Check that your parametrization matches the graph of the function.

3. Challenge

Find a vector function \(\vec{r}(t)\) that:

• Parametrizes the part of the parabola \(y = x^2\) from \(P(-1,1)\) to \(Q(1,1)\).
• Bounces back and forth as \(t\) increases from \(t = -10\) to \(t = 10\).

EXERCISE 3 — Circles, Ellipses, Helices

Background & Notes

A circle \(x^2 + y^2 = R^2\) in the \(xy\)-plane can be parametrized using polar coordinates:

\[\vec{r}(t) = \big< R \cos(ct + k), R \sin(ct + k) \big>\]

Where \(R > 0\) is a constant.

Assumptions

Let \(\vec{r}_A(t)\), \(\vec{r}_B(t)\), \(\vec{r}_C(t)\) be vector functions parametrizing the same circle:

\[\begin{align*}
\vec{r}_A(t) &= \big< 2 \cos t, 2 \sin t \big>\\
\vec{r}_B(t) &= \big< 2 \cos 3t, 2 \sin 3t \big>\\
\vec{r}_C(t) &= \big< 2 \cos (t + \pi/2), 2 \sin (t + \pi/2) \big>
\end{align*}\]

Instructions

1. Graph & Analyze

Go to graph 6, and in the control window, identify the space curves corresponding to each of the three vector functions:

• \(\vec{r}_A(t)\) is the top space curve (red).
• \(\vec{r}_B(t)\) is the middle space curve (blue).
• \(\vec{r}_C(t)\) is the bottom space curve (green).

Select the box for one of the vector functions, and use the t-slider associated with that vector function to observe its motion as it traces out the circle.

Based on your observations, answer the following questions:

• As \(t\) increases, do these vector functions travel clockwise or counter-clockwise around the circle?
• When \(t = 0\), where is each vector function located on the circle?
• What is the radius of the circle? Can you write the equation for this circle in Cartesian coordinates?

2. Experiment

In graph 6, select the vector function \(\vec{r}_A(t)\) at the top of the control window (red).

Multiply the \(t\)-variable by -1 in both coordinate functions, so that \(x(t) = 2 \cos(-t)\), \(y(t) = 2 \sin(-t)\).

How does this change the motion of the vector function as \(t\) increases?

Next, select the vector function \(\vec{r}_C(t)\) at the bottom of the control window (green).

Change the constant \(\pi/2\) in each coordinate function, so that \(x(t) = 2 \cos(t + pi)\) and \(y(t) = 2 \sin(t + pi)\).

How does this change the location of the vector function on the circle when \(t = 0\)?

3. Challenge

Construct a vector function \(\vec{r}(t)\) that parametrizes the circle:

\[x^2 + y^2 = 25\]

so that:

• As \(t\) increases, the direction of travel around the circle is counter-clockwise.
• When \(t = 0\), the vector function \(\vec{r}(0)\) is located on the positive \(x\)-axis.

4. Friends of Circles — Ellipses & Helices

An ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) can be parametrized by the vector function:

\[\vec{r}(t) = \big< a \cos t, b \sin t \big>\]

where \(a, b > 0\) are constants.

A circular helix centered about the \(z\)-axis can be parametrized by the vector function:

\[\vec{r}(t) = \big< R \cos(2\pi t), R \sin(2\pi t), m t\big>\]

where \(R, m\) are constants, \(R > 0\).

Ellipses and helices are both types of “generalized circles”.

When are ellipses, circles? When is a helix a circle?

BIG IDEAS

Vector Equations of Lines, Revisited (exercise 1)

Vector functions \(\vec{r}(t)\) can be used to represent position as a function of time.

A curve is a 1-dimensional set of points in a higher-dimensional space.

In the same way that different drivers will travel the same road at different speeds, in different directions, with different starting points, etc, a curve can be parametrized by (infinitely many) different vector functions.

Parametrizing Graphs of Functions (exercise 2)

Graphs of functions can be parametrized using the relationship between the coordinate functions.

If \(y = f(x)\), one way to parametrize its graph is:

\[\vec{r}(t) = \big< t, f(t) \big>\]

Similarly, the graph of \(x = g(y)\) can be parametrized:

\[\vec{r}(t) = \big< g(t), t \big>\]

Circles, Ellipses, Helices (exercise 3)

A circle \(x^2 + y^2 = R^2\), oriented counter-clockwise, can be parametrized by the vector function:

\[\vec{r}(t) = \big< R \cos(c t + k), R \sin(c t + k) \big>\]

where \(R, c, k\) are constants, with \(R, c > 0\).

An ellipse \(\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1\), oriented counter-clockwise, can be parametrized by the vector function:

\[\vec{r}(t) = \big< a \cos(c t + k), b \sin(c t + k) \big>\]

where \(a, b, c, k\) are constants, with \(a, b, c > 0\)

A circular helix centered about \(z\)-axis can be parametrized by the vector function:

\[\vec{r}(t) = \big< R \cos( 2\pi c t), R \sin( 2\pi c t), m t \big>\]

where \(R, c, m\) are constants, with \(R, c > 0\).