Learning Outcomes
- Sketch the graph of a two-dimensional parametric/vector equation, and convert such equations into equations of only \(x\) and \(y\text{.}\)
Section 7.1 Parametric/vector equations (CO1)
Subsection 7.1.1 Activities
Activity 7.1.1.
Consider how we might graph the equation \(y=2-x^2\) in the \(xy\)-plane.
(a)
Complete the following chart of \(xy\) values by plugging each \(x\) value into the equation to produce its \(y\) value.
| \(x\) | \(y\) |
| \(-2\) | |
| \(-1\) | \(1\) |
| \(0\) | |
| \(1\) | |
| \(2\) |
(b)
Plot each point \((x,y)\) in your chart in the \(xy\) plane.
(c)
Connect the dots to obtain a reasonable sketch of the equation’s graph.
Activity 7.1.2.
Suppose that we are told that at after \(t\) seconds, an object is located at the \(x\)-coordinate given by \(x=t-2\) and the \(y\)-coordinate given by \(y=-t^2+4t-2\text{.}\)
(a)
Complete the following chart of \(txy\) values by plugging each \(t\) value into the equations to produce its \(x\) and \(y\) values.
| \(t\) | \(x\) | \(y\) |
| \(0\) | ||
| \(1\) | \(-1\) | \(1\) |
| \(2\) | ||
| \(3\) | ||
| \(4\) |
(b)
Plot each point \((x,y)\) in your chart in the \(xy\) plane, labeling it with its \(t\) value.
(c)
Connect the dots to obtain a reasonable sketch of the equation’s graph.
Definition 7.1.3.
Graphs in the \(xy\) plane can be described by parametric equations \(x=f(t)\) and \(y=g(t)\text{,}\) where plugging in different values of \(t\) into the functions \(f\) and \(g\) produces different points of the graph.
The \(t\)-values may be thought of representing the moment of time when an object is located at a particular position, and the graph may be thought of as the path the object travels throughout time.
Activity 7.1.4.
Earlier we obtained the same graphs for the \(xy\) equation \(y=2-x^2\) and the parametric equations \(x=t-2\) and \(y=-t^2+4t-2\text{.}\) Do the following steps to find out why.
(a)
Which of the following equations describes \(t\) in terms of \(x\text{?}\)
- \(\displaystyle t=x-2\)
- \(\displaystyle t=x+2\)
- \(\displaystyle t=2x\)
- \(\displaystyle t=-2x\)
(b)
Which of these is the result of plugging this choice in for \(t\) in the parametric equation for \(y\text{?}\)
- \(\displaystyle y=-x+2^2+4x+2-2\)
- \(\displaystyle y=-(x+2)^2+4(x+2)-2\)
- \(\displaystyle y=-x^2+2^2+4x+4\cdot2-2\)
(c)
Show how to simplify this choice to obtain the equation \(y=2-x^2\text{.}\)
Fact 7.1.5.
One method of graphing parametric equations \(x=f(t)\) and \(y=g(t)\) is to combine them into a single equation only involving \(x\) and \(y\text{,}\) and using your usual graphing techniques.
Activity 7.1.6.
Parametric equations have the advantage of describing paths that cannot be described by a function \(y=h(x)\text{.}\) One such example is the graph of \(x=3\sin(\pi t)\) and \(y=-3\cos(\pi t)\text{.}\) (Use technology or the approximation \(\sqrt 2\approx 0.707\) to approximate coordinates as needed.)
(a)
Complete the folowing table.
| \(t\) | \(x\) | \(y\) |
| \(0\) | ||
| \(1/4\) | ||
| \(1/2\) | ||
| \(3/4\) | \(2.12\) | \(2.12\) |
| \(1\) | ||
| \(5/4\) | ||
| \(3/2\) | ||
| \(7/4\) | ||
| \(2\) |
(b)
Plot these \((x,y)\) points in the \(xy\) plane and connect the dots to draw a sketch of the graph.
(c)
What do you obtain by plugging the parametric equations into the expression \(x^2+y^2\text{?}\)
- \(\displaystyle x^2+y^2=-6\sin(\pi x)\cos(\pi x)\)
- \(\displaystyle x^2+y^2=9\)
- \(\displaystyle x^2+y^2=6\sin(\pi x)\cos(\pi x)\)
- \(\displaystyle x^2+y^2=0\)
(d)
Which of these describes the \(xy\) equation and graph given by these parametric equations?
- a parabola
- a line
- a circle
- a square
(e)
The graph of these parametric equations cannot be described by a function. Why?
- The graph fails the vertical line test.
- The graph fails the horizontal line test.
- The graph doesn’t extend vertically to \(+\infty\text{.}\)
- The graph doesn’t extend horizontally to \(-\infty\text{.}\)
Definition 7.1.7.
The parametric equations \(x=f(t)\) and \(y=g(t)\) are sometimes written in the form of the vector equation \(\vec r=\tuple{f(t),g(t)}\text{.}\)
For example, the parametric equations \(x=3\sin(\pi t)\) and \(y=-3\cos(\pi t)\) may be combined into the single vector equation \(\vec r=\tuple{3\sin(\pi t),-3\cos(\pi t)}\text{.}\)
Activity 7.1.8.
Consider the vector equation \(\vec r=\tuple{2t-3,-6t+13}\text{.}\)
(a)
What are the corresponding parametric equations?
- \(x=2t-3\) and \(y=-6t+13\)
- \(y=2t-3\) and \(x=-6t+13\)
- \(\displaystyle xy=2t-3-6t+13\)
- Vector equations cannot be converted into parametric equations.
(b)
Draw a table of \(t\text{,}\) \(x\text{,}\) and \(y\) values with \(t=0,1,2,3,4\text{.}\)
(c)
Plot these \((x,y)\) points in the plane and connect the dots to sketch the graph of this vector equation.
(d)
Solve for \(t\) in terms of \(x\) and plug into the \(y\) parametric equation to show that this is the vector equation for the line \(y=-3x+4\text{.}\)
