Chapter 8.4, Problem 1E

1. (a) The parametric equations x = f(t) and y = g(t) give the coordinates of a point (x, y) = (f(t), g(t)) for appropriate values of t. The variable t is called a __________.
2. (b) Suppose that the parametric equations x = t, y = t², t ≥ 0, model the position of a moving object at time t. When t = 0, the object is at $(\boxed{},\boxed{})$, and when t = 1, the object is at $(\boxed{},\boxed{})$.
3. (c) If we eliminate the parameter in part (b), we get the equation y = __________. We see from this equation that the path of the moving object is a __________.

To determine

To fill: The variable $t$ for which the parametric equation is defined.

Answer to Problem 1E

The variable $t$ in parametric equations $x=f\left(t\right)$ and $y=g\left(t\right)$ is called a parameter.

Explanation of Solution

The two functions $f\left(t\right)$ and $g\left(t\right)$ are defined on an interval $I$, then the set of coordinates $\left(f\left(t\right),g\left(t\right)\right)$ form a plane curve

The equations $x=f\left(t\right)$ and $y=g\left(t\right)$ are the parametric equations of the curve formed where $t$ is the parameter.

To determine

To fill: The position coordinates $\left(x,y\right)$ of a moving object at time $t=0$ and time $t=1$.

Answer to Problem 1E

The position coordinate $\left(x,y\right)$ of a moving object with parametric equations $x=t$ and $y=t^2$ at time $t=0$ is $\left(0,0\right)$ and at time $t=1$ is $\left(1,1\right)$.

Explanation of Solution

The two parametric equations are $x=t$ and $y=t^2$.

Substitute $0$ for $t$ in the above equations.

The value of $x$ is $0$ and the value of $y$ is $0$.

Clearly, the position coordinate $\left(x,y\right)$ of a moving object at time $t=0$ is $\left(0,0\right)$.

Substitute $1$ for $t$ in the above equations.

The value of $x$ is $1$ and the value of $y$ is $1$.

Clearly, the position coordinate $\left(x,y\right)$ of a moving object at time $t=1$ is $\left(1,1\right)$.

Thus, the position coordinate $\left(x,y\right)$ of a moving object with parametric equations $x=t$ and $y=t^2$ at time $t=0$ is $\left(0,0\right)$ and at time $t=1$ is $\left(1,1\right)$.

To determine

To fill: The equation after elimination of the parameter and the path formed by the equation.

Answer to Problem 1E

The equation after elimination of parameter is $y=x^2$ and the path of the equation is parabolic.

Explanation of Solution

From part (b) the two parametric equations are $x=t$ and $y=t^2$.

Eliminate the parameter in the above parametric equations.

Substitute $t$ for $x$ in the equation $y=t^2$.

$y=x^2$

Thus, the required equation is $y=x^2$.

The below graph is of the equation $y=x^2$,

Figure (1)

Clearly, the above graph is of a parabola.

Thus, the path of the moving object is parabolic.