Functions Involving Parameters, Vectors, and Matrices

b≠c and e=f

b=c and e=f

b=0 and e=0

b=1 and e=1

A curved line on an xy-plane where t-values increase then decrease along it

A series of horizontal line segments on an xy-plane with gaps between them

A straight-line segment on an xy-plane where t-values increase uniformly along it

Multiple separate points scattered across an xy-plane corresponding to different t-values

Y-axis symmetry

X-axis symmetry

Origin symmetry

Rotational symmetry

It follows an upward-opening parabola.

It moves along a circle of radius 5.

It moves along an ellipse centered at the origin.

It travels back and forth on a straight line.

$f(x) = ax^2 + bx + c$

$y = mx + b$

$x(t), y(t)$

$(x, y) \mid x = f(y)$

Force

Time

Speed

Distance

They eliminate the need for technology such as calculators or computers.

They always give exact solutions.

Numerical methods simplify algebraic manipulation significantly.

They provide approximate solutions when analytical methods are complex.

Smooth transitions without interruption which can indicate consistent behaviors

Discontinuous motion can represent changes in direction or speed even if not physically possible

Discontinued paths are usually preferred in engineering and science since they are simpler to analyze

Discontinuities require advanced calculations for precise models which are not necessary

Alteration in spatial dimensions only without time consideration.

Modification in geometric shapes being described by parameters.

Creation of new variables unrelated to time.

Progression of time throughout the motion.

$\frac{\pi}{2}$

$\frac{\pi}{5}$

$\pi$

$2\pi$