Functions Involving Parameters, Vectors, and Matrices

$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

$\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$

$\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}$

$\begin{pmatrix} -1 & -1 \\ 0 & 0 \end{pmatrix}$

f(x) = (a + t)x + (d + p)

f(x) = (at + c)x + (bt + d)

f(x) = (t + a)x + (d + b)

f(x) = a(t + x) + b(d + x)

Point P moves closer to the origin, whereas Q moves away.

Both points get further apart from each other.

They remain in the same place.

Both points move towards each other.

The eigenvectors' corresponding scalar multiples

All points equidistant from origin

Any point along the line $y = x$

Coordinates where both elements are equal

x = 0

x = -2

x = 4

x = 2

Jump discontinuity, due to inconsistent limits from left and right.

Infinite discontinuity stemming from the numerator approaching infinity faster than the denominator.

There’s a removable point of discontinuity, as indicated by algebraic simplification.

Jump discontinuity, due to absence of real numerator roots matching the denominator.

M is a symmetric Matrix reflecting vectors across a main diagonal

M is a singular Matrix creating a dependent vector from an independent one

M is a diagonal Matrix and the transformation is a scale transformation with factor $m$

M is an anti-diagonal Matrix representing rotation by 90 degrees

Infinite discontinuity

Jump discontinuity

Oscillating discontinuity

Removable discontinuity

They indicate non-removable points of discontinuity.

They suggest places where limits do not exist on either side.

They show locations where j(x) crosses its horizontal asymptote.

They always correspond with zeroes in the derivative of j(x).

The difference between corresponding entries from both matrices.

The sum of corresponding entries from both matrices.

The dot product of corresponding row from first matrix and column from second matrix.

The product of corresponding entries from both matrices.