Unit 4: Functions Involving Parameters, Vectors, and Matrices

Parameter

A value that acts like a constant within one situation but can change between situations; changing it produces a different graph/model.

Family of functions

A collection of functions generated by varying a parameter in a formula (e.g., different values of a give different parabolas).

Input variable (independent variable)

The variable (like x or t) that changes along a single graph and determines the output of the function.

Parameter vs. input variable

The input variable varies along the curve; a parameter changes the entire curve (a different member of the family).

Quadratic vertex form

A quadratic written as f(x)=a(x−h)^2+k, which makes shifts and stretching/reflection easier to see.

Vertex (in vertex form)

For f(x)=a(x−h)^2+k, the vertex is (h,k) and does not change when only a changes.

Vertical stretch/compression (quadratic)

In f(x)=a(x−h)^2+k, |a|>1 makes the parabola narrower (stretch) and 0<|a|<1 makes it wider (compression).

Parabola opening direction

In f(x)=a(x−h)^2+k, a>0 opens upward and a<0 opens downward.

Exponential growth model

A model like P(t)=P0(1+r)^t where growth depends on an initial value and a growth rate parameter.

Initial value (P0)

In P(t)=P0(1+r)^t, the starting amount at t=0; a parameter that sets the initial height/scale of the model.

Growth rate (r)

In P(t)=P0(1+r)^t, the parameter controlling how fast the quantity grows or decays per time step.

Growth factor (1+r)

The base of the exponential in P(t)=P0(1+r)^t; increasing r increases the growth factor and speeds up growth.

Parametric representation

A way to describe a curve by writing both x and y as functions of a parameter (often time).

Parametric equations

Equations of the form x=f(t) and y=g(t) that produce points (x(t),y(t)) as t varies.

Coordinate functions

The separate functions x(t) and y(t) in a parametric description; both depend on the same parameter t.

Trace (parametric curve)

The set of points (x(t),y(t)) produced as t varies; the point moves along and “traces” the curve.

Orientation (direction of tracing)

The direction a parametric curve is traveled as t increases; this is part of the parametric description.

Eliminating the parameter

Converting parametric equations to a single x–y relation by solving for t in one equation and substituting into the other.

Cartesian equation

A single equation relating x and y (no parameter), often found by eliminating the parameter from parametric equations.

Circle parametrization

A radius-r circle centered at the origin can be written as x=r cos(t), y=r sin(t).

Pythagorean identity (trig)

cos^2(t)+sin^2(t)=1; used to eliminate t from x=r cos(t), y=r sin(t) to get x^2+y^2=r^2.

Parameter interval restriction

Restricting t (e.g., 0≤t≤2) restricts which portion of the curve is traced, even if the Cartesian equation describes more.

Planar motion model

Interpreting parametric equations with t as time, where x(t) and y(t) give an object’s position over time.

Position (in parametrics)

The location of a particle at time t given by the point (x(t),y(t)).

Position vector

A vector from the origin to the position: p(t)=⟨x(t),y(t)⟩ (or x(t)i+y(t)j).

Displacement vector

Change in position from t=a to t=b: ⟨x(b)−x(a), y(b)−y(a)⟩.

Average velocity vector

Displacement divided by elapsed time: ⟨(x(b)−x(a))/(b−a), (y(b)−y(a))/(b−a)⟩.

Direction of motion (component test)

Determined by whether x(t) and y(t) increase or decrease as t increases (e.g., increasing x and decreasing y means right and down).

Horizontal extrema (parametric motion)

The maximum and minimum values of x(t) on the given time interval (furthest left/right points).

Vertical extrema (parametric motion)

The maximum and minimum values of y(t) on the given time interval (highest/lowest points).

Implicitly defined relation

An equation in x and y not solved for one variable in terms of the other; it may represent multiple curves or fail the vertical line test.

Vertical line test

A graph represents y as a function of x only if every vertical line intersects it at most once.

Parametrization verification (by substitution)

To check a parametrization, substitute x(t) and y(t) into the original equation and confirm it holds for all allowed t.

Vector

A quantity with magnitude and direction; in the plane it is often represented by components.

Vector components (component form)

Writing a vector as ⟨a,b⟩ meaning move a units horizontally and b units vertically.

Vector from P to Q

For P(x1,y1) and Q(x2,y2), the vector PQ is ⟨x2−x1, y2−y1⟩.

Magnitude of a vector

For v=⟨a,b⟩, the length is |v|=√(a^2+b^2).

Scalar multiplication (vectors)

Multiplying v=⟨a,b⟩ by k gives k⟨a,b⟩=⟨ka,kb⟩, scaling length by |k| and reversing direction if k<0.

Unit vector

A vector of magnitude 1 giving direction; for nonzero v, v̂ = (1/|v|)v.

Dot product

For u=⟨a,b⟩ and v=⟨c,d⟩, u·v=ac+bd; connects algebra to angles and perpendicularity.

Perpendicular vectors (dot product test)

If u·v=0 and both vectors are nonzero, then u and v are perpendicular.

Matrix (dimensions)

A rectangular array of numbers with size n×m (n rows, m columns) used for data, systems, and transformations.

Matrix multiplication (defined and how)

(m×k)(k×n) is defined and produces an m×n matrix; entries come from row-by-column products (not element-wise).

Identity matrix (2×2)

I=[[1,0],[0,1]]; multiplying by I leaves a vector unchanged (I x = x).

Determinant (2×2)

For A=[[a,b],[c,d]], det(A)=ad−bc; |det(A)| is the area scale factor, det(A)=0 means not invertible, and a negative det flips orientation.

Inverse of a 2×2 matrix

If det(A)≠0 for A=[[a,b],[c,d]], then A^{-1}=(1/(ad−bc))[[d,−b],[−c,a]].

Linear transformation (in R^2)

A mapping L that preserves addition and scalar multiplication: L(u+v)=L(u)+L(v) and L(c v)=cL(v); it keeps the origin fixed and can be represented by a 2×2 matrix.

Composition of transformations (order matters)

If A is applied first and then B, the combined matrix is BA; in general AB≠BA, so changing order can change the result.

Transition matrix

A matrix whose entries encode how amounts/percentages move between states each time step; repeated multiplication models the system over time.

State vector

A vector listing the amounts (or proportions) in each state at a given time; multiplying by a transition matrix updates it to the next time step.