Function Parameters and Sinusoidal Functions

Flashcards covering parameters of different function types and the characteristics of sinusoidal functions.

Parameters of Linear Functions

Two parameters (m and b) are needed to describe a linear function (y = mx + b).

Parameters of Quadratic Functions

Three parameters (a, h, and k) are needed to describe a quadratic function in vertex form (y = a(x-h)^2 + k).

Parameters of Exponential Functions

Two parameters (a and b in ab^x or a and k in ae^(kx)) are needed to describe an exponential function.

Hardness of Exponential Functions

Exponential functions are considered hard to deal with because they require the use of logarithms to solve equations.

Parent Function for Linear Functions

All linear functions are transformations of the parent function y = x.

Parent Function for Quadratic Functions

All quadratic functions are transformations of the parent function y = x^2.

Parent Function for Exponential Functions

All exponential functions are transformations of the parent function y = e^x.

Parameters of Sinusoidal Functions

Four parameters (A, B, H, K) are needed to describe a sinusoidal function (e.g., A sin(B(t-H)) + K or A cos(B(t-H)) + K).

Factors Making Sinusoidal Functions Hard

Sinusoidal functions are hard due to more parameters, multiple possible parent functions, and the need for geometry (not just algebra) to solve equations.

Parameter 'A' in Sinusoidal Functions

The 'A' parameter stretches/compresses and possibly reflects the graph from the horizontal axis, affecting the amplitude.

Parameter 'K' in Sinusoidal Functions

The 'K' parameter shifts the graph vertically, affecting the midline.

Parameter 'B' in Sinusoidal Functions

The 'B' parameter stretches/compresses the graph horizontally, affecting the period.

Parameter 'H' in Sinusoidal Functions

The 'H' parameter shifts the graph horizontally (right or left), affecting the starting point.

Period of a Sinusoidal Function

The period of a sinusoidal function is calculated as 2

p / |B|.

Parameters of Linear Functions

Two parameters (m and b) are needed to describe a linear function (y = mx + b).

Parameters of Quadratic Functions

Three parameters (a, h, and k) are needed to describe a quadratic function in vertex form (y = a(x-h)^2 + k).

Parameters of Exponential Functions

Two parameters (a and b in ab^x or a and k in ae^(kx)) are needed to describe an exponential function.

Hardness of Exponential Functions

Exponential functions are considered hard to deal with because they require the use of logarithms to solve equations.

Parent Function for Linear Functions

All linear functions are transformations of the parent function y = x.

Parent Function for Quadratic Functions

All quadratic functions are transformations of the parent function y = x^2.

Parent Function for Exponential Functions

All exponential functions are transformations of the parent function y = e^x.

Parameters of Sinusoidal Functions

Four parameters (A, B, H, K) are needed to describe a sinusoidal function (e.g., A sin(B(t-H)) + K or A cos(B(t-H)) + K).

Factors Making Sinusoidal Functions Hard

Sinusoidal functions are hard due to more parameters, multiple possible parent functions, and the need for geometry (not just algebra) to solve equations.

Parameter 'A' in Sinusoidal Functions

The 'A' parameter stretches/compresses and possibly reflects the graph from the horizontal axis, affecting the amplitude.

Parameter 'K' in Sinusoidal Functions

The 'K' parameter shifts the graph vertically, affecting the midline.

Parameter 'B' in Sinusoidal Functions

The 'B' parameter stretches/compresses the graph horizontally, affecting the period.

Parameter 'H' in Sinusoidal Functions

The 'H' parameter shifts the graph horizontally (right or left), affecting the starting point.

Period of a Sinusoidal Function

The period of a sinusoidal function is calculated as 2\pi / |B|._

Role of 'm' in Linear Functions

The 'm' parameter in y = mx + b represents the slope of the line, indicating its steepness and direction.

Role of 'b' in Linear Functions

The 'b' parameter in y = mx + b represents the y-intercept, which is the point where the line crosses the y-axis (when x=0).

Role of 'a' in Quadratic Functions (Vertex Form)

The 'a' parameter in y = a(x-h)^2 + k determines the vertical stretch or compression of the parabola and its direction of opening (upwards if a > 0, downwards if a < 0).

Role of 'h' in Quadratic Functions (Vertex Form)

The 'h' parameter in y = a(x-h)^2 + k represents the horizontal shift of the parabola, and it is the x-coordinate of the vertex.

Role of 'k' in Quadratic Functions (Vertex Form)

The 'k' parameter in y = a(x-h)^2 + k represents the vertical shift of the parabola, and it is the y-coordinate of the vertex.

Role of 'a' in Exponential Functions

The 'a' parameter in exponential functions (y = ab^x or y = ae^{kx}) represents the initial value or the y-intercept (when x = 0).

Role of 'b' in Exponential Functions (y = ab^x)

The 'b' parameter in y = ab^x represents the growth or decay factor per unit of x. If b > 1, it's growth; if 0 < b < 1, it's decay.

Role of 'k' in Exponential Functions (y = ae^{kx})

The 'k' parameter in y = ae^{kx} represents the continuous growth or decay rate. If k > 0, it's continuous growth; if k < 0, it's continuous decay.