Advanced Algebra – Transformations of Functions

These flashcards cover parent functions, the general transformation equation, specific parameters (a, b, h, k), and their effects on graph position, size, and orientation based on lecture notes.

What is a function rule?

An algebraic expression showing what operation(s) is/are being performed on $x$.

What is the parent function rule for a Linear function?

$$f(x) = x$$

What is the parent function rule for a Quadratic function?

$$f(x) = x^2$$

What is the parent function rule for an Absolute Value function?

$$f(x) = |x|$$

What is the parent function rule for a Square Root function?

$$f(x) = \sqrt{x}$$

What is the parent function rule for a Cubic function?

$$f(x) = x^3$$

What is the parent function rule for an Exponential function?

$$f(x) = b^x$$

What key graph feature is shared by Quadratic, Absolute Value, and Cubic functions?

The vertex.

What is the key graph feature of a Square Root function?

The endpoint.

What is the key graph feature of an Exponential function?

The asymptote.

Define the transformation of a function.

When we move or resize the graph of a function.

What is the general equation for transformations of a function?

$$f(x) = a[-b(x - h)] + k$$

In the general transformation equation, what does the parameter $k$ represent?

Vertical shift (up or down).

In the general transformation equation, what does the parameter $h$ represent?

Horizontal shift (left or right).

What effect does a positive $k$ value have on a graph?

A vertical shift Up.

What effect does a negative $k$ value have on a graph?

A vertical shift Down.

What effect does the sequence $(x + h)$ have on a graph's horizontal position?

A horizontal shift Left.

What effect does the sequence $(x - h)$ have on a graph's horizontal position?

A horizontal shift Right.

What type of transformation occurs if $a > 1$?

Vertical stretch (away from the x-axis).

What type of transformation occurs if $0 < a < 1$?

Vertical shrink (towards the x-axis).

What type of transformation occurs if $b > 1$?

Horizontal shrink (towards the y-axis).

What type of transformation occurs if $0 < b < 1$?

Horizontal stretch (away from the y-axis).

What does a negative $a$ value ($-a$) indicate?

A vertical reflection over the x-axis.

What does a negative $b$ value ($-b$) indicate?

A horizontal reflection over the y-axis.

What is the transformation for $f(x) = x^2 + 4$?

Vertical shift up 4.

What is the transformation for $g(x) = |x + 4|$?

Horizontal shift Left 4.

What is the transformation for $p(x) = 4\sqrt{x}$?

Vertical stretch by 4.

What is the transformation for $h(x) = (4x)^2$?

Horizontal shrink by 4.

What is the transformation for $g(x) = -x^3$?

Vertical reflection over the x-axis.

What is the transformation for $f(x) = \sqrt{-x}$?

Horizontal reflection over the y-axis.

What are the transformations for $k(x) = 3\sqrt{x} - 2$?

Vertical stretch by 3 and a vertical shift down 2.

What are the domain and range for $p(x) = \sqrt{-x} + 1$?

Domain: $(-\infty, 0]$, Range: $[1, \infty)$.

What is the new asymptote for the function $n(x) = 2^x - 5$?

$$y = -5$$

What are the transformations for $h(x) = (-(x - 3))^3$?

Vertical reflection over the x-axis, Horizontal reflection over the y-axis, and Horizontal shift Right 3.