Study Guide for Functions and Transformations

Curriculum Dot Point 1.2.22: This section covers the examination of translations and the graphs of functions represented by $y = f(x) + c$ and $y = f(x - c)$ .
Curriculum Dot Point 1.2.23: This section covers the examination of dilations and the graphs of functions represented by $y = k f(x)$ and $y = f(kx)$ .
Worksheet Reference: Associated materials and worksheets can be found on OneNote under the "Functions Transformations" heading.

Definition of a Function: A function is defined as a relation in which every possible input value results in exactly one unique output value.
Relational Language: We describe this relationship by saying, "the output is a function of the input."
Naming Conventions: Function notation allows for a variety of names beyond the standard identifier "y." Common notation includes identifiers such as $f(x)$ and $g(x)$ .
The Function Machine Model: * Function Name: The letter $f$ represents the name assigned to the function. * Input Variable: The $x$ written in brackets indicates that the variable used for input values is $x$ . * Operational Meaning: The expression $f(x)$ indicates that the specific "f" function is being applied to the input value of "x."

Types of Transformations: There are three primary types of transformations, often remembered by the acronym DRT: * Dilations * Reflections * Translations
The Position Rule (Inside vs. Outside): * Transformations Outside the Function: Affect the output and result in vertical (up and down) movements. An example provided is $2f(x) + 3$ . * Transformations Inside the Function: Affect the input directly and result in horizontal (left and right) movements. An example provided is $f(2x + 3)$ .

Mechanism: Adding or subtracting a constant to the entire function $f(x)$ translates the curve along the y-axis.
Directionality: * Adding a constant moves the graph up. * Subtracting a constant moves the graph down.
Example Case: * Base function: $f(x) = x^2$ * Transformed function: $f(x) + 2 = x^2 + 2$
Key Characteristic: Because the transformation is applied outside the function, it behaves as expected (up for addition, down for subtraction) and affects vertical positioning rather than horizontal.

Mechanism: Adding or subtracting a constant directly to the $x$ value (replacing $x$ with $x \pm c$ ) translates the curve left or right by $c$ units.
Directionality (Inverse Logic): * Adding C ( $f(x + c)$ ): Moves the function to the left. * Subtracting C ( $f(x - c)$ ): Moves the function to the right.
Reasoning for Inverse Movement: For a function like $(x - 2)^2$ , the value of $x$ must be positive $2$ in order for the function to reach zero ( $f(x) = 0$ ), showing that the graph has shifted to the right.
Example of Application: * Given function: $f(x) = x^3 - x^2 + 4x$ * Required: Determine the new function $g(x)$ when the graph is translated $c$ units to the left. * Result: $g(x) = f(x + c) = (x + c)^3 - (x + c)^2 + 4(x + c)$

Mechanism: Multiplying the entire function by a constant $k$ dilates the curve by a factor of $k$ parallel to the y-axis.
Coordinate Change: To obtain the new transformed points, the y-coordinate of every point on the original curve is multiplied by the factor $k$ .
Vector of Movement: Because the transformation occurs outside the function, the movement is vertical (upward/downward stretch or compression) rather than horizontal.

Mechanism: Replacing the variable $x$ with the term $kx$ dilates the curve parallel to the x-axis.
Dilation Factor: The curve is dilated by a factor of $\frac{1}{k}$ .
Coordinate Change: The x-coordinate of every point on the original curve is multiplied by $\frac{1}{k}$ to find the new transformed point.
Example Case: * Original: $f(x) = x^2$ * Transformed: $g(x) = (2x)^2$ * Calculation: In this instance, every $x$ value in $f(x)$ is multiplied by $\frac{1}{2}$ to generate the points for $g(x)$ . * Constant Values: The y-values for each point remain unchanged.
Vector of Movement: Because the transformation is inside the function, the movement is horizontal (left-right) as opposed to vertical.

Reflection in the x-axis ( $-f(x)$ ): * Mechanism: Multiplying the entire function by $-1$ reflects the curve across the x-axis. * Type: This is an "outside" transformation, meaning it results in vertical (up-down) flipping.
Reflection in the y-axis ( $f(-x)$ ): * Mechanism: Replacing the input variable $x$ with $-x$ reflects the curve across the y-axis. * Type: This is an "inside" transformation, meaning it results in horizontal (left-right) flipping.