Transformations: Videos & Practice Problems

Transformations of functions involve changing their position or shape through three basic types: reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, while a shift moves the function to a new location. A stretch squeezes the function vertically. The function notation changes accordingly: reflections become negative, shifts are represented as $f(x-h)+k)$, and stretches involve a constant multiplied by the function. Understanding these transformations is crucial for analyzing function behavior.

Intro to Transformations

Transformations of functions involve manipulating a function to change its position or shape. There are three primary types of transformations: reflections, shifts, and stretches. Understanding these transformations can simplify the concept significantly.

A reflection occurs when a function is flipped over a specific axis. For example, reflecting a function over the x-axis changes its output to negative, represented mathematically as \( f(x) \) becoming \( -f(x) \). This means that every point on the graph is mirrored across the x-axis.

A shift involves moving a function horizontally or vertically. The general form for a shift is given by \( f(x - h) + k \), where \( h \) indicates the horizontal shift and \( k \) indicates the vertical shift. For instance, if a function is shifted to the right by 3 units and up by 2 units, it would be expressed as \( f(x - 3) + 2 \).

A stretch occurs when a function is vertically stretched or compressed. This is represented by multiplying the function by a constant \( c \), resulting in \( c \cdot f(x) \). If \( c > 1 \), the function is stretched; if \( 0 < c < 1 \), it is compressed. This transformation alters the steepness of the graph.

To illustrate these transformations, consider the function \( f(x) = |x| \). If we apply the transformations, we can analyze the following functions:

• For \( p(x) = |x - 3| + 2 \), this represents a shift transformation, moving the graph 3 units to the right and 2 units up.
• For \( q(x) = -|x| \), this is a reflection over the x-axis, flipping the graph upside down.
• For \( r(x) = -2|x| \), this function combines both a reflection and a vertical stretch, as the negative sign indicates a reflection and the factor of 2 indicates a vertical stretch.

In summary, transformations of functions can be categorized into reflections, shifts, and stretches, each with specific mathematical representations. Recognizing these transformations helps in understanding how the graph of a function changes in response to different manipulations.

Here’s what students ask on this topic:

What are the three basic types of transformations in functions?

The three basic types of transformations in functions are reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, such as reflecting over the x-axis. A shift involves moving a function to a new position without altering its shape, represented as f(x - h) + k, where h is the horizontal shift and k is the vertical shift. A stretch involves squeezing or expanding a function vertically, represented by a constant multiplied by f(x), such as c * f(x).

How does a reflection transformation affect the function notation?

A reflection transformation affects the function notation by making the function negative. For example, if the original function is f(x), reflecting it over the x-axis would result in the function becoming -f(x). This indicates that the graph of the function has been flipped over the x-axis.

What is the function notation for a shift transformation?

The function notation for a shift transformation is f(x - h) + k. In this notation, h represents the horizontal shift, and k represents the vertical shift. For example, if the original function is f(x), shifting it horizontally by h units and vertically by k units would result in the function f(x - h) + k.

How do you identify a stretch transformation in a function?

A stretch transformation in a function is identified by a constant multiplied by the function. For example, if the original function is f(x), a vertical stretch would be represented as c * f(x), where c is the constant responsible for the stretch. If c > 1, the function is stretched vertically, and if 0 < c < 1, the function is compressed vertically.

Can multiple transformations occur simultaneously in a function? Provide an example.

Yes, multiple transformations can occur simultaneously in a function. For example, consider the function g(x) = -2|x|. This function combines a reflection and a vertical stretch. The negative sign indicates a reflection over the x-axis, and the constant 2 indicates a vertical stretch. Thus, the graph of g(x) is both flipped and stretched compared to the original function f(x) = |x|.