Transformations: Videos & Practice Problems

Transformations of functions include reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, changing its sign, while shifts move the function horizontally or vertically, represented by $f(x-h+k)$. Stretches and shrinks alter the graph's shape based on constants multiplied inside or outside the function. Understanding these transformations is crucial for analyzing the domain and range of functions, as they can significantly change these properties.

Introduction to Transformations

Transformations of functions are essential concepts in mathematics that 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 study of functions significantly.

A reflection occurs when a function is flipped over a specific axis. For instance, reflecting a function over the x-axis changes its output to the negative of the original function, represented mathematically as f(x) → -f(x). This transformation effectively "folds" the graph over the axis.

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

A stretch transformation occurs when a function is vertically stretched or compressed. This is represented as f(x) → c * f(x), where c is a constant. If c is greater than 1, the function is stretched; if c is between 0 and 1, the function is compressed.

To illustrate these transformations, consider the function f(x) = |x|. If we analyze the transformations of this function, we can match various transformed functions to their corresponding graphs. For example, the function p(x) = |x - 3| + 2 represents a shift transformation, moving the graph to the right by 3 units and up by 2 units. The function q(x) = -|x| represents a reflection over the x-axis, flipping the graph upside down. Lastly, a function like r(x) = -2|x| combines both a reflection and a vertical stretch, as indicated by the negative sign and the constant factor of 2.

In summary, understanding these basic transformations—reflections, shifts, and stretches—enables students to analyze and manipulate functions effectively. Recognizing how these transformations affect the function's graph is crucial for mastering more complex mathematical concepts.

Reflections of Functions

Reflection transformations are a key concept in understanding how functions behave when their graphs are manipulated. A reflection occurs when a graph appears to be folded over a specific axis, either the x-axis or the y-axis. This folding changes the positions of points on the graph, which can be visualized as creasing a piece of paper along the chosen axis.

When reflecting a function over the x-axis, the y-values of the points on the graph change signs. For example, if a point on the graph is at (−1, 1), after reflection over the x-axis, it will move to (−1, −1). This transformation can be expressed mathematically as:

f(x) → −f(x)

Here, the output values (y-values) are negated, indicating that the graph is flipped vertically.

Conversely, reflecting a function over the y-axis involves changing the signs of the x-values while keeping the y-values the same. For instance, a point at (−1, 1) would reflect to (1, 1). This transformation is represented as:

f(x) → f(−x)

In this case, the input values (x-values) are negated, indicating a horizontal flip of the graph.

To illustrate these concepts, consider the function f(x) = x + 2. If we want to find the reflection of this function over the x-axis, we apply the transformation:

h(x) = −f(x) = −(x + 2) = −x − 2

This new function h(x) represents the reflected graph, which can be sketched by folding the original graph downwards along the x-axis. The resulting graph will show the original points inverted in the vertical direction.

Understanding these transformations is crucial for analyzing and sketching functions accurately. By mastering reflections, students can better grasp how different transformations affect the shape and position of graphs in a coordinate plane.

Example 1

To understand the transformation of a function through reflection, consider the function \( f(x) \), represented by a curve. When we reflect this function over the x-axis, we create a new function \( g(x) \). This reflection can be visualized as folding the graph of \( f(x) \) along the x-axis, similar to how one would fold a sheet of paper. The key concept here is that every point on the graph of \( f(x) \) will have its y-coordinate inverted, resulting in the new function \( g(x) = -f(x) \).

For example, if a point on the graph of \( f(x) \) is at \( (a, b) \), after reflection, the corresponding point on \( g(x) \) will be at \( (a, -b) \). This means that all positive values of \( f(x) \) will become negative in \( g(x) \), and vice versa. The overall shape of the graph will be preserved, but it will be flipped vertically.

In summary, reflecting a function over the x-axis involves negating its output values, leading to the transformation \( g(x) = -f(x) \). This process allows us to visualize how the graph changes while maintaining the same x-coordinates, effectively flipping the graph upside down.

Written below (green dotted curve) is a graph of the function $f\left(x\right)=\sqrt{x-2}$.If g(x) (blue solid curve) is a reflection of f(x) about the y-axis what is the equation for g(x)?

• A. $g\left(x\right)=\sqrt{-x-2}$
• B. $g\left(x\right)=\sqrt{-x}-2$
• C. $g\left(x\right)=\sqrt{x-2}$
• D. $g\left(x\right)=\sqrt{x}-2$

Answer: (A)

Shifting A Functions

Transformations of functions involve moving graphs either vertically or horizontally from their original positions. Understanding shifts is crucial, as they can be represented by specific numbers added to or subtracted from the function. The general form for shifts can be expressed as f(x) \rightarrow f(x - h) + k, where h indicates the horizontal shift and k indicates the vertical shift.

When considering vertical shifts, the graph moves up or down without changing the x-values. For instance, if a parabola is shifted from a y-value of 0 to 2, the transformation can be represented as f(x) + 2. Here, a positive k results in an upward shift, while a negative k would shift the graph downward.

In contrast, horizontal shifts involve moving the graph left or right, affecting the x-values while keeping the y-values constant. For example, if a graph shifts from 0 to 3, it can be expressed as f(x - 3). A negative h indicates a shift to the right, while a positive h indicates a shift to the left. This can seem counterintuitive, as a negative sign typically suggests a leftward movement. However, when the transformation is written as f(x - (-2)), it simplifies to f(x + 2), demonstrating a leftward shift.

To apply these concepts, consider the transformation f(x - 2 + 3). Here, h is 2 (indicating a shift to the right) and k is 3 (indicating a shift upward). Thus, the graph shifts 2 units to the right and 3 units up from its original position. The resulting function retains its shape but is repositioned according to the specified transformations.

Understanding these shifts is essential for graphing functions accurately and interpreting their transformations effectively.

The green dotted line in the graph below represents the function $f\left(x\right)$. The blue solid line represents the function $g\left(x\right)$, which is the function $f\left(x\right)$after it has gone through a shift transformation. Find the equation for $g\left(x\right)$.

• A. $g\left(x\right)=f\left(x-2\right)+3$
• B. $g\left(x\right)=f\left(x-2\right)-3$
• C. $g\left(x\right)=f\left(x+2\right)-3$
• D. $g\left(x\right)=f\left(x\right)-3$

Answer: (C)

Graphs of Shifted and Reflected Functions

In the study of function transformations, understanding how to graphically represent shifts and reflections is essential. A transformation can involve both shifting a graph and reflecting it, which may seem complex at first but can be simplified through systematic analysis.

When reflecting a function over the x-axis, the output of the function becomes negative. For example, if the original function is denoted as f(x), the reflected function is represented as -f(x). Shifting a function involves moving it horizontally or vertically. A horizontal shift to the left by h units is represented as f(x + h), while a vertical shift upwards by k units is shown as f(x) + k.

Combining these transformations requires careful notation. For instance, if a function is reflected and then shifted, the combined transformation can be expressed as g(x) = -f(x + h) + k, where h and k represent the horizontal and vertical shifts, respectively. This notation allows for a clear understanding of how the original function has been altered.

To illustrate, consider the absolute value function f(x) = |x|. If this function is reflected over the x-axis and shifted 2 units to the left, the transformation can be expressed as follows: first, reflect the function to get -f(x), resulting in -|x|. Then, apply the horizontal shift, leading to g(x) = -|x + 2|. This final equation encapsulates both transformations, demonstrating how the graph has been modified.

Understanding these transformations is crucial for graphing functions accurately and interpreting their behavior under various modifications. By mastering the combination of reflections and shifts, students can confidently approach more complex function transformations in their studies.

Example 2

To solve the problem involving the transformation of the function \( f(x) = x^3 \), we need to apply two specific transformations: a reflection over the x-axis and a vertical shift downward by 2 units.

First, reflecting a function over the x-axis changes the function from \( f(x) \) to \( -f(x) \). Therefore, applying this transformation to our original function gives us:

\[ h_1(x) = -f(x) = -x^3 \]

Next, we need to account for the vertical shift downward by 2 units. A downward shift is represented by subtracting the value of the shift from the function. In this case, since we are shifting down 2 units, we subtract 2 from our already transformed function:

\[ h(x) = h_1(x) - 2 = -x^3 - 2 \]

Thus, the equation for the transformed function \( h(x) \) is:

\[ h(x) = -x^3 - 2 \]

Now, to sketch the graph of \( h(x) \), we start with the graph of the original function \( f(x) = x^3 \). The graph of \( x^3 \) is a cubic curve that passes through the origin and has a characteristic shape that increases steeply in the positive direction and decreases in the negative direction.

After reflecting this graph over the x-axis, the curve will now decrease steeply in the positive direction and increase in the negative direction. This reflection can be visualized as flipping the graph upside down.

Next, we shift the entire graph down by 2 units. This means every point on the graph of \( -x^3 \) will move down 2 units. The new center of the graph will be at the point (0, -2) instead of (0, 0).

In summary, the final graph of \( h(x) = -x^3 - 2 \) will have the same shape as the original cubic function but will be inverted and shifted down, resulting in a curve that passes through the point (0, -2) and reflects the transformations applied.

Stretches and Shrinks of Functions

Transformations of functions are essential in understanding how graphs behave under various modifications. One key transformation is the stretch or shrink, which involves multiplying the function by a constant. This transformation can be categorized into vertical and horizontal stretches or shrinks, depending on where the constant is applied.

A vertical stretch occurs when a constant greater than 1 is multiplied outside the function. For example, if we have a function f(x) and we apply a transformation like 2f(x), the graph will stretch vertically by a factor of 2. Conversely, if a constant between 0 and 1 is used, such as (1/2)f(x), the graph will undergo a vertical shrink, reducing its height by half.

On the other hand, horizontal stretches and shrinks depend on constants multiplied inside the function. For instance, if we have f(1/2x), this represents a horizontal stretch because the constant (1/2) is less than 1. To visualize this, you can think of needing to double the x-values to return to the original function. In contrast, if the constant is greater than 1, such as f(2x), the graph will compress horizontally, bringing points closer to the y-axis.

To summarize, the behavior of the graph under these transformations can be understood as follows: a vertical stretch occurs with constants greater than 1 outside the function, while a vertical shrink occurs with constants between 0 and 1 outside the function. For horizontal transformations, a stretch happens with constants between 0 and 1 inside the function, and a shrink occurs with constants greater than 1 inside the function. This understanding is crucial for accurately sketching transformed graphs and predicting their behavior.

Example 3

In exploring the function \( f(x) = c \cdot x^2 \), we analyze how different constants \( c \) affect the graph of the parabola. The standard form \( f(x) = x^2 \) serves as our baseline for comparison. When \( c = 2 \), the function transforms to \( f(x) = 2 \cdot x^2 \). By substituting various values of \( x \), we can identify key points on the graph. For \( x = 0 \), we find \( f(0) = 2 \cdot 0^2 = 0 \), establishing the point (0, 0). For \( x = -1 \), we calculate \( f(-1) = 2 \cdot (-1)^2 = 2 \cdot 1 = 2 \), resulting in the point (-1, 2). Similarly, for \( x = 1 \), \( f(1) = 2 \cdot 1^2 = 2 \), giving us the point (1, 2). The graph of \( f(x) = 2 \cdot x^2 \) appears narrower than the standard parabola, indicating a horizontal shrink, which occurs when the constant \( c \) is greater than 1.

Next, we consider the case where \( c = \frac{1}{2} \), leading to the function \( f(x) = \frac{1}{2} \cdot x^2 \). Again, substituting \( x = 0 \) yields \( f(0) = \frac{1}{2} \cdot 0^2 = 0 \), maintaining the point (0, 0). For \( x = 2 \), we compute \( f(2) = \frac{1}{2} \cdot 2^2 = \frac{1}{2} \cdot 4 = 2 \), resulting in the point (2, 2). The same calculation for \( x = -2 \) gives \( f(-2) = \frac{1}{2} \cdot (-2)^2 = 2 \), confirming the point (-2, 2). The graph of \( f(x) = \frac{1}{2} \cdot x^2 \) appears wider than the standard parabola, indicating a horizontal stretch, which occurs when the constant \( c \) is between 0 and 1.

It is important to note that these transformations can visually resemble vertical stretches or compressions. A horizontal shrink (when \( c > 1 \)) may appear as a vertical stretch, while a horizontal stretch (when \( 0 < c < 1 \)) may look like a vertical compression. Understanding these relationships helps in visualizing the effects of different constants on the graph of quadratic functions.

Domain and Range of Function Transformations

Transformations of functions, such as reflections, shifts, and stretches or shrinks, can significantly alter the domain and range of a function. Understanding how to find the domain and range after a transformation is crucial for analyzing functions effectively.

To determine the domain of a transformed function, visualize the graph being compressed along the x-axis. For example, if a function originally has a domain from -3 to 3, after a transformation, it may change to a new domain, such as from -2 to 4. Similarly, the range can be found by compressing the graph along the y-axis. If the original range was also from -3 to 3, a transformation might adjust it to a new range, such as from -1 to 5.

Consider the function \( f(x) = x^2 \), which is a standard parabola centered at the origin. When transformed to \( g(x) = (x - 3)^2 + 2 \), we can identify the transformation parameters. Here, \( h = 3 \) indicates a shift to the right, and \( k = 2 \) indicates a shift upward. Thus, the vertex of the parabola moves from (0, 0) to (3, 2).

The domain of the transformed function \( g(x) \) remains all real numbers, expressed as \( (-\infty, \infty) \), since the parabola extends infinitely in both directions. However, the range changes due to the upward shift; it now starts from the new vertex at \( y = 2 \) and extends to infinity, represented as \( [2, \infty) \).

This example illustrates that while the domain may remain unchanged, the range can be affected by transformations. Understanding these concepts is essential for effectively working with transformed functions in various mathematical contexts.

The green dotted curve below is a graph of the function $f\left(x\right)$. Find the domain and range of $g\left(x\right)$(the blue solid curve), which is a transformation of $f\left(x\right)$.

• A. Dom: $\left[1,4\right]$ , Ran: $\left[-5,-1\right]$
• B. Dom: $\left[1,5\right]$ , Ran: $\left[-5,1\right]$
• C. Dom: $\left[-1,3\right]$ , Ran: $\left[-2,4\right]$
• D. Dom: $\left[-2,3\right]$ , Ran: $\left[2,4\right]$

Answer: (B)

Here’s what students ask on this topic:

What are the three basic types of transformations in trigonometry?

The three basic types of transformations in trigonometry are reflections, shifts, and stretches. A reflection occurs when a function is folded over an axis, changing its sign. Shifts move the function horizontally or vertically, represented by $f(x-h)+k$. Stretches and shrinks alter the graph's shape based on constants multiplied inside or outside the function. Understanding these transformations is crucial for analyzing the domain and range of functions, as they can significantly change these properties.

How does a reflection transformation affect a function's graph?

A reflection transformation affects a function's graph by flipping it over a specific axis. If the reflection is over the x-axis, the y-values of the function change signs, transforming $f\left(x\right)$ into $-f\left(x\right)$. If the reflection is over the y-axis, the x-values change signs, transforming $f\left(x\right)$ into $f(-x)$. This transformation changes the graph's orientation but not its shape.

What is the difference between a vertical and horizontal shift in function transformations?

A vertical shift moves a function's graph up or down, while a horizontal shift moves it left or right. In the function notation $f(x-h)+k$, $h$ represents the horizontal shift and $k$ represents the vertical shift. A positive $h$ shifts the graph to the right, while a negative $h$ shifts it to the left. A positive $k$ shifts the graph up, and a negative $k$ shifts it down.

How do you determine the domain and range of a transformed function?

To determine the domain and range of a transformed function, observe the new graph after the transformation. The domain is found by projecting the graph onto the x-axis, and the range is found by projecting it onto the y-axis. For example, if a function $f\left(x\right)$ is shifted to the right by 2 units and up by 3 units, the domain and range will adjust accordingly. Always consider how the transformation affects the graph's position and shape to accurately determine the new domain and range.

What is the effect of a vertical stretch or shrink on a function's graph?

A vertical stretch or shrink affects a function's graph by altering its height. If a constant $c$ is multiplied outside the function, $cf\left(x\right)$, and $c$ is greater than 1, the graph stretches vertically. If $c$ is between 0 and 1, the graph shrinks vertically. For example, multiplying $f\left(x\right)$ by 2 will double its height, while multiplying by 0.5 will halve it.