Transformation of Functions

Curves and Their Representations

A curve in mathematics is a set of points \((x, y)\) in the coordinate plane that satisfy a specific relationship between the variables \(x\) and \(y\). This relationship can be described in two main ways: explicit form and implicit form. These representations allow us to understand, analyze, and graph curves systematically.

When a curve is given in explicit form, the dependent variable \(y\) is expressed directly as a function of the independent variable \(x\). This is written as:

\[ y = f(x). \]

For example, the equation \(y = x^2\) represents a parabola that opens upwards, while \(y = \sin(x)\) represents the periodic sine wave oscillating around the x-axis. Explicit forms are straightforward because for each value of \(x\) in the domain, there is exactly one corresponding value of \(y\). However, explicit forms cannot always represent curves when \(y\) is not uniquely determined for every \(x\).

In the implicit form, the relationship between \(x\) and \(y\) is described as an equation involving both variables, without solving explicitly for one variable in terms of the other. This is written as:

\[ f(x, y) = 0. \]

For example, the equation \(x^2 + y^2 = 1\) represents a circle centered at the origin with radius 1. Unlike explicit forms, implicit forms can describe more complex curves, including those where \(y\) may take multiple values for the same \(x\). For instance, the same circle \(x^2 + y^2 = 1\) can only be partially represented in explicit form as:

\[ y = \sqrt{1 - x^2} \quad \text{(upper semicircle)}, \quad y = -\sqrt{1 - x^2} \quad \text{(lower semicircle)}. \]

Here, the implicit form is more natural because it captures the entire circle in one equation.

Explicit and implicit forms are connected. Any explicit equation \(y = f(x)\) can be written in implicit form as \(f(x, y) = y - f(x) = 0\). For example, the parabola \(y = x^2\) can also be written implicitly as \(x^2 - y = 0\). However, not all implicit equations can be converted into a single explicit form. For example, \(x^2 + y^2 = 1\) cannot be expressed as a single function \(y = f(x)\) because \(y\) has two possible values for most \(x\).

Let’s look at a few examples to clarify:

Parabola:
- Explicit form: \(y = x^2\).
  Here, \(y\) is uniquely determined for every \(x\).
- Implicit form: \(x^2 - y = 0\).
  This represents the same parabola but in a different form.
Circle:
- Implicit form: \(x^2 + y^2 = 1\).
  This describes the entire circle.
- Explicit form: \(y = \sqrt{1 - x^2}\) (upper semicircle) and \(y = -\sqrt{1 - x^2}\) (lower semicircle).
  The explicit form requires splitting the curve into two parts.
Ellipse:
- Implicit form: \(\frac{x^2}{4} + \frac{y^2}{9} = 1\).
  This describes an ellipse centered at the origin.
- Explicit form: \(y = \pm\sqrt{9\left(1 - \frac{x^2}{4}\right)}\).
  Again, the explicit form splits the curve into two branches.

Implicit forms are particularly useful when the curve is complex or symmetric, making it hard to write as a single explicit function. For example, the hyperbola \(x^2 - y^2 = 1\) describes two disconnected branches. Explicitly, this must be written as \(y = \pm\sqrt{x^2 - 1}\), but the implicit form captures both branches in a single equation.

In summary, explicit forms are simple and practical when \(y\) is uniquely determined by \(x\), while implicit forms are more flexible and better suited for describing curves where \(y\) may take multiple values for the same \(x\). Understanding these two forms allows us to analyze and represent curves effectively, whether they are simple like a parabola or complex like a circle or hyperbola.

Transformation of Graphs of Functions

In this section, we aim to understand how the equation of a curve changes under various transformations of its graph and vice versa. Given a curve represented by an explicit equation \( y = f(x) \) or an implicit form \( F(x, y) = 0 \), we explore the effects of specific transformations—such as shifting, reflecting, stretching, or compressing—on both the graph and the equation.

For example:

Shifting: Moving the graph upwards, downwards, leftwards, or rightwards alters the equation in predictable ways. Conversely, a given modification in the equation implies a corresponding shift in the graph.
Reflection: Flipping the graph across the \(x\)-axis or \(y\)-axis results in changes to the signs of specific terms in the equation.
Scaling: Stretching or compressing the graph vertically or horizontally impacts the coefficients or arguments of the function.
Other transformations: Operations such as rotations, inversions, or combining multiple transformations also lead to specific changes in the equation.

By the end of this section, you will be able to:

Identify the transformation applied to a graph and determine its effect on the corresponding equation.
Predict the graphical change caused by modifying the equation of the function.
Apply these transformations systematically to understand and manipulate mathematical curves.

Through this exploration, you will gain deeper insight into the interplay between equations and their geometric representations, enhancing your ability to analyze and visualize functions effectively.

Shifting of Graphs

Shifting refers to the translation of a graph either horizontally or vertically, which changes the position of the graph without altering its shape. For a given curve represented by \( y = f(x) \) or \( F(x, y) = 0 \), shifts can be understood by analyzing how the inputs or outputs of the function are modified.

Horizontal Shifts: The graph moves along the \(x\)-axis.
- Rightward Shift (Forward Shift):
  
  When \( x \) is replaced by \( x - a \) in \( F(x, y) = 0 \) (where \( a > 0 \)), the new equation becomes \( F(x - a, y) = 0 \). For the same output, the input to the function must now be larger by \( a \). This implies that the graph shifts \( a \) units forward (in the positive \(x\)-direction).
- Leftward Shift (Backward Shift):
  
  When \( x \) is replaced by \( x + a \) in \( F(x, y) = 0 \) (where \( a > 0 \)), the new equation becomes \( F(x + a, y) = 0 \). For the same output, the input to the function must now be smaller by \( a \). This results in the graph shifting \( a \) units backward (in the negative \(x\)-direction).
Vertical Shifts: The graph moves along the \(y\)-axis.
- Upward Shift:
  
  When \( y \) is replaced by \( y - a \) in \( F(x, y) = 0 \) (where \( a > 0 \)), the new equation becomes \( F(x, y - a) = 0 \). To maintain the same output, the \(y\)-coordinate must increase by \( a \). This shifts the graph \( a \) units upward.
- Downward Shift:
  
  When \( y \) is replaced by \( y + a \) in \( F(x, y) = 0 \) (where \( a > 0 \)), the new equation becomes \( F(x, y + a) = 0 \). To maintain the same output, the \(y\)-coordinate must decrease by \( a \). This shifts the graph \( a \) units downward.

These shifts occur because the transformations adjust the input or output values of the function, altering where the graph achieves the same outputs as before. Understanding these changes allows us to predict the behavior of the graph based on modifications to the function's equation.

Vertical Shifts for Explicit Equations

When the equation of a curve is given in explicit form \( y = f(x) \), vertical shifts modify the \(y\)-coordinate directly, resulting in the following:

Upward Shift:

Replacing \( y \) with \( y - a \) (where \( a > 0 \)) gives:

\[ y - a = f(x) \implies y = f(x) + a. \]

This shifts the graph upward by \( a \) units, as adding \( a \) increases the output \( y \) for the same \( x \).
Downward Shift:

Replacing \( y \) with \( y + a \) (where \( a > 0 \)) gives:

\[ y + a = f(x) \implies y = f(x) - a. \]

This shifts the graph downward by \( a \) units, as subtracting \( a \) decreases the output \( y \) for the same \( x \).

Thus, in explicit form, vertical shifts are directly visible as \( y = f(x) + a \) for an upward shift and \( y = f(x) - a \) for a downward shift.

Example

Draw the graph of \(f(x) = |x+2|-1\)

Solution:

To graph the function \( f(x) = |x+2| - 1 \), we start with the base function \( f(x) = |x| \) and apply transformations step-by-step:

Base Graph (\( f(x) = |x| \)):
- The graph of \( f(x) = |x| \) is a "V" shape with its vertex at the origin \((0, 0)\), symmetric about the \(y\)-axis.
Horizontal Shift (\( f(x) = |x+2| \)):
- Replacing \( x \) with \( x+2 \) shifts the graph 2 units to the left.
- The vertex of the new graph moves from \((0, 0)\) to \((-2, 0)\).
Vertical Shift (\( f(x) = |x+2| - 1 \)):
- Subtracting 1 from \( f(x+2) \) shifts the graph 1 unit downward.
- The vertex of the final graph moves from \((-2, 0)\) to \((-2, -1)\).

Final Graph

The resulting graph is a "V" shape:

The vertex is at \((-2, -1)\).
The slopes of the lines remain the same as the base graph (\( \pm 1 \)).

Example

Draw the graph of \(y = x^2-4x + 5\)

Solution:

First we write \(y = x^2 - 4x + 5\) in vertex-centric form using completing the square method as \(y = (x-2)^2 + 1\), then we use triansformations to draw the final graph:

Base Graph (\(y = x^2\)):
- The blue curve represents the original function \(y = x^2\), a parabola with its vertex at \((0, 0)\).
Horizontal Shift (\(y = (x-2)^2\)):
- The red curve represents the function \(y = (x-2)^2\), obtained by replacing \(x\) with \(x-2\).
- This transformation shifts the parabola 2 units to the right, moving the vertex to \((2, 0)\).
Vertical Shift (\(y = (x-2)^2 + 1\)):
- The magenta curve represents the function \(y = (x-2)^2 + 1\), obtained by adding 1 to \((x-2)^2\).
- This transformation shifts the red parabola 1 unit upward, moving the vertex to \((2, 1)\).

Reflection of Graphs

Reflections are transformations that flip a graph about the \(x\)-axis or \(y\)-axis. This is achieved mathematically by replacing one of the variables in the equation with its negation. Let the curve \(C\) be represented by \(F(x, y) = 0\) or \(y = f(x)\). The effect of these transformations is described rigorously as follows.

Reflection Across the \(y\)-Axis

When we replace \(x\) with \(-x\) in the equation \(F(x, y) = 0\), the resulting equation becomes:

\[ F(-x, y) = 0. \]

If a point \((x_0, y_0)\) lies on the curve \(F(x, y) = 0\), it satisfies the equation \(F(x_0, y_0) = 0\). Replacing \(x_0\) with \(-x_0\), the point \((-x_0, y_0)\), which is the reflection of \((x_0, y_0)\) across the \(y\)-axis, satisfies the equation \(F(-x_0, y_0) = 0\). Therefore, the curve represented by \(F(-x, y) = 0\) is the reflection of \(F(x, y) = 0\) across the \(y\)-axis. Geometrically, replacing \(x\) with \(-x\) flips the graph horizontally, with every point \((x, y)\) mapping to \((-x, y)\), ensuring symmetry about the \(y\)-axis.

In the explicit case, when the equation is given as \(y = f(x)\), replacing \(x\) with \(-x\) yields:

\[ y = f(-x). \]

The graph of \(y = f(-x)\) is the reflection of \(y = f(x)\) across the \(y\)-axis, as every point \((x_0, y_0)\) on \(y = f(x)\) corresponds to \((-x_0, y_0)\) on \(y = f(-x)\).

Reflection Across the \(x\)-Axis

When we replace \(y\) with \(-y\) in the equation \(F(x, y) = 0\), the resulting equation becomes:

\[ F(x, -y) = 0. \]

If a point \((x_0, y_0)\) lies on the curve \(F(x, y) = 0\), it satisfies the equation \(F(x_0, y_0) = 0\). Replacing \(y_0\) with \(-y_0\), the point \((x_0, -y_0)\), which is the reflection of \((x_0, y_0)\) across the \(x\)-axis, satisfies the equation \(F(x_0, -y_0) = 0\). Thus, the curve represented by \(F(x, -y) = 0\) is the reflection of \(F(x, y) = 0\) across the \(x\)-axis. Geometrically, replacing \(y\) with \(-y\) flips the graph vertically, with every point \((x, y)\) mapping to \((x, -y)\), ensuring symmetry about the \(x\)-axis.

In the explicit case, when the equation is given as \(y = f(x)\), replacing \(y\) with \(-y\) gives:

\[ -y = f(x) \implies y = -f(x). \]

The graph of \(y = -f(x)\) is the reflection of \(y = f(x)\) across the \(x\)-axis, as every point \((x_0, y_0)\) on \(y = f(x)\) corresponds to \((x_0, -y_0)\) on \(y = -f(x)\).

Thus, replacing \(x\) with \(-x\) reflects the graph across the \(y\)-axis, as it reverses the sign of the \(x\)-coordinate while keeping the \(y\)-coordinate unchanged. Replacing \(y\) with \(-y\) reflects the graph across the \(x\)-axis, as it reverses the sign of the \(y\)-coordinate while keeping the \(x\)-coordinate unchanged. These transformations rigorously ensure that the reflected graphs satisfy the modified equations, preserving symmetry about the respective axes.

Example

Draw the graph of \(y = -(x-2)^2 + 3\)

Solution:

Base Graph (\(y = x^2\)):
- The blue curve represents the upward-opening parabola with its vertex at the origin \((0, 0)\).
Reflection Across the \(x\)-Axis (\(y = -x^2\)):
- The red curve is obtained by reflecting \(y = x^2\) across the \(x\)-axis, resulting in a downward-opening parabola with the same vertex \((0, 0)\).
Horizontal Shift and Reflection (\(y = -(x-2)^2\)):
- The green curve is a horizontally shifted version of \(y = -x^2\), with the vertex moved to \((2, 0)\).
Vertical Shift (\(y = -(x-2)^2 + 3\)):
- The magenta curve is obtained by shifting \(y = -(x-2)^2\) vertically upwards by 3 units, moving the vertex to \((2, 3)\).

Modulus Transformation: Replacing x by |x|

When \(x\) is replaced by \(|x|\) in the equation \(F(x, y) = 0\), the graph of the resulting function \(F(|x|, y) = 0\) can be constructed using the following steps:

Right Side (\(x \geq 0\)):

For \(x \geq 0\), \(|x| = x\), so the graph of \(F(|x|, y) = 0\) coincides with the original graph of \(F(x, y) = 0\) in this region. This means the portion of the graph to the right of the \(y\)-axis remains unchanged.
Left Side (\(x < 0\)):

For \(x < 0\), \(|x| = -x\). This means the graph on the left of the \(y\)-axis is determined by reflecting the portion of the graph on the right (\(x \geq 0\)) across the \(y\)-axis.

Construction of the Graph:

Step 1: Start with the original graph of \(F(x, y) = 0\). Retain the portion of the graph to the right of the \(y\)-axis (\(x \geq 0\)) as is.
Step 2: Remove the portion of the graph to the left of the \(y\)-axis (\(x < 0\)), if any.
Step 3: Reflect the retained portion of the graph (for \(x \geq 0\)) across the \(y\)-axis. The reflected portion now represents the behavior of the graph for \(x < 0\).

The resulting graph, combining the retained and reflected portions, is the graph of \(F(|x|, y) = 0\).

Replacing \(x\) with \(|x|\) ensures that for every positive value of \(x\), the graph is mirrored symmetrically on the left side of the \(y\)-axis. The graph becomes symmetric with respect to the \(y\)-axis, with the right side (\(x \geq 0\)) dictating the entire shape of the graph.

Important:

The graph of \( f(|x|, y) = 0 \) consists of two portions:

The right portion of the graph is described by the equation \( f(x, y) = 0 \) for \( x \geq 0 \).
The left portion of the graph is described by the equation \( f(-x, y) = 0 \), which is the reflection of the right portion across the \(y\)-axis.

Example

Draw the graph of \( 3|x| + 2y = 6 \)

Solution:

Graph \(3x + 2y = 6\):
- Rewrite as \(y = -\frac{3}{2}x + 3\).
- Plot \((0, 3)\) (intercept on \(y\)-axis) and \((2, 0)\) (intercept on \(x\)-axis).
- Draw a straight line through these points for all \(x \in \mathbb{R}\).
Graph \(3|x| + 2y = 6\):
- For \(x \geq 0\), the graph coincides with \(3x + 2y = 6\).
- For \(x < 0\), replace \(x\) with \(-x\), giving \(y = \frac{3}{2}x + 3\), which is the reflection of the right-hand side across the \(y\)-axis.
Combine:
- Retain the portion for \(x \geq 0\) and reflect it for \(x < 0\) to get a "V"-shaped graph symmetric about the \(y\)-axis.

Modulus Transformation: Replacing y with |y|

When \(y\) is replaced by \(|y|\) in the equation \(f(x, y) = 0\), the graph of the new function \(f(x, |y|) = 0\) is constructed as follows. First, remove any part of the graph of \(f(x, y) = 0\) that lies below the \(x\)-axis, as \(|y|\) cannot be negative. Then, reflect the remaining portion of the graph above the \(x\)-axis to complete the symmetric shape. The graph above the \(x\)-axis remains unchanged, while the reflected part creates symmetry about the \(x\)-axis.

The result is a graph of \(f(x, |y|) = 0\) that is symmetric with respect to the \(x\)-axis, where all negative \(y\)-values in the original graph are replaced with their corresponding positive values.

This can be best understood using an explicit form \(y = f(x)\). The graph of \(|y| = f(x)\) can be constructed as follows:

Begin with the graph of \(y = f(x)\).
Remove any portion of the graph that lies below the \(x\)-axis, as \(|y|\) cannot be negative.
Reflect the remaining portion of the graph (above the \(x\)-axis) across the \(x\)-axis to form the graph of \(|y| = f(x)\).

The resulting graph is symmetric about the \(x\)-axis. For \(x\)-values where \(f(x) \geq 0\), \(|y| = f(x)\) corresponds to \(y = f(x)\) and \(y = -f(x)\). This transformation visually represents how the modulus operation replaces negative \(y\)-values with their positive counterparts while keeping positive values unchanged.

This can be best understood using the explicit form of the curve \(y = f(x)\). The equation \(|y| = f(x)\) is defined only for those values of \(x\) where \(f(x) \geq 0\), as the left-hand side \(|y|\) must satisfy \(|y| \geq 0\). Therefore, we first remove any portion of the graph of \(y = f(x)\) that lies below the \(x\)-axis, as it corresponds to \(f(x) < 0\), which is invalid for \(|y| = f(x)\).

Next, the equation \(|y| = f(x)\) implies \(y = \pm f(x)\). This means for any value of \(x\) where \(f(x) \geq 0\), there are two corresponding values of \(y\): \(y = f(x)\) and \(y = -f(x)\). Geometrically, this creates two branches of the graph: one above the \(x\)-axis, given by \(y = f(x)\), and the other below the \(x\)-axis, given by \(y = -f(x)\). These two branches are reflections of each other in the \(x\)-axis, forming a symmetric graph about the \(x\)-axis.

Example

Draw the graph of \(|x|+|y| = 1\)

Solution:

\(x + y = 1\):
- This is a straight line passing through \((1, 0)\) and \((0, 1)\).
- Plot this directly as it does not involve modulus.
\(|x| + y = 1\):
- Start with \(x + y = 1\) for \(x \geq 0\).
- Reflect this line across the \(y\)-axis to account for \(|x|\), creating a "V"-shaped graph symmetric about the \(y\)-axis.
\(|x| + |y| = 1\):
- Start with \(|x| + y = 1\) for \(y \geq 0\) (a "V" shape above the \(x\)-axis).
- Reflect this portion across the \(x\)-axis to include \(|y|\), forming a diamond-shaped graph symmetric about both the \(x\)- and \(y\)-axes.

Modulus Transformation: y = f(x) to y = |f(x)|

When the function is explicitly defined as \(y = f(x)\), the graph of \(y = |f(x)|\) can be constructed as follows:

Start with the Graph of \(y = f(x)\):

Identify the portions of the graph where \(f(x) \geq 0\) (above the \(x\)-axis) and where \(f(x) < 0\) (below the \(x\)-axis). These portions determine how the transformation will affect the graph.
Transform Negative Portions:

The modulus operation \(|f(x)|\) ensures that all negative \(y\)-values become positive. Therefore, reflect the portions of the graph of \(y = f(x)\) that lie below the \(x\)-axis (where \(f(x) < 0\)) across the \(x\)-axis. This reflection geometrically replaces \(f(x)\) with \(-f(x)\) for those \(x\)-values where \(f(x) < 0\).
Retain Positive Portions:

The portions of the graph where \(f(x) \geq 0\) remain unaffected by the transformation, as \(|f(x)| = f(x)\) in this region.

The final graph is symmetric about the \(x\)-axis for the regions where \(f(x) < 0\) and remains identical to \(f(x)\) for the regions where \(f(x) \geq 0\). This ensures the entire graph lies above or on the \(x\)-axis, as required by the modulus operation.

Horizontal Scaling Transformation

When \(x\) is replaced by \(ax\) (\(a > 0\)) in the equation \(f(x, y) = 0\), the new equation \(f(ax, y) = 0\) describes a transformation known as scaling. This transformation affects the horizontal coordinates of points on the graph, stretching or compressing it in the horizontal direction depending on the value of \(a\).

Effect on Points

Let \((p, q)\) be a point on the original curve \(f(x, y) = 0\). After the scaling transformation, this point moves to a new position:

\[ \left(\frac{p}{a}, q\right), \]

where the \(y\)-coordinate remains unchanged, but the \(x\)-coordinate is scaled by \(1/a\).

Overall Effect

For \(0 < a < 1\): The graph is stretched horizontally, as all \(x\)-coordinates are divided by \(a\), making the graph wider.
For \(a > 1\): The graph is compressed horizontally, as all \(x\)-coordinates are divided by \(a\), making the graph narrower.

This transformation can be visualized as pulling or pushing a rubber sheet horizontally, while keeping the vertical distances unchanged.

Invariant Points

Points that lie on the \(y\)-axis (\(x = 0\)) remain unchanged because their \(x\)-coordinate is zero, and dividing by \(a\) has no effect. These points are called invariant points under the scaling transformation.

Example

Show the transformation of replacing \(x\) by \(2x\) in \(y = \frac{1}{6}(x+2)(x-2)(x-3)\)

Solution:

Original Curve \(y = \frac{1}{6}(x+2)(x-2)(x-3)\):
- The roots are \((-2, 0)\), \((2, 0)\), and \((3, 0)\).
- The curve cuts the \(y\)-axis at the invariant point \((0, 2)\), as the \(y\)-intercept does not change under the scaling transformation.
Scaling Transformation (\(x \to 2x\)):
- Replacing \(x\) with \(2x\) gives the new equation:
  
  \[ y = \frac{1}{6}(2x+2)(2x-2)(2x-3). \]
- The graph is horizontally compressed:
  - Roots move closer to the \(y\)-axis:
  - \((-2, 0)\) becomes \((-1, 0)\),
  - \((2, 0)\) becomes \((1, 0)\),
  - \((3, 0)\) becomes \((\frac{3}{2}, 0)\).
  - This happens with all points, the point \((4, 2)\) becomes \((2, 2)\), showing compression.
Invariant Point:
- The point where the curve cuts the \(y\)-axis, \((0, 2)\), remains unchanged, as the \(x\)-coordinate is zero and is unaffected by scaling.
Visual Effect:
- The transformed graph appears compressed toward the \(y\)-axis while maintaining its vertical structure and invariant point \((0, 2)\). The \(y\)-values remain unchanged.

Example

Observe a similar contraction in the graph of \(y=4-||x-2|-2|\) when \(x\) is replaced by \(2x\).

The original equation \(y = 4 - ||x - 2| - 2|\) represents a "zigzag" graph with symmetry about \(x = 2\). The transformation \(x \to 2x\) compresses the graph horizontally by a factor of \(2\), while preserving its overall structure and vertical features.

Effect of Scaling:

Replacing \(x\) with \(2x\) results in the new equation:

\[ y = 4 - ||2x - 2| - 2|. \]

This scaling transformation affects the \(x\)-coordinates of points on the graph, halving them, while the \(y\)-coordinates remain unchanged.

Key Points Before and After Transformation:

Invariant Point:

The graph cuts the \(y\)-axis at \((0, 0)\), which remains unchanged because the \(y\)-intercept does not depend on \(x\).
Other Points:
- \((2, 2)\) becomes \((1, 2)\).
- \((4, 4)\) remains \((2, 4)\), as it lies on the line of symmetry.
- \((8, 0)\) becomes \((4, 0)\).
- \((-4, 0)\) becomes \((-2, 0)\).

The graph contracts horizontally, bringing all \(x\)-coordinates closer to the \(y\)-axis by a factor of \(2\). The zigzag structure remains intact, maintaining symmetry about \(x = 2\) and preserving the relative positions of the points vertically.

This transformation demonstrates how scaling impacts the horizontal spread of the graph while keeping its key vertical features, like the \(y\)-values and invariant points, unchanged.

Streching

Show the transformation of \(y = 2(x+1)(x-1)(x-2)\) to \(y = 2\left(\frac{x}{2}+1\right)\left(\frac{x}{2}-1\right)\left(\frac{x}{2}-2\right) \) by replacing \(x\) by \(\frac{x}{2}\)

Scaling Transformation:
- Replacing \(x\) with \(\frac{x}{2}\) scales the graph horizontally by a factor of \(a = \frac{1}{2}\), where \(a < 1\).
- This results in an overall horizontal stretching of the graph.
Effect on Key Points:
- The roots of the original curve \((-1, 0)\), \((1, 0)\), and \((2, 0)\) are scaled to \((-2, 0)\), \((2, 0)\), and \((4, 0)\), respectively.
- The point \((0, 4)\), where the graph cuts the y-axis is invariant, that is, it remains as it is.
Geometric Interpretation:
- The graph expands horizontally as the \(x\)-coordinates are scaled by a factor of \(2\), while the overall shape and \(y\)-values remain unchanged.
- This creates a "stretched" version of the original graph.

This transformation emphasizes how scaling by \(x \to \frac{x}{2}\) stretches the graph along the horizontal axis while preserving its symmetry and vertical alignment.

Example

Transformation of \(y = \sin(x)\) to \(y = \sin(3x)\)

Transformation Process:
- Replacing \(x\) with \(3x\) compresses the graph horizontally by a factor of \(3\).
- The equation becomes \(y = \sin(3x)\).
Effect on Roots:
- Roots of \(y = \sin(x)\): \((0, 0), (\pi, 0), (2\pi, 0)\).
- Under the transformation \(x \to 3x\), the roots become:
  - \((0, 0)\) remains unchanged.
  - \((\pi, 0)\) transforms to \((\pi/3, 0)\).
  - \((2\pi, 0)\) transforms to \((2\pi/3, 0)\).
Range and Extrema:
- The range of \(\sin(x)\) (\([-1, 1]\)) is unchanged.
- Maximum and minimum values (\(1\) and \(-1\)) occur at the same \(y\)-values.
Visual Effect:
- The \(y = \sin(3x)\) graph oscillates faster, completing three cycles within the interval \([0, 2\pi]\) compared to one cycle for \(y = \sin(x)\).

This transformation compresses the \(x\)-intervals by a factor of \(3\) while preserving the range, symmetry, and amplitude.

Effect on Range

Scaling \(x\) by a factor of \(a > 0\), such that \(x \to ax\), modifies the function \(y = f(x)\) to \(y = f(ax)\), causing a horizontal compression (if \(a > 1\)) or stretching (if \(0 < a < 1\)) of the graph. This transformation does not affect the range, as the \(y\)-values for corresponding inputs remain unchanged. Similarly, the maximum and minimum values of the function remain the same because scaling affects only the horizontal spacing of points and not their vertical heights. Domain is definitely affected

Vertical Scaling Transformation

When \(y\) is replaced by \(ay\), where \(a > 0\), the transformation scales the graph vertically. The equation becomes \(f(x, y) \to f(x, ay) = 0\), and every point \((p, q)\) on the original graph transforms to \(\left(p, \frac{q}{a}\right)\).

Effect on Domain and Range:

The domain of the function remains unchanged because \(x\)-values are not affected by this transformation.
The range is scaled inversely by \(a\). If the original range is \([m, M]\), it becomes \(\left[\frac{m}{a}, \frac{M}{a}\right]\).
The maximum value of the function \(M\) transforms to \(\frac{M}{a}\), and the minimum value \(m\) transforms to \(\frac{m}{a}\).

Invariant Points:

Points where the graph intersects the \(x\)-axis (\(y = 0\)) remain invariant, as scaling \(y\) does not affect zero values.

Geometric Interpretation:

If \(a > 1\), the graph undergoes vertical compression, making it appear "shorter."
If \(0 < a < 1\), the graph undergoes vertical stretching, making it appear "taller."

This transformation affects only the vertical properties of the graph, leaving the horizontal structure, including the domain, unchanged.

Vertical Scaling of Explicit Functions

When \(y = f(x)\) is transformed to \(y = af(x)\) with \(a > 0\), the graph stretches or compresses vertically. This happens because we are multiplying output of \(f\) with \(a\). Altering the output results in the change along \(y-axis\).

If \(a > 1\): The graph is stretched (taller).
If \(0 < a < 1\): The graph is compressed (shorter).
The range scales to \([am, aM]\), where \(m\) and \(M\) are the original minimum and maximum values.
Roots (where \(f(x) = 0\)) remain invariant, as multiplying by \(a\) does not change zero.
The domain is unaffected, as the \(x\)-coordinates remain the same.

This transformation alters vertical height without affecting horizontal structure.

Example

Consider the circle \(x^2 + y^2 = 1\). Replacing \(y\) with \(2y\) transforms the equation into \(x^2 + 4y^2 = 1\). This results in a vertical scaling of the graph, compressing it along the \(y\)-axis.

In the original circle, the range of \(y\) is \([-1, 1]\). After the transformation, the range becomes \([-0.5, 0.5]\) because each \(y\)-coordinate is scaled by \(1/2\). The domain, however, remains \([-1, 1]\), as \(x\)-values are unaffected.

Points where the graph intersects the \(x\)-axis, such as \((\pm1, 0)\), remain unchanged since \(y = 0\) is invariant under vertical scaling. The circle is thus transformed into an ellipse, with its major axis along the \(x\)-direction and its minor axis vertically compressed.

Example

Consider the function \(y = \cos(x)\) and its vertical scaling transformations \(y = \frac{1}{2}\cos(x)\) and \(y = 2\cos(x)\). When a function is scaled vertically, its graph is stretched or compressed along the \(y\)-axis, while the \(x\)-intercepts remain unchanged because \(y = 0\) is invariant under multiplication.

For \(y = \frac{1}{2}\cos(x)\), the amplitude is reduced from \(1\) to \(\frac{1}{2}\), resulting in vertical compression. Similarly, for \(y = 2\cos(x)\), the amplitude increases from \(1\) to \(2\), causing vertical stretching. Despite these changes, the \(x\)-intercepts at \(\frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots\) remain fixed, as the scaling factor does not affect points where \(y = 0\).

The range of \(y = \cos(x)\), originally \([-1, 1]\), changes to \([-\frac{1}{2}, \frac{1}{2}]\) for \(y = \frac{1}{2}\cos(x)\) and \([-2, 2]\) for \(y = 2\cos(x)\).

Greatest Integer Function Transformation

Replacing \(x\) by \([\![x]\!]\):

When \(x\) is replaced by \([\![x]\!]\), the greatest integer less than or equal to \(x\), in the equation \(f(x, y) = 0\), the resulting function is \(f([\![x]\!], y) = 0\). This transformation modifies the graph of the original function by segmenting it into intervals, where each interval corresponds to a constant \(x\)-value determined by \([\![x]\!]\).

Explanation of the Transformation

The graph is divided into intervals of the form \(n \leq x < n+1\), where \(n \in \mathbb{Z}\). Within each interval, the value of \(x\) is replaced by the constant \(n\), and the corresponding \(y\)-values are computed as \(f(n)\). This results in horizontal segments for each interval, as the value of \(x\) does not vary within an interval.

Steps to Draw the Graph

Divide the Domain: Draw vertical lines at all integer values of \(x\). These lines divide the graph into intervals \([n, n+1)\).
Replace \(x\): In each interval, replace \(x\) with the constant integer \(n\), resulting in \(y = f(n)\). The graph in each interval is a horizontal line segment at height \(y = f(n)\).
Breaks in the graph: At each integer value of \(x\), the graph exhibits breaks as jumps, called discontinuities, as the value of \(y\) changes to \(f(n+1)\).

Example: Graph of \(y = \sin([\![x]\!])\)

To construct the graph of \(y = \sin([\![x]\!])\), start with the graph of \(y = \sin(x)\). For each interval \([n, n+1)\), replace \(x\) with \(n\). The value of \(y\) becomes constant at \(y = \sin(n)\) within each interval.

For \(0 \leq x < 1\), \(y = \sin(0) = 0\).
For \(1 \leq x < 2\), \(y = \sin(1)\).
For \(2 \leq x < 3\), \(y = \sin(2)\), and so on.

The graph consists of horizontal segments corresponding to these values, with vertical jumps at integer points. This results in a "stepwise" graph that represents the behavior of the greatest integer function applied to the \(x\)-coordinate of the original curve.\

Replacing \(y\) by \([\![y]\!]\):

When \(y\) is replaced by \([\![y]\!]\) in \(y = f(x)\), the resulting function is \([\![y]\!] = f(x)\). This transformation only retains those values of \(x\) for which \(f(x)\) is an integer. Since \([\![y]\!] = k\) (\(k \in \mathbb{Z}\)) implies \(y \in [k, k+1)\), the graph of \([\![y]\!] = f(x)\) is constructed as follows:

Identify Integer Points: Determine the points on the curve \(y = f(x)\) where \(f(x)\) takes integer values. These are the only \(x\)-values that contribute to the graph of \([\![y]\!] = f(x)\).
Vertical Line Segments: At each such point, draw a vertical line segment of length \(1\) starting from the integer value \(k = f(x)\) and extending upward to \(k+1\).

This transformation effectively "discretizes" the continuous curve of \(y = f(x)\), breaking it into vertical segments that correspond to integer values of the function.

For example: Let us draw the graph of \([\![y]\!] = 2\sin(x)\).

Identify integer points: Determine where \(2\sin(x)\) is an integer (\(-2, -1, 0, 1, 2\)).
Plot vertical segments: At each point where \(2\sin(x)\) is an integer, draw a vertical line segment of length 1 unit upwards starting from the integer value \([\![2\sin(x)]\!]\).

For example:

At \(x = 0, \pi, 2\pi\), \(2\sin(x) = 0\), so draw a vertical segment from \(0\) to \(1\).
At \(x = \frac{\pi}{2}, \frac{5\pi}{2}\), \(2\sin(x) = 2\), so draw a vertical segment from \(2\) to \(3\).
At \(x = \frac{3\pi}{2}\), \(2\sin(x) = -2\), so draw a vertical segment from \(-2\) to \(-1\).

Result:

The graph is stepwise, with each vertical segment indicating the integer value of \(2\sin(x)\) in its respective interval.

Drawing the Graph of y = [f(x)]

When working with a function defined explicitly as \(y = f(x)\), we sometimes want to visualize how applying the greatest integer function modifies its graph. In this section, we explore how to construct the graph of \(y = [f(x)]\), where \([f(x)]\) represents the greatest integer less than or equal to \(f(x)\). This transformation produces a stepwise graph, where each step corresponds to the integer part of the original function \(f(x)\).

The greatest integer function \([f(x)]\):

Rounds \(f(x)\) down to the nearest integer.
Only preserves the integer part of \(f(x)\), truncating any fractional value.

For example:

If \(f(x) = 1.7\), then \([f(x)] = 1\).
If \(f(x) = -0.3\), then \([f(x)] = -1\).

Procedure to Draw \(y = [f(x)]\)

Start with \(y = f(x)\):
- First, plot the graph of the explicit function \(y = f(x)\). This serves as the base graph for constructing the stepwise transformation.
Draw horizontal guide lines:
- Draw horizontal lines at integer values of \(y\) (\(y = 0, \pm 1, \pm 2, \dots\)) across the graph. These lines divide the \(y\)-axis into regions corresponding to integer intervals of \(f(x)\).
Identify intersection points:
- Identify the points where the graph of \(f(x)\) intersects these horizontal lines. These points correspond to the values of \(x\) where \(f(x)\) equals an integer.
Divide into intervals:
- Between two consecutive intersection points \(x_k\) and \(x_{k+1}\), the values of \(f(x)\) lie within a single integer interval \(n \leq f(x) < n+1\), where \(n\) is an integer.
Draw horizontal segments:
- For each interval \(x_k \leq x < x_{k+1}\), draw a horizontal line segment at \(y = n\), where \(n\) is the greatest integer less than or equal to \(f(x)\).
Repeat for all intervals:
- Repeat the process across the entire domain of \(f(x)\) to complete the stepwise graph of \(y = [f(x)]\).

The Floor Analogy

Imagine the graph of \(y = f(x)\) as a path moving through a building, where horizontal lines at integer \(y\)-values (\(y = 0, \pm1, \pm2, \dots\)) represent the floors of the building. Now, think of applying the greatest integer function \([f(x)]\) as causing any portion of the graph of \(f(x)\) between two consecutive integer floors to "fall" to the lower floor.

This analogy helps us visualize how the continuous graph of \(f(x)\) transforms into the stepwise graph of \(y = [f(x)]\).

Graphing y = f({x})

The function \( y = f(\{x\}) \) involves the fractional part of \( x \), denoted as \( \{x\} \), which is defined as \( \{x\} = x - \lfloor x \rfloor \), where \( \lfloor x \rfloor \) represents the greatest integer less than or equal to \( x \). This means \( \{x\} \) always lies within the interval \([0, 1)\). Consequently, the graph of \( y = f(\{x\}) \) is periodic, repeating the portion of \( y = f(x) \) from \([0, 1)\) across all intervals \( [n, n+1) \), where \( n \in \mathbb{Z} \).

Procedure

Draw the graph of \( y = f(x) \):

Begin by plotting the graph of \( y = f(x) \) over its domain. This serves as the base function from which \( y = f(\{x\}) \) is derived.
Isolate the portion in \( [0, 1) \):

Restrict \( x \) to the interval \( [0, 1) \). Keep only the part of the graph within this interval and remove the rest.
Repeat across all intervals:

Copy the isolated portion of \( y = f(x) \) from \( [0, 1) \) and repeat it in each interval \( [n, n+1) \), where \( n \in \mathbb{Z} \). This means for every \( n \), the portion of the graph in \( [0, 1) \) is shifted horizontally by \( n \).

Graphing y = {f(x)}

The graph of \( y = \{f(x)\} \), where \( \{f(x)\} \) represents the fractional part of \( f(x) \), can be obtained from the graph of \( y = f(x) \). The fractional part function retains only the decimal part of \( f(x) \), effectively mapping \( f(x) \) to the interval \([0, 1)\). This transformation involves "cutting" the graph of \( f(x) \) into sections and "folding" them into the specified range.

Steps to Draw the Graph of \( y = \{f(x)\} \)

Draw the Graph of \( y = f(x) \):
- Begin with the complete graph of \( y = f(x) \).
Add Horizontal Guidelines:
- Draw horizontal lines at integer values \( y = 1, 2, 3, \dots \) and \( y = -1, -2, -3, \dots \) to act as boundaries.
Cut and Retain the Fractional Part:
- Identify the portion of \( f(x) \) that lies within each interval \( [n, n+1) \), where \( n \in \mathbb{Z} \).
- Discard the integer part of \( f(x) \), retaining only the "decimal" part. This maps \( f(x) \) to \( y \in [0, 1) \).
- Basically, cut the portion in \( [n, n+1) \) and drop it to x-axis.
Repeat Periodically:
- Repeat the mapped section in every interval \( [n, n+1) \) along the \( x \)-axis.