MTH120 College Algebra Chapter 3.5

 3.5 Transformation of Functions:

Transformation of functions involves making changes to the graphs of functions to create new graphs. These changes are often achieved through operations such as shifting, stretching, compressing, reflecting, and combining functions. Here are some common transformations of functions:

1. Vertical Shift (Translation):

   • Adding or subtracting a constant to the function shifts the graph vertically. For example, $�(�)+�$ shifts the graph $�$ units up, while $�(�)-�$ shifts it $�$ units down.

2. Horizontal Shift (Translation):

   • Adding or subtracting a constant inside the function's argument shifts the graph horizontally. For example, $�(�+�)$ shifts the graph $�$ units to the left, while $�(�-�)$ shifts it $�$ units to the right.

3. Vertical Stretch/Compression:

   • Multiplying the function by a constant $�$ changes the vertical scale. If $�>1$, it stretches the graph vertically. If $0<�<1$, it compresses it vertically.

4. Horizontal Stretch/Compression:

   • Multiplying the function's argument by a constant $�$ changes the horizontal scale. If $�>1$, it compresses the graph horizontally. If $0<�<1$, it stretches it horizontally.

5. Reflections:

   • Reflecting a function across the x-axis is achieved by multiplying it by -1. Reflecting it across the y-axis is done by changing the sign of the independent variable, e.g., $�(-�)$.

6. Combining Functions:

   • Combining two or more functions through addition, subtraction, multiplication, or division can create complex graphs.

7. Piecewise Functions:

   • Defining a function differently for different intervals creates a piecewise function with different behaviors on different segments.

8. Absolute Value Functions:

   • The absolute value function $\mathrm{\mid }�\mathrm{\mid }$ reflects negative values to positive values, resulting in a V-shaped graph.

9. Shifted Absolute Value Functions:

   • Adding or subtracting a constant inside the absolute value, such as $\mathrm{\mid }�-�\mathrm{\mid }$, shifts the V-shaped graph horizontally.

10. Exponential Functions:

    • Changing the base of an exponential function, such as ${�}^{�}$ where $�>1$, stretches or compresses the graph.

11. Logarithmic Functions:

    • Changing the base of a logarithmic function, such as ${\log }_{�}(�)$, stretches or compresses the graph.

12. Trigonometric Functions:

    • Altering the amplitude, period, phase shift, and vertical shift of trigonometric functions like sine and cosine can transform their graphs.

These transformations allow you to manipulate functions to model various real-world phenomena, adapt existing models to new situations, and explore mathematical relationships in depth. They are essential tools in mathematics, physics, engineering, and many other fields.

Graphing functions using vertical and horizontal shifts involves modifying the original function's graph by shifting it up/down (vertical shift) or left/right (horizontal shift). These shifts are achieved by adding or subtracting constants to the function's formula or by altering the argument of the function. Here's how you can graph functions with these shifts:

Vertical Shift (Translation):

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Determine the Vertical Shift: Decide whether you want to shift the graph upward or downward. If you want to shift it up by $�$ units, you'll add $�$ to the function, resulting in $�(�)+�$. If you want to shift it down by $�$ units, you'll subtract $�$, resulting in $�(�)-�$.

3. Graph the Original Function: First, graph the original function $�(�)$ without any shifts. This serves as your starting point.

4. Apply the Vertical Shift: If you're shifting up, move the graph $�$ units up from the original graph. If you're shifting down, move it $�$ units down. This will be your new graph.

Horizontal Shift (Translation):

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Determine the Horizontal Shift: Decide whether you want to shift the graph left or right. If you want to shift it to the left by $�$ units, you'll replace $�$ with $�+�$ inside the function, resulting in $�(�+�)$. If you want to shift it to the right by $�$ units, you'll replace $�$ with $�-�$, resulting in $�(�-�)$.

3. Graph the Original Function: First, graph the original function $�(�)$ without any shifts. This serves as your starting point.

4. Apply the Horizontal Shift: If you're shifting to the left, move the graph $�$ units to the left from the original graph. If you're shifting to the right, move it $�$ units to the right. This will be your new graph.

Let's look at an example:

Suppose you have the function $�(�)={�}^2$, and you want to graph $�(�)=(�-2{)}^2+3$.

1. Start with the Original Function: The original function is $�(�)={�}^2$.

2. Determine the Horizontal and Vertical Shifts:

   • There's a horizontal shift to the right by $2$ units ($�-2$).
   • There's a vertical shift up by $3$ units ($+3$).

3. Graph the Original Function: First, graph $�(�)={�}^2$ as your starting point.

4. Apply the Shifts:

   • Shift the graph $2$ units to the right.
   • Shift the resulting graph $3$ units up.

The final graph of $�(�)=(�-2{)}^2+3$ will be the shifted parabola, moved $2$ units to the right and $3$ units up from the graph of $�(�)={�}^2$.

This process allows you to easily visualize how shifts in the function's formula affect the position of its graph on the coordinate plane.

Adding a constant to a function involves modifying the function's formula by adding a fixed number (the constant) to its output (the dependent variable) for every input (the independent variable). This shift is typically a vertical shift, which moves the graph of the function up or down.

Mathematically, if you have a function $�(�)$, you can create a new function $�(�)$ by adding a constant $�$ to it:

$�(�)=�(�)+�$

Here's how this works:

1. Original Function: Start with the original function $�(�)$.

2. Determine the Constant Shift: Decide whether you want to shift the graph upward or downward. If you want to shift it up, the constant $�$ is positive; if you want to shift it down, the constant $�$ is negative.

3. Modify the Function Formula: Add the constant $�$ to the function formula:

   • If shifting upward ($�$ is positive): $�(�)=�(�)+�$
   • If shifting downward ($�$ is negative): $�(�)=�(�)-\mathrm{\mid }�\mathrm{\mid }$

4. Graph the Original Function: First, graph the original function $�(�)$. This will serve as your starting point.

5. Apply the Constant Shift: Move the entire graph of the original function either up or down by $\mathrm{\mid }�\mathrm{\mid }$ units, depending on the sign of $�$.

Let's illustrate this with an example:

Suppose you have the function $�(�)={�}^2$, and you want to create a new function $�(�)$ by shifting $�(�)$ up by 3 units:

1. Original Function: $�(�)={�}^2$

2. Determine the Constant Shift: We want to shift the graph upward, so $�=3$.

3. Modify the Function Formula: Add the constant $3$ to the function:

   $�(�)=�(�)+3={�}^2+3$

4. Graph the Original Function: First, graph $�(�)={�}^2$.

5. Apply the Constant Shift: Shift the entire graph of $�(�)$ upward by $3$ units. The new graph $�(�)={�}^2+3$ will be located $3$ units above the original graph.

This process allows you to see how adding a constant to a function affects the position of its graph. In this case, it results in a vertical shift of the graph.

To represent a vertical shift for a tabular function, you can create a new row in the table that corresponds to the shifted function. Here are the steps to do this:

1. Start with the Original Tabular Function: Begin with the table of values for the original function. This table should have two columns: one for the $�$ values and another for the $�$ values (the function's outputs).

2. Determine the Vertical Shift: Decide whether you want to shift the graph upward or downward. If you want to shift it up, you'll add a constant $�$ to the original $�$ values; if you want to shift it down, you'll subtract $�$ from the original $�$ values.

3. Create a New Row: Add a new row to the table to represent the shifted function.

4. Modify the $�$ Values: In the new row, modify the $�$ values to reflect the vertical shift:

   • If shifting upward ($�$ is positive): Add $�$ to each of the original $�$ values.
   • If shifting downward ($�$ is negative): Subtract $\mathrm{\mid }�\mathrm{\mid }$ from each of the original $�$ values.

5. Fill in the $�$ Values: The $�$ values in the new row should be the same as those in the original function since the horizontal position of the points does not change with a vertical shift.

6. Graph the Shifted Function: Plot the points from the new row on a coordinate plane. These points will represent the graph of the shifted function.

Here's an example to illustrate this process:

Suppose you have the original tabular function:

$�$	$�$
1	2
2	4
3	6

And you want to create a new row in the table to represent a vertical shift of 3 units upward:

1. Original Tabular Function:

   $�$	$�$
   1	2
   2	4
   3	6

2. Determine the Vertical Shift: We want to shift upward by 3 units, so $�=3$.

3. Create a New Row: Add a new row to the table for the shifted function.

4. Modify the $�$ Values: In the new row, modify the $�$ values by adding 3 to each of the original $�$ values.

   $�$	$�$
   1	2
   2	4
   3	6
   1	5
   2	7
   3	9

5. Fill in the $�$ Values: The $�$ values in the new row are the same as in the original function.

6. Graph the Shifted Function: Plot the points from the new row on a coordinate plane. These points represent the graph of the shifted function, which has been shifted 3 units upward compared to the original function.

This process allows you to visualize how a vertical shift affects the graph of a tabular function by adding or subtracting a constant to the $�$ values.

Identifying horizontal shifts in a function involves recognizing how changes in the function's formula or expression affect the horizontal position (left or right) of its graph. Here's how you can identify horizontal shifts in a function:

1. Identify the Original Function: Start with the original function, denoted as $�(�)$, which you want to analyze for horizontal shifts.

2. Inspect the Formula or Expression:

   • Look at the function's formula or expression. Identify any terms or factors that involve $�$.
   • Pay attention to whether there are any additions or subtractions inside the function's argument (i.e., inside the parentheses or square roots). These can indicate horizontal shifts.

3. Determine the Direction of Shift:

   • To identify a horizontal shift, ask yourself whether the effect of the formula or expression will move the graph to the right or to the left.
   • If a term is added inside the function's argument, it will shift the graph to the left.
   • If a term is subtracted inside the function's argument, it will shift the graph to the right.

4. Identify the Shift Amount (Magnitude):

   • Determine the amount of the horizontal shift by examining the numerical value associated with the addition or subtraction.
   • If there's no numerical value, it's often a shift of one unit to the left or right.

5. Check for Multiple Shifts: Be aware that a function may involve more than one shift. In such cases, identify each shift separately and consider their combined effect on the graph.

6. Practice and Recognize Patterns: With practice, you'll become more adept at recognizing common functions and their associated shifts. For example, a function of the form $�(�-�)$ will involve a shift of $�$ units to the right, while $�(�+�)$ will involve a shift of $�$ units to the left.

Let's look at a few examples:

Example 1:

Given the function $�(�)=(�-3{)}^2$, you can recognize that there's a horizontal shift to the right by $3$ units because of the subtraction inside the parentheses.

Example 2:

For $�(�)=\sqrt{�+2}$, there's a horizontal shift to the left by $2$ units because of the addition inside the square root.

Example 3:

Consider $ℎ(�)=\sin (�-�\mathrm{/}4)$. This function involves a horizontal shift to the right by $�\mathrm{/}4$ radians due to the subtraction inside the sine function.

By analyzing the formula or expression of a function, you can determine the direction and amount of any horizontal shifts, which is crucial for understanding how the function's graph moves on the coordinate plane.

To represent a horizontal shift for a tabular function, you can create a new row in the table that corresponds to the shifted function. Here are the steps to do this:

1. Start with the Original Tabular Function: Begin with the table of values for the original function. This table should have two columns: one for the $�$ values and another for the $�$ values (the function's outputs).

2. Determine the Horizontal Shift: Decide whether you want to shift the graph to the left or to the right. If you want to shift it to the left, you'll subtract a constant $�$ from the original $�$ values; if you want to shift it to the right, you'll add $�$ to the original $�$ values.

3. Create a New Row: Add a new row to the table to represent the shifted function.

4. Modify the $�$ Values: In the new row, modify the $�$ values to reflect the horizontal shift:

   • If shifting to the left ($�$ is positive): Subtract $�$ from each of the original $�$ values.
   • If shifting to the right ($�$ is negative): Add $\mathrm{\mid }�\mathrm{\mid }$ to each of the original $�$ values.

5. Fill in the $�$ Values: The $�$ values in the new row should be the same as those in the original function since the vertical position of the points does not change with a horizontal shift.

6. Graph the Shifted Function: Plot the points from the new row on a coordinate plane. These points will represent the graph of the shifted function.

Here's an example to illustrate this process:

Suppose you have the original tabular function:

$�$	$�$
1	2
2	4
3	6

And you want to create a new row in the table to represent a horizontal shift of 2 units to the right:

1. Original Tabular Function:

   $�$	$�$
   1	2
   2	4
   3	6

2. Determine the Horizontal Shift: We want to shift to the right, so $�=2$.

3. Create a New Row: Add a new row to the table for the shifted function.

4. Modify the $�$ Values: In the new row, modify the $�$ values by adding $2$ to each of the original $�$ values.

   $�$	$�$
   1	2
   2	4
   3	6
   3	2
   4	4
   5	6

5. Fill in the $�$ Values: The $�$ values in the new row are the same as in the original function.

6. Graph the Shifted Function: Plot the points from the new row on a coordinate plane. These points represent the graph of the shifted function, which has been shifted 2 units to the right compared to the original function.

This process allows you to visualize how a horizontal shift affects the graph of a tabular function by adding or subtracting a constant to the $�$ values.

Combining vertical and horizontal shifts involves modifying a function to shift its graph both vertically (up or down) and horizontally (left or right). These shifts are achieved by adding or subtracting constants to the function's formula or altering the argument of the function. Here's how you can combine vertical and horizontal shifts:

Vertical Shift (Translation):

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Determine the Vertical Shift: Decide whether you want to shift the graph upward or downward. If you want to shift it up by $�$ units, you'll add $�$ to the function, resulting in $�(�)+�$. If you want to shift it down by $�$ units, you'll subtract $�$, resulting in $�(�)-�$.

Horizontal Shift (Translation):

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Determine the Horizontal Shift: Decide whether you want to shift the graph left or right. If you want to shift it to the left by $�$ units, you'll replace $�$ with $�+�$ inside the function, resulting in $�(�+�)$. If you want to shift it to the right by $�$ units, you'll replace $�$ with $�-�$, resulting in $�(�-�)$.

Combining Both Shifts:

To combine both vertical and horizontal shifts, you simply apply the vertical shift first and then the horizontal shift, or vice versa, depending on your desired order of shifts. For example:

1. Vertical Shift First, Then Horizontal Shift:

   • Vertical Shift Upward by ${�}_1$ units: $�(�)+{�}_1$
   • Horizontal Shift to the Right by ${�}_2$ units: $(�(�)+{�}_1)-{�}_2$ or $(�(�)-{�}_2)+{�}_1$

2. Horizontal Shift First, Then Vertical Shift:

   • Horizontal Shift to the Right by ${�}_2$ units: $�(�-{�}_2)$
   • Vertical Shift Upward by ${�}_1$ units: $(�(�-{�}_2)+{�}_1)$ or $(�(�+{�}_1)-{�}_2)$

The choice of order depends on the specific problem or context. Different orders of shifts can lead to different final expressions for the transformed function.

For example, if you have the function $�(�)={�}^2$ and you want to apply a vertical shift of $3$ units up and a horizontal shift of $2$ units to the left, you can do it in either order:

• Vertical Shift First, Then Horizontal Shift:

  • $�(�)=(�(�)+3)-2={�}^2+1$

• Horizontal Shift First, Then Vertical Shift:

  • $ℎ(�)=�(�-2)+3=(�-2{)}^2+3$

Both $�(�)$ and $ℎ(�)$ represent the same final graph, but they are expressed differently due to the order of shifts.

Graphing functions using reflections about the axes involves modifying the original function's graph by reflecting it across the x-axis, y-axis, or both. These reflections are achieved by changing the signs of the function's output (y-values), input (x-values), or both. Here's how you can graph functions with reflections:

Reflection Across the X-Axis:

To reflect a function across the x-axis, you change the sign of its output (y-values). This is done by multiplying the function by -1:

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Apply the Reflection: Multiply the entire function by -1 to reflect it across the x-axis:

   • $-�(�)$

Reflection Across the Y-Axis:

To reflect a function across the y-axis, you change the sign of its input (x-values). This is done by adding a negative sign to the independent variable inside the function's argument:

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Apply the Reflection: Replace $�$ with $-�$ inside the function to reflect it across the y-axis:

   • $�(-�)$

Reflection Across Both Axes:

To reflect a function across both the x-axis and y-axis, you can combine the two reflections by applying both transformations:

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Apply the Reflections: First, reflect it across the x-axis by multiplying by -1, and then reflect the result across the y-axis by replacing $�$ with $-�$:

   • $-�(-�)$

Graphing the Reflected Function:

After applying the reflections, you can graph the reflected function by plotting points based on the modified formula. Keep in mind that these reflections create symmetry in the graph. For example, reflecting a function across the x-axis will make the graph symmetric with respect to the x-axis, while reflecting it across the y-axis will make it symmetric with respect to the y-axis.

Let's look at an example:

Suppose you have the function $�(�)={�}^2$, and you want to reflect it across the x-axis:

1. Start with the Original Function: $�(�)={�}^2$

2. Apply the Reflection Across the X-Axis: Reflect the function across the x-axis by multiplying by -1:

   $-�(�)=-{�}^2$

Now, graphing the reflected function $-�(�)=-{�}^2$, you'll see that it is the same as the original function $�(�)={�}^2$ but upside down due to the reflection across the x-axis. This reflection creates symmetry with respect to the x-axis.

To reflect the graph of a function both vertically and horizontally, you can apply two transformations: a vertical reflection and a horizontal reflection. These transformations will flip the graph both vertically (across the x-axis) and horizontally (across the y-axis). Here's how you can do it:

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Apply the Vertical Reflection (across the x-axis): To reflect the graph vertically, multiply the entire function by -1:

   • $-�(�)$

3. Apply the Horizontal Reflection (across the y-axis): To reflect the graph horizontally, replace $�$ with $-�$ inside the function's argument:

   • $-�(-�)$

Now, the function $-�(-�)$ represents the graph of the original function reflected both vertically and horizontally.

Graphing the Reflected Function:

After applying both reflections, you can graph the reflected function by plotting points based on the modified formula. Keep in mind that these reflections create symmetry in the graph. The graph of $-�(-�)$ will be symmetric with respect to both the x-axis and the y-axis.

Let's illustrate this with an example:

Suppose you have the function $�(�)={�}^3$, and you want to reflect its graph both vertically and horizontally:

1. Start with the Original Function: $�(�)={�}^3$

2. Apply the Vertical Reflection (across the x-axis): Reflect the function vertically by multiplying by -1:

   • $-�(�)=-{�}^3$

3. Apply the Horizontal Reflection (across the y-axis): Reflect the function horizontally by replacing $�$ with $-�$ inside the function's argument:

   • $-�(-�)=-(-�{)}^3={�}^3$

Now, the function ${�}^3$ represents the graph of the original function reflected both vertically and horizontally.

Graphing the reflected function ${�}^3$, you'll see that it is the same as the original function $�(�)={�}^3$ but with a 180-degree rotation. This reflection creates symmetry with respect to both the x-axis and the y-axis.

Even and odd functions are special types of functions that exhibit specific symmetry properties. You can determine whether a function is even, odd, or neither by examining its algebraic expression and graph. Here's how to identify even and odd functions:

Even Function:

A function $�(�)$ is even if it satisfies the following property:

$�(�)=�(-�)$

In other words, a function is even if it is symmetric with respect to the y-axis. Here's how to determine if a function is even:

1. Start with the Function: Begin with the algebraic expression of the function, let's say $�(�)$.

2. Apply the Even Function Property: Substitute $-�$ for $�$ in the function. If the resulting expression is equal to the original function, the function is even.

3. Check the Graph: You can also check the graph of the function. If the graph is symmetric with respect to the y-axis, it's an even function. This means that if you fold the graph along the y-axis, the two halves will match.

Odd Function:

A function $�(�)$ is odd if it satisfies the following property:

$�(�)=-�(-�)$

In other words, a function is odd if it is symmetric with respect to the origin (the point $(0,0)$). Here's how to determine if a function is odd:

1. Start with the Function: Begin with the algebraic expression of the function, let's say $�(�)$.

2. Apply the Odd Function Property: Substitute $-�$ for $�$ in the function and add a negative sign to the entire expression. If the resulting expression is equal to the original function with a negative sign, the function is odd.

3. Check the Graph: You can also check the graph of the function. If the graph is symmetric with respect to the origin (i.e., it's symmetric when rotated by 180 degrees around the origin), it's an odd function.

Neither Even nor Odd:

If a function does not satisfy the properties of an even or odd function, it is considered neither even nor odd.

Here are some common examples:

• Even Functions: $�(�)={�}^2$, $�(�)=\cos (�)$ (cosine function), $�(�)={�}^{{�}^2}$

• Odd Functions: $�(�)={�}^3$, $�(�)=\sin (�)$ (sine function), $�(�)=\frac{1}{�}$

• Neither Even nor Odd: $�(�)=�$, $�(�)={�}^2+1$, $�(�)=\ln (�)$

Determining whether a function is even, odd, or neither is important in mathematics, as it can simplify calculations and provide insights into the function's behavior and properties.

Graphing functions using stretches and compressions involves modifying the original function's graph by stretching or compressing it vertically or horizontally. These transformations change the shape and size of the graph while preserving its basic characteristics. Here's how to graph functions with stretches and compressions:

Vertical Stretch or Compression:

A vertical stretch or compression of a function $�(�)$ is achieved by multiplying the entire function by a constant $�$. If $�>1$, it results in a vertical stretch, making the graph taller; if $0<�<1$, it results in a vertical compression, making the graph shorter.

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Determine the Stretch/Compression Factor: Decide whether you want to stretch or compress the graph vertically and determine the value of $�$.

3. Apply the Stretch/Compression: Multiply the entire function by the constant $�$:

   • $��(�)$ for a vertical stretch (if $�>1$)
   • $��(�)$ for a vertical compression (if $0<�<1$)

Horizontal Stretch or Compression:

A horizontal stretch or compression of a function $�(�)$ is achieved by multiplying the input (independent variable $�$) by a constant $�$. If $�>1$, it results in a horizontal compression, making the graph narrower; if $0<�<1$, it results in a horizontal stretch, making the graph wider.

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Determine the Stretch/Compression Factor: Decide whether you want to stretch or compress the graph horizontally and determine the value of $�$.

3. Apply the Stretch/Compression: Replace $�$ with $��$ inside the function's argument:

   • $�(��)$ for a horizontal stretch (if $0<�<1$)
   • $�(��)$ for a horizontal compression (if $�>1$)

Graphing the Stretched or Compressed Function:

After applying the stretch or compression, you can graph the transformed function by plotting points based on the modified formula. Keep in mind that these transformations affect the size and shape of the graph while preserving its essential features and symmetry.

Let's look at an example:

Suppose you have the function $�(�)={�}^2$, and you want to apply a vertical stretch by a factor of $2$ and a horizontal compression by a factor of $0.5$:

1. Start with the Original Function: $�(�)={�}^2$

2. Determine the Stretch/Compression Factors:

   • Vertical Stretch: $�=2$ (because $�>1$)
   • Horizontal Compression: $�=0.5$ (because $0<�<1$)

3. Apply the Stretch/Compression:

   • Vertical Stretch: $2�(�)=2{�}^2$
   • Horizontal Compression: $�(0.5�)=(0.5�{)}^2=0.25{�}^2$

Graphing the transformed function $0.25{�}^2$, you'll see that it is narrower (horizontally compressed) and taller (vertically stretched) compared to the original function ${�}^2$.

These transformations allow you to modify the appearance of a function's graph while preserving its basic features and behavior.

Vertical stretches and compressions are transformations applied to a function's graph to change its vertical size. These transformations involve multiplying the function's output (y-values) by a constant factor. Vertical stretches make the graph taller, while vertical compressions make it shorter. Here's how vertical stretches and compressions work:

Vertical Stretch:

A vertical stretch of a function $�(�)$ is achieved by multiplying the entire function by a constant $�$ where $�>1$. The effect is that the graph becomes "stretched" vertically, making it taller compared to the original.

Mathematically, for a vertical stretch:

$�=��(�)$

• $�>1$: This factor stretches the graph vertically.

Vertical Compression:

A vertical compression of a function $�(�)$ is achieved by multiplying the entire function by a constant $�$ where $0<�<1$. The effect is that the graph becomes "compressed" vertically, making it shorter compared to the original.

Mathematically, for a vertical compression:

$�=��(�)$

• $0<�<1$: This factor compresses the graph vertically.

To apply vertical stretches and compressions to a function:

1. Start with the Original Function: Begin with the formula of the original function, $�(�)$.

2. Determine the Stretch/Compression Factor: Decide whether you want to stretch or compress the graph vertically and determine the value of $�$.

3. Apply the Stretch/Compression: Multiply the entire function by the constant $�$ to achieve the desired vertical transformation.

4. Graph the Transformed Function: Plot the points of the transformed function based on the modified formula. The graph will show the vertical stretching or compression effect.

Let's look at an example:

Suppose you have the function $�(�)={�}^2$, and you want to apply a vertical stretch by a factor of $3$:

1. Start with the Original Function: $�(�)={�}^2$

2. Determine the Stretch Factor: You want to stretch the graph vertically, so $�=3$ (because $�>1$).

3. Apply the Vertical Stretch: Multiply the function by $3$:

   $�=3{�}^2$

4. Graph the Transformed Function: Plot the points of the transformed function $3{�}^2$, and you'll see that the graph is taller compared to the original ${�}^2$ graph.

In summary, vertical stretches and compressions modify the vertical size of a function's graph while preserving its shape and basic features. The choice of $�$ determines the degree of stretching or compression applied to the graph.

To create a table for a vertical compression of a tabular function, you'll modify the original $�$ values (outputs) to reflect the vertical compression effect. A vertical compression involves multiplying the original $�$ values by a constant factor $0<�<1$. Here are the steps:

1. Start with the Original Tabular Function: Begin with the table of values for the original function, which should have two columns: one for the $�$ values and another for the $�$ values.

2. Determine the Compression Factor: Decide on the value of $�$ for the vertical compression, where $0<�<1$. This factor determines how much the graph will be compressed vertically.

3. Create a New Row: Add a new row to the table to represent the vertically compressed function.

4. Modify the $�$ Values: In the new row, modify the $�$ values by multiplying each of the original $�$ values by the compression factor $�$. This represents the vertical compression:

   ${�}_{\text{compressed}}=�\cdot {�}_{\text{original}}$

   Substitute the original $�$ values from the original functions' table into this equation to compute the vertically compressed $�$ values.

5. Fill in the $�$ Values: The $�$ values in the new row should be the same as those in the original function since the horizontal position of the points does not change with a vertical compression.

6. Graph the Vertically Compressed Function: Plot the points from the new row on a coordinate plane. These points represent the graph of the vertically compressed function.

Here's an example to illustrate this process:

Suppose you have the original tabular function:

$�$	$�$
1	2
2	4
3	6

And you want to create a new row in the table to represent a vertical compression by a factor of $0.5$:

1. Original Tabular Function:

   $�$	$�$
   1	2
   2	4
   3	6

2. Determine the Compression Factor: $�=0.5$ (because $0<�<1$).

3. Create a New Row: Add a new row to the table for the vertically compressed function.

4. Modify the $�$ Values: In the new row, modify the $�$ values by multiplying each original $�$ value by $0.5$:

   $�$	$�$
   1	2
   2	4
   3	6
   1	1
   2	2
   3	3

5. Fill in the $�$ Values: The $�$ values in the new row are the same as in the original function.

6. Graph the Vertically Compressed Function: Plot the points from the new row on a coordinate plane. These points represent the graph of the vertically compressed function, which is half as tall as the original function.

This process allows you to visualize how a vertical compression affects the graph of a tabular function by reducing the height of each point.

Horizontal stretches and compressions are transformations applied to a function's graph to change its horizontal size. These transformations involve multiplying the input (independent variable $�$) by a constant factor. Horizontal stretches make the graph wider, while horizontal compressions make it narrower. Here's how horizontal stretches and compressions work:

Horizontal Stretch:

A horizontal stretch of a function $�(�)$ is achieved by multiplying the input $�$ by a constant $�$ where $�>1$. The effect is that the graph becomes "stretched" horizontally, making it wider compared to the original.

Mathematically, for a horizontal stretch:

$�=�(��)$

• $�>1$: This factor stretches the graph horizontally.

Horizontal Compression:

A horizontal compression of a function $�(�)$ is achieved by multiplying the input $�$ by a constant $�$ where $0<�<1$. The effect is that the graph becomes "compressed" horizontally, making it narrower compared to the original.

Mathematically, for a horizontal compression:

$�=�(��)$

• $0<�<1$: This factor compresses the graph horizontally.

To apply horizontal stretches and compressions to a function:

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Determine the Stretch/Compression Factor: Decide whether you want to stretch or compress the graph horizontally and determine the value of $�$.

3. Apply the Stretch/Compression: Replace $�$ with $��$ inside the function's argument to achieve the desired horizontal transformation.

4. Graph the Transformed Function: Plot the points of the transformed function based on the modified formula. The graph will show the horizontal stretching or compression effect.

Let's look at an example:

Suppose you have the function $�(�)=\sin (�)$, and you want to apply a horizontal stretch by a factor of $2$:

1. Start with the Original Function: $�(�)=\sin (�)$

2. Determine the Stretch Factor: You want to stretch the graph horizontally, so $�=2$ (because $�>1$).

3. Apply the Horizontal Stretch: Replace $�$ with $2�$ inside the function:

   $�=\sin (2�)$

4. Graph the Transformed Function: Plot the points of the transformed function $\sin (2�)$, and you'll see that the graph is wider (horizontally stretched) compared to the original $\sin (�)$ graph.

In summary, horizontal stretches and compressions modify the horizontal size of a function's graph while preserving its shape and basic features. The choice of $�$ determines the degree of stretching or compression applied to the graph.

Performing a sequence of transformations on a function involves applying multiple transformations in a specific order to modify the original function's graph. These transformations can include translations (horizontal and vertical shifts), stretches, compressions, reflections, and more. To successfully perform a sequence of transformations, follow these steps:

1. Start with the Original Function: Begin with the formula of the original function, let's say $�(�)$.

2. Determine the Order of Transformations: Decide on the order in which you want to apply the transformations. The order matters because some transformations may affect others differently depending on their sequence.

3. Apply Each Transformation Step by Step: For each transformation in the sequence, perform the following steps: a. Identify the specific transformation you want to apply (e.g., vertical shift, horizontal stretch). b. Determine the values of any relevant parameters (e.g., the amount of shift or stretch factor). c. Apply the transformation to the current function formula.

4. Combine Multiple Transformations: If you are applying multiple transformations in succession, apply each transformation to the result of the previous transformation. Continue this process until you have applied all the desired transformations.

5. Graph the Transformed Function: Once you have applied all the transformations in the sequence, graph the resulting function by plotting points based on the modified formula. This graph will represent the function after all the transformations.

6. Check for Accuracy: Review the graph to ensure that it accurately reflects the desired sequence of transformations and that it matches the expected result.

Example of a Sequence of Transformations:

Suppose you have the original function $�(�)={�}^2$ and you want to apply the following sequence of transformations in this order:

1. Horizontal stretch by a factor of $2$.
2. Vertical compression by a factor of $0.5$.
3. Horizontal shift to the right by $3$ units.
4. Vertical shift upward by $4$ units.

Here's how you can perform this sequence of transformations:

1. Start with the Original Function: $�(�)={�}^2$

2. Determine the Order of Transformations: Apply the transformations in the specified order.

3. Apply Each Transformation Step by Step:

   a. Horizontal Stretch by a factor of $2$:

   • $�(�)=(2�{)}^2=4{�}^2$

   b. Vertical Compression by a factor of $0.5$:

   • $�(�)=0.5\cdot 4{�}^2=2{�}^2$

   c. Horizontal Shift to the Right by $3$ units:

   • $�(�)=2(�-3{)}^2$

   d. Vertical Shift Upward by $4$ units:

   • $�(�)=2(�-3{)}^2+4$

4. Graph the Transformed Function: Plot the points of the transformed function $2(�-3{)}^2+4$.

This sequence of transformations has transformed the original function $�(�)={�}^2$ into the function $2(�-3{)}^2+4$. The graph of the transformed function represents the result of all the applied transformations.

Here are four examples of sequences of transformations applied to functions:

Example 1: Vertical Stretch and Horizontal Shift

Start with the function $�(�)=\sin (�)$.

1. Vertical Stretch by a factor of $2$.
2. Horizontal Shift to the right by $�\mathrm{/}4$ units.

Resulting function: $�(�)=2\sin (�-�\mathrm{/}4)$

Example 2: Horizontal Compression, Vertical Stretch, and Vertical Shift

Start with the function $�(�)={�}^3$.

1. Horizontal Compression by a factor of $1\mathrm{/}2$.
2. Vertical Stretch by a factor of $3$.
3. Vertical Shift upward by $2$ units.

Resulting function: $�(�)=3(1\mathrm{/}2�{)}^3+2$

Example 3: Horizontal Reflection and Vertical Compression

Start with the function $�(�)=\sqrt{�}$.

1. Horizontal Reflection (reflect across the y-axis).
2. Vertical Compression by a factor of $0.5$.

Resulting function: $�(�)=-0.5\sqrt{-�}$

Example 4: Vertical Shift, Horizontal Stretch, and Vertical Compression

Start with the function $�(�)=\cos (�)$.

1. Vertical Shift downward by $2$ units.
2. Horizontal Stretch by a factor of $2�$.
3. Vertical Compression by a factor of $0.5$.

Resulting function: $�(�)=0.5\cos (2��)-2$

These examples demonstrate sequences of transformations that modify the original functions in various ways, including shifts, stretches, compressions, and reflections. Each transformation affects the graph in a specific manner, resulting in a new function with distinct characteristics.