Glossary

Additive Transformations (Translations)

Criticality: 3

Changes to a function's graph that involve adding or subtracting constants, resulting in a shift of the graph's position without altering its shape or size.

Example:

Changing $f(x) = \sin(x)$ to $g(x) = \sin(x) + 2$ is an additive transformation, moving the entire sine wave up by 2 units.

Combining Transformations

Criticality: 3

Applying multiple transformations (translations, dilations, reflections) sequentially to a single function to achieve a complex alteration of its graph.

Example:

To model a bouncing ball, you might start with a parabola, then apply a combining transformations to reflect it, stretch it, and shift it to simulate its path.

Dilations

Criticality: 3

Transformations that stretch or compress a function's graph either vertically or horizontally, changing its size or proportion.

Example:

If a spring's oscillation is $f(t)$ , then $g(t) = 2f(t)$ represents a dilation that doubles the amplitude of its movement.

Domain and Range Changes

Criticality: 2

Alterations to the set of possible input values (domain) or output values (range) of a function as a result of transformations.

Example:

If $f(x) = \sqrt{x}$ has a domain of $[0, \infty)$ , then $g(x) = \sqrt{-x}$ will have its domain and range changes such that its domain becomes $(-\infty, 0]$ .

Horizontal Dilations

Criticality: 3

Scales a function's graph horizontally by a factor of $|1/b|$ when the input is multiplied by 'b', i.e., $g(x) = f(bx)$.

Example:

If a song's tempo is $f(t)$ , then $g(t) = f(2t)$ makes the song play twice as fast, demonstrating a horizontal dilation (compression).

Horizontal Shrink

Criticality: 2

A type of horizontal dilation where the graph becomes narrower, occurring when the input is multiplied by a constant 'b' such that $|b| > 1$.

Example:

If a video's playback speed is $f(t)$ , then $g(t) = f(2t)$ results in a horizontal shrink, making the video play twice as fast.

Horizontal Stretch

Criticality: 2

A type of horizontal dilation where the graph becomes wider, occurring when the input is multiplied by a constant 'b' such that $0 < |b| < 1$.

Example:

If a musical note's frequency is $f(t)$ , then $g(t) = f(0.5t)$ causes a horizontal stretch, making the note last twice as long.

Horizontal Translations

Criticality: 3

Shifts a function's graph left or right by adding or subtracting a constant 'h' inside the function, i.e., $g(x) = f(x + h)$, where $h>0$ shifts left and $h<0$ shifts right.

Example:

If a sound wave is described by $f(t)$ , delaying its start by 2 seconds, $g(t) = f(t-2)$ , is a horizontal translation to the right.

Multiplicative Transformations

Criticality: 3

Changes to a function's graph that involve multiplying by constants, resulting in stretches, compressions, or reflections.

Example:

If $f(x)$ models the intensity of a light, then $g(x) = 0.5f(x)$ is a multiplicative transformation that halves the intensity.

Order of Transformations

Criticality: 3

The specific sequence in which multiple transformations must be applied to a function to correctly obtain the transformed graph, typically horizontal changes before vertical changes.

Example:

When transforming $f(x)$ to $g(x) = -2f(x+1)-3$ , applying the horizontal shift first, then the vertical stretch/reflection, and finally the vertical shift, follows the correct order of transformations.

Reflection over x-axis

Criticality: 3

A transformation that flips the graph of a function vertically across the x-axis, occurring when the entire function is multiplied by -1, i.e., $g(x) = -f(x)$.

Example:

If a mountain's profile is $f(x)$ , then $g(x) = -f(x)$ creates an inverted image, which is a reflection over x-axis.

Reflection over y-axis

Criticality: 3

A transformation that flips the graph of a function horizontally across the y-axis, occurring when the input variable is multiplied by -1, i.e., $g(x) = f(-x)$.

Example:

If a car's acceleration is $f(t)$ , then $g(t) = f(-t)$ describes its acceleration if time were reversed, which is a reflection over y-axis.

Reflections

Criticality: 3

Transformations that flip a function's graph across an axis, changing its orientation.

Example:

If a mountain range's profile is $f(x)$ , then $g(x) = -f(x)$ creates a mirrored image, which is a reflection across the x-axis.

Transformations of Functions

Criticality: 3

Processes that alter the graph of a function by shifting, stretching, compressing, or reflecting it without changing its fundamental shape or type.

Example:

If you have the graph of $y = x^2$ , applying a transformation might turn it into $y = (x-2)^2 + 3$ , moving it right and up.

Vertical Dilations

Criticality: 3

Scales a function's graph vertically by a factor of $|a|$ when the function is multiplied by 'a', i.e., $g(x) = af(x)$.

Example:

If a rubber band's stretch is $f(x)$ , then $g(x) = 3f(x)$ means it stretches three times as much, illustrating a vertical dilation.

Vertical Shrink

Criticality: 2

A type of vertical dilation where the graph becomes shorter, occurring when the function is multiplied by a constant 'a' such that $0 < |a| < 1$.

Example:

If a signal's strength is $f(t)$ , then $g(t) = 0.5f(t)$ results in a vertical shrink, reducing the signal's amplitude by half.

Vertical Stretch

Criticality: 2

A type of vertical dilation where the graph becomes taller, occurring when the function is multiplied by a constant 'a' such that $|a| > 1$.

Example:

If a spring's natural oscillation is $f(t)$ , then $g(t) = 2f(t)$ causes a vertical stretch, making the oscillations twice as high.

Vertical Translations

Criticality: 3

Shifts a function's graph up or down by adding or subtracting a constant 'k' outside the function, i.e., $g(x) = f(x) + k$.

Example:

If a roller coaster's height is modeled by $f(x)$ , adding 50 feet to its design, $g(x) = f(x) + 50$ , represents a vertical translation upward.

Glossary

Additive Transformations (Translations)

Criticality: 3

Changes to a function's graph that involve adding or subtracting constants, resulting in a shift of the graph's position without altering its shape or size.

Example:

Changing $f(x) = \sin(x)$ to $g(x) = \sin(x) + 2$ is an additive transformation, moving the entire sine wave up by 2 units.

Combining Transformations

Criticality: 3

Applying multiple transformations (translations, dilations, reflections) sequentially to a single function to achieve a complex alteration of its graph.

Example:

To model a bouncing ball, you might start with a parabola, then apply a combining transformations to reflect it, stretch it, and shift it to simulate its path.

Dilations

Criticality: 3

Transformations that stretch or compress a function's graph either vertically or horizontally, changing its size or proportion.

Example:

If a spring's oscillation is $f(t)$ , then $g(t) = 2f(t)$ represents a dilation that doubles the amplitude of its movement.

Domain and Range Changes

Criticality: 2

Alterations to the set of possible input values (domain) or output values (range) of a function as a result of transformations.

Example:

If $f(x) = \sqrt{x}$ has a domain of $[0, \infty)$ , then $g(x) = \sqrt{-x}$ will have its domain and range changes such that its domain becomes $(-\infty, 0]$ .

Horizontal Dilations

Criticality: 3

Scales a function's graph horizontally by a factor of $|1/b|$ when the input is multiplied by 'b', i.e., $g(x) = f(bx)$.

Example:

If a song's tempo is $f(t)$ , then $g(t) = f(2t)$ makes the song play twice as fast, demonstrating a horizontal dilation (compression).

Horizontal Shrink

Criticality: 2

A type of horizontal dilation where the graph becomes narrower, occurring when the input is multiplied by a constant 'b' such that $|b| > 1$.

Example:

If a video's playback speed is $f(t)$ , then $g(t) = f(2t)$ results in a horizontal shrink, making the video play twice as fast.

Horizontal Stretch

Criticality: 2

A type of horizontal dilation where the graph becomes wider, occurring when the input is multiplied by a constant 'b' such that $0 < |b| < 1$.

Example:

If a musical note's frequency is $f(t)$ , then $g(t) = f(0.5t)$ causes a horizontal stretch, making the note last twice as long.

Horizontal Translations

Criticality: 3

Shifts a function's graph left or right by adding or subtracting a constant 'h' inside the function, i.e., $g(x) = f(x + h)$, where $h>0$ shifts left and $h<0$ shifts right.

Example:

If a sound wave is described by $f(t)$ , delaying its start by 2 seconds, $g(t) = f(t-2)$ , is a horizontal translation to the right.

Multiplicative Transformations

Criticality: 3

Changes to a function's graph that involve multiplying by constants, resulting in stretches, compressions, or reflections.

Example:

If $f(x)$ models the intensity of a light, then $g(x) = 0.5f(x)$ is a multiplicative transformation that halves the intensity.

Order of Transformations

Criticality: 3

The specific sequence in which multiple transformations must be applied to a function to correctly obtain the transformed graph, typically horizontal changes before vertical changes.

Example:

Reflection over x-axis

Criticality: 3

A transformation that flips the graph of a function vertically across the x-axis, occurring when the entire function is multiplied by -1, i.e., $g(x) = -f(x)$.

Example:

If a mountain's profile is $f(x)$ , then $g(x) = -f(x)$ creates an inverted image, which is a reflection over x-axis.

Reflection over y-axis

Criticality: 3

A transformation that flips the graph of a function horizontally across the y-axis, occurring when the input variable is multiplied by -1, i.e., $g(x) = f(-x)$.

Example:

If a car's acceleration is $f(t)$ , then $g(t) = f(-t)$ describes its acceleration if time were reversed, which is a reflection over y-axis.

Reflections

Criticality: 3

Transformations that flip a function's graph across an axis, changing its orientation.

Example:

If a mountain range's profile is $f(x)$ , then $g(x) = -f(x)$ creates a mirrored image, which is a reflection across the x-axis.

Transformations of Functions

Criticality: 3

Processes that alter the graph of a function by shifting, stretching, compressing, or reflecting it without changing its fundamental shape or type.

Example:

If you have the graph of $y = x^2$ , applying a transformation might turn it into $y = (x-2)^2 + 3$ , moving it right and up.

Vertical Dilations

Criticality: 3

Scales a function's graph vertically by a factor of $|a|$ when the function is multiplied by 'a', i.e., $g(x) = af(x)$.

Example:

If a rubber band's stretch is $f(x)$ , then $g(x) = 3f(x)$ means it stretches three times as much, illustrating a vertical dilation.

Vertical Shrink

Criticality: 2

A type of vertical dilation where the graph becomes shorter, occurring when the function is multiplied by a constant 'a' such that $0 < |a| < 1$.

Example:

If a signal's strength is $f(t)$ , then $g(t) = 0.5f(t)$ results in a vertical shrink, reducing the signal's amplitude by half.

Vertical Stretch

Criticality: 2

A type of vertical dilation where the graph becomes taller, occurring when the function is multiplied by a constant 'a' such that $|a| > 1$.

Example:

If a spring's natural oscillation is $f(t)$ , then $g(t) = 2f(t)$ causes a vertical stretch, making the oscillations twice as high.

Vertical Translations

Criticality: 3

Shifts a function's graph up or down by adding or subtracting a constant 'k' outside the function, i.e., $g(x) = f(x) + k$.

Example:

If a roller coaster's height is modeled by $f(x)$ , adding 50 feet to its design, $g(x) = f(x) + 50$ , represents a vertical translation upward.