Easy Algebra 2 - 2.6 function transformations

1. Formula
   Observation

   The handwritten notation uses $y=$ at the start of the parent-function example and again in the worked transformation equation.

2. Audio
   Observation

   The speaker says “whether that’s written as y equals or f of x,” linking $y$ to the dependent-variable notation.

y

Dependent variable or output value of a function.

Real-valued outputs are implied by the graphing context; no explicit domain is stated.

1. Formula
   Observation

   The board shows $f(x)$ beside the parent-function example and inside the transformed equation $y=f(x+4)-3$.

2. Audio
   Observation

   The speaker refers to “f of x” as an alternative way to write the function.

$f(x)$

Function notation for the value of the function $f$ at input $x$; here used first as the parent function and then as the base expression being transformed.

1. Formula
   Observation

   The parent-function example is written as $x^2$, and the horizontal shift is inserted into $f(x+4)$.

2. Audio
   Observation

   The speaker discusses moving left/right relative to the input position and writes the shift inside the parentheses with $x$.

x

Independent variable or input value of the function.

1. Formula
   Observation

   The board shows $y=f(x)=x^2$ as the example parent function.

2. Audio
   Observation

   The speaker says “let’s imagine we had something like x squared” and calls it the simplest version.

$x^2$

Quadratic parent-function rule used as the concrete example of a parent function.

1. Formula
   Observation

   The worked equation becomes $y=f(x+4)-3$ after the speaker explains the left shift.

2. Audio
   Observation

   The speaker says “if I want to move it left, I use plus.”

+4

Horizontal-shift term inserted inside the function argument to represent a translation 4 units left.

1. Formula
   Observation

   The final boxed equation is $y=f(x+4)-3$.

2. Audio
   Observation

   The speaker says “three down is just minus three.”

-3

Vertical-shift term appended outside the function to represent a translation 3 units down.

1. Formula
   Observation

   The board repeatedly shows the base relation written as $y = f(x)$.

$y = f(x)$

The original function relation before applying the listed transformations.

General function notation; no explicit domain is stated in the clip.

1. Formula
   Observation

   The variable x appears inside the transformed expressions such as $f(x+4)$, $f(x-5)$, and $f(-x-5)$.

2. Audio
   Observation

   The instructor says horizontal movement goes inside the parentheses and uses x with a number.

x

The input variable of the function, modified for horizontal translations and reflection across the y-axis.

Input variable in the function argument.

1. Formula
   Observation

   The function name f is used throughout the worked examples.

f

The given function being transformed.

Unspecified general function.

1. Formula
   Observation

   Part (A) text includes translate 4 units left, and the worked formula contains $x+4$.

2. Audio
   Observation

   The instructor refers to moving left and using the opposite sign inside the parentheses.

4

The horizontal shift amount in part (A).

Positive constant used in a translation.

1. Formula
   Observation

   Part (A) text includes 3 down, and the worked formula ends with -3.

2. Audio
   Observation

   The instructor discusses up/down adjustments as outside-the-function changes.

3

The vertical shift amount in part (A).

Positive constant used in a vertical translation.

1. Formula
   Observation

   Parts (B) and (C) both include translate 5 units right, and the worked formulas contain x-5.

2. Audio
   Observation

   The instructor says five units to the right and then uses the opposite sign inside the parentheses.

5

The horizontal shift amount in parts (B) and (C).

Positive constant used in a translation.

1. Audio
   Observation

   The speaker says, “most functions will have what is called a parent function,” and defines it as “the simplest form of a function.”

2. Formula
   Observation

   The board writes $y=f(x)=x^2$ as the example.

3. Diagram
   Observation

   A simple upward-opening parabola is sketched next to the formula.

A parent function is the simplest representative form of a family of functions. In this clip, the quadratic parent function is illustrated by $y=f(x)=x^2$, whose basic graph is a parabola.

1. Used as the starting function before transformations are applied.

2. The example shown is quadratic, but the spoken definition is general.

1. Audio
   Observation

   The speaker lists possible changes: shift left/right, up/down, reflect across the x- or y-axis, stretch vertically, or compress vertically.

2. Formula
   Observation

   The board text reads “Transformations = Shifts left, right, up, down / reflections / Vertical stretches + Compressions.”

The clip introduces transformations as operations applied to a parent function’s graph. The named categories are translations (left, right, up, down), reflections, and vertical stretches or compressions.

1. These are presented as general ways to modify the graph of a parent function.

2. Only horizontal and vertical translations are actually solved within this clip.

1. Parent function

1. Audio
   Observation

   The speaker explicitly warns that moving left is not represented by a negative sign outside the function; instead, “if I want to move it left, I use plus.”

2. Formula
   Observation

   The worked equation inserts the shift inside the argument as $f(x+4)$.

For a horizontal translation, the shift amount is placed inside the function argument with $x$. In this clip, translating 4 units left is encoded as $x+4$, so the transformed expression begins as $f(x+4)$.

1. Applies to horizontal shifts of the graph of $y=f(x)$.

2. The clip demonstrates only the left-shift case explicitly.

1. Parent function
2. Types of function transformations introduced

1. Audio
   Observation

   The speaker says “three down is just minus three.”

2. Formula
   Observation

   The final equation appends $-3$ outside the function: $y=f(x+4)-3$.

For a vertical translation, the shift amount is added or subtracted outside the function. In this clip, moving 3 units down is represented by appending $-3$, giving $f(x+4)-3$.

1. Applies to vertical shifts of the graph of $y=f(x)$.

2. The clip demonstrates the downward case explicitly.

1. Parent function
2. Types of function transformations introduced

1. Audio
   Observation

   The instructor explains that when moving left or right, the change goes inside the parentheses and the plus or minus is opposite of what you would expect.

2. Formula
   Observation

   Translate 4 units left is written as $f(x+4)$; translate 5 units right is written as $f(x-5)$.

For a horizontal shift, modify the input variable inside the function argument. A shift to the left by h units corresponds to replacing x by $x+h$, while a shift to the right by h units corresponds to replacing x by x-h. The instructor emphasizes that the sign inside the parentheses is opposite the intuitive direction word.

1. The transformation is applied to the input variable x.

2. The shift amount is a positive constant h.

1. $y = f(x)$
2. x

1. Audio
   Observation

   The instructor contrasts horizontal changes with up/down changes and treats vertical shifts as easier adjustments outside the function.

2. Formula
   Observation

   Part (A) shows 3 down as an appended -3 outside the function: $f(x+4)-3$.

For a vertical shift, add or subtract a constant outside the function. Moving down by k units subtracts k from the whole function value, while moving up by k units adds k. In the worked example, 3 down is represented by appending -3 after $f(x+4)$.

1. The transformation is applied after evaluating the function.

2. The shift amount is a positive constant k.

1. $y = f(x)$

1. Audio
   Observation

   The instructor says that if you are reflecting across the x-axis, something on top flips over to the bottom, so you put a negative sign in front of the entire function.

2. Formula
   Observation

   Part (B) changes from $f(x-5)$ to -f(x-5).

3. Diagram
   Observation

   An orange sketch shows a curve above a horizontal line flipping to below it.

Reflecting a graph across the x-axis changes the sign of every output value. Algebraically, this is done by placing a negative sign in front of the entire function, not inside the argument. The instructor visually describes this as the top of the graph flipping over to the bottom.

1. The reflection is across the x-axis.

2. The negative sign applies to the whole function value.

1. $y = f(x)$
2. -f($\cdots$)

1. Audio
   Observation

   The instructor says that when reflecting across the y-axis, you only make the x negative.

2. Formula
   Observation

   Part (C) changes from $f(x-5)$ to $f(-x-5)$.

Reflecting a graph across the y-axis changes the sign of the input variable only. Algebraically, this means replacing x by -x inside the function argument while leaving the outside of the function unchanged. In the worked example, the translated expression $f(x-5)$ becomes $f(-x-5)$.

1. The reflection is across the y-axis.

2. The negative sign is applied only to the input variable x.

1. $y = f(x)$
2. x
3. -x

1. Audio
   Observation

   The speaker states that when moving left, one should use plus rather than minus, calling this the key point that causes issues.

2. Formula
   Observation

   The worked solution writes $f(x+4)$ for a translation 4 units left.

To translate the graph of $y=f(x)$ four units left, replace $x$ by $x+4$, producing $y=f(x+4)$.

1. The starting graph has equation $y=f(x)$.

2. The intended transformation is a horizontal translation 4 units left.

For the specific example shown in the clip.

1. Audio
   Observation

   The speaker says “three down is just minus three.”

2. Formula
   Observation

   The final boxed equation is $y=f(x+4)-3$.

To translate the graph of $y=f(x)$ three units down, append $-3$ to the function, producing $y=f(x)-3$; combined with the left shift, the result is $y=f(x+4)-3$.

1. The starting graph has equation $y=f(x)$.

2. The intended transformation includes a vertical translation 3 units down.

For the specific example shown in the clip.

1. Audio
   Observation

   The instructor states that for left/right movement, the plus or minus inside the parentheses is opposite of what you would expect.

2. Formula
   Observation

   Left 4 becomes $x+4$, and right 5 becomes x-5 in the worked examples.

When translating a function horizontally, the sign inserted with x inside the parentheses is opposite the verbal direction: left uses + and right uses -.

1. The transformation is a horizontal translation of $f(x)$.

2. The shift amount is positive.

For the horizontal translations demonstrated in this clip.

1. Audio
   Observation

   The instructor says that reflecting across the x-axis means putting a negative sign in front of the entire function.

2. Formula
   Observation

   The worked result for part (B) is -f(x-5).

To reflect a graph across the x-axis, multiply the entire function by -1.

1. The reflection is across the x-axis.

For the function-transformation setting shown in the clip.

1. Audio
   Observation

   The instructor says that when reflecting across the y-axis, you only make the x negative.

2. Formula
   Observation

   The worked result for part (C) is $f(-x-5)$.

To reflect a graph across the y-axis, replace x by -x inside the function argument.

1. The reflection is across the y-axis.

For the function-transformation setting shown in the clip.

1. Audio
   Observation

   The speaker builds the answer step by step from the base idea $y=f(x)$, first handling the left shift and then the down shift.

2. Formula
   Observation

   The board shows the progression from $y=f(x)$ to $y=f(x+4)-3$.

1. Expression

   $$y=f(x)$$
   Explanation

   Start from the given parent-form equation for the untransformed graph.

   Justification

   This is the base function specified in the problem statement on the board.

   Shown in the video
2. Expression

   $$y=f(x+4)$$
   Explanation

   Apply the horizontal translation 4 units left by inserting $+4$ inside the function argument.

   Justification

   The speaker explicitly states that moving left uses plus inside the parentheses.

   Shown in the video
3. Expression

   $$y=f(x+4)-3$$
   Explanation

   Apply the vertical translation 3 units down by subtracting 3 outside the function.

   Justification

   The speaker states that “three down is just minus three.”

   Shown in the video

The transformed equation for translating $y=f(x)$ four units left and three units down is $y=f(x+4)-3$.

1. Formula
   Observation

   The board shows the prompt translate 5 units right, reflect across the x-axis, and the worked answer develops into -f(x-5).

2. Audio
   Observation

   The instructor narrates the right shift first and then the x-axis reflection.

1. Expression

   $$y=f(x)$$
   Explanation

   Start from the original function relation shown on the board.

   Justification

   Base function given in the problem setup.

   Shown in the video
2. Expression

   $$f(x-5)$$
   Explanation

   Apply the horizontal translation 5 units to the right by subtracting 5 from x inside the argument.

   Justification

   Horizontal translation rule: right by h gives x-h.

   Shown in the video
3. Expression

   $$-f(x-5)$$
   Explanation

   Apply the reflection across the x-axis by placing a negative sign in front of the whole function.

   Justification

   Reflection across the x-axis changes the sign of the output values.

   Shown in the video

The transformed equation for part (B) is $y=-f(x-5)$.

1. Formula
   Observation

   The board shows the prompt translate 5 units right, reflect across the y-axis, and the worked answer develops into $f(-x-5)$.

2. Audio
   Observation

   The instructor repeats the right-shift step and then says to make only the x negative for the y-axis reflection.

1. Expression

   $$y=f(x)$$
   Explanation

   Start again from the original function relation.

   Justification

   Base function given in the problem setup.

   Shown in the video
2. Expression

   $$f(x-5)$$
   Explanation

   Apply the horizontal translation 5 units to the right.

   Justification

   Horizontal translation rule: right by h gives x-h.

   Shown in the video
3. Expression

   $$f(-x-5)$$
   Explanation

   Apply the reflection across the y-axis by changing only the x term to -x inside the argument.

   Justification

   Reflection across the y-axis replaces x by -x.

   Shown in the video

The transformed equation for part (C) is $y=f(-x-5)$.

1. Caption evidence
   Observation

   The board text reads “(Ex) Write the equation for the transformation of the graph $y=f(x)$” and “(A) translate 4 units left, 3 down.”

2. Audio
   Observation

   The speaker reads the prompt aloud and solves part A.

3. Formula
   Observation

   The final handwritten answer is boxed as $y=f(x+4)-3$.

Write the equation for the transformation of the graph $y=f(x)$ when it is translated 4 units left and 3 units down.

1. Original graph: $y=f(x)$.

2. Transformation: 4 units left.

3. Transformation: 3 units down.

Find the equation of the transformed graph.

1. Expression

   $$y=f(x)$$
   Explanation

   Begin with the original function notation given in the problem.

   Justification

   This is the starting equation stated on the board.

   Shown in the video
2. Expression

   $$y=f(x+4)$$
   Explanation

   Encode the leftward horizontal shift by adding 4 inside the argument.

   Justification

   The speaker identifies this as the key rule: left means plus inside $f(\cdot)$.

   Shown in the video
3. Expression

   $$y=f(x+4)-3$$
   Explanation

   Encode the downward vertical shift by subtracting 3 outside the function.

   Justification

   The speaker directly states that 3 down corresponds to $-3$.

   Shown in the video

$y=f(x+4)-3$

The final equation matches the boxed handwritten result on the board.

1. Caption evidence
   Observation

   The board shows “(B) translate 5 units right, reflect across the x-axis.”

2. Audio
   Observation

   The speaker says “Let’s try the second one. Letter B. Translate…” but the clip ends before the solution is completed.

1. No worked steps or final equation for part B appear within the provided clip.

2. The exact intended answer is not shown in this segment.

Write the equation for the transformation of the graph $y=f(x)$ when it is translated 5 units right and reflected across the x-axis.

1. Original graph: $y=f(x)$.

2. Transformation: 5 units right.

3. Transformation: reflect across the x-axis.

Find the equation of the transformed graph.

1. Expression

   $$\text{(prompt shown only)}$$
   Explanation

   The problem statement is visible on the board, but no solution steps are completed in this clip.

   Justification

   The video reaches the end of the segment while introducing part B.

   Shown in the video

Not completed within this clip.

Only the prompt is present; no final answer is displayed before the clip ends.

1. Caption evidence
   Observation

   The board shows “(C) translate 5 units right, reflect across the y-axis.”

2. Audio
   Observation

   The speaker does not solve part C in this clip.

1. No worked steps or final equation for part C appear within the provided clip.

2. The exact intended answer is not shown in this segment.

Write the equation for the transformation of the graph $y=f(x)$ when it is translated 5 units right and reflected across the y-axis.

1. Original graph: $y=f(x)$.

2. Transformation: 5 units right.

3. Transformation: reflect across the y-axis.

Find the equation of the transformed graph.

1. Expression

   $$\text{(prompt shown only)}$$
   Explanation

   The problem statement is visible on the board, but no solution steps are completed in this clip.

   Justification

   The segment ends before part C is addressed.

   Shown in the video

Not completed within this clip.

Only the prompt is present; no final answer is displayed before the clip ends.

1. Formula
   Observation

   The board shows (A) translate 4 units left, 3 down and the completed answer $y=f(x+4)-3$.

2. Audio
   Observation

   The instructor discusses left/right movement going inside the parentheses and up/down movement as an outside adjustment.

Given $y=f(x)$, write the transformed equation for a translation 4 units left and 3 units down.

1. Original function: $y=f(x)$

2. Horizontal shift: 4 units left

3. Vertical shift: 3 units down

Produce the transformed formula.

1. Expression

   $$f(x+4)$$
   Explanation

   Move 4 units left by adding 4 inside the function argument.

   Justification

   Horizontal translation rule: left by h gives $x+h$.

   Shown in the video
2. Expression

   $$f(x+4)-3$$
   Explanation

   Move 3 units down by subtracting 3 outside the function.

   Justification

   Vertical translation rule: down by k gives -k outside the function.

   Shown in the video

$y=f(x+4)-3$

The final boxed expression on the board matches the stated transformations.

1. Formula
   Observation

   The board shows (B) translate 5 units right, reflect across the x-axis and the completed answer $y=-f(x-5)$.

2. Audio
   Observation

   The instructor explicitly says to go opposite for the right shift and then put a negative sign in front of the entire function for the x-axis reflection.

Given $y=f(x)$, write the transformed equation after translating 5 units right and reflecting across the x-axis.

1. Original function: $y=f(x)$

2. Horizontal shift: 5 units right

3. Reflection: across the x-axis

Produce the transformed formula.

1. Expression

   $$f(x-5)$$
   Explanation

   Translate 5 units right by subtracting 5 from x inside the argument.

   Justification

   Horizontal translation rule: right by h gives x-h.

   Shown in the video
2. Expression

   $$-f(x-5)$$
   Explanation

   Reflect across the x-axis by multiplying the whole function by -1.

   Justification

   Reflection across the x-axis changes the sign of all outputs.

   Shown in the video

$y=-f(x-5)$

The final boxed expression on the board is -f(x-5).

1. Formula
   Observation

   The board shows (C) translate 5 units right, reflect across the y-axis and the completed answer $y=f(-x-5)$.

2. Audio
   Observation

   The instructor says that for the y-axis reflection, only the x becomes negative.

Given $y=f(x)$, write the transformed equation after translating 5 units right and reflecting across the y-axis.

1. Original function: $y=f(x)$

2. Horizontal shift: 5 units right

3. Reflection: across the y-axis

Produce the transformed formula.

1. Expression

   $$f(x-5)$$
   Explanation

   Translate 5 units right by subtracting 5 from x inside the argument.

   Justification

   Horizontal translation rule: right by h gives x-h.

   Shown in the video
2. Expression

   $$f(-x-5)$$
   Explanation

   Reflect across the y-axis by replacing x with -x inside the argument.

   Justification

   Reflection across the y-axis changes the sign of the input variable only.

   Shown in the video

$y=f(-x-5)$

The final boxed expression on the board is $f(-x-5)$.

1. Diagram
   Observation

   The page header reads “2-6 Functions, Equations, and Graphs” and “FAMILIES OF FUNCTIONS.” Below it are handwritten notes: “Parent function” and “Transformations = Shifts left, right, up, down / reflections / Vertical stretches + Compressions.”

2. Animation
   Observation

   The speaker writes $y=f(x)=x^2$ next to “Parent function.”

1. Title text

2. Handwritten section heading

3. Parent-function note

4. Transformations list

1. The parent-function example $y=f(x)=x^2$ is added during the opening explanation.

1. The overall lesson topic remains families of functions and transformations.

The visual organization establishes that the lesson will define a parent function and then classify common transformations of its graph.

1. Diagram
   Observation

   A small upward-opening parabola is drawn beside $y=f(x)=x^2$.

2. Audio
   Observation

   The speaker says the example “would be a parabola, nice and simple.”

1. Parabola sketch

2. Formula $y=f(x)=x^2$

1. A simple U-shaped curve is added to visualize the parent function.

1. The sketch represents the untransformed parent graph only.

The drawing links the algebraic rule $x^2$ to its standard graphical shape, a parabola.

1. Animation
   Observation

   Under part (A), the equation is built progressively until the final boxed form $y=f(x+4)-3$ appears.

2. Diagram
   Observation

   The final answer is enclosed in a pink box.

1. Problem text for (A)

2. Handwritten equation

3. Pink box around the final answer

1. The equation develops from the base form to the fully transformed form $y=f(x+4)-3$.

1. The original prompt remains “translate 4 units left, 3 down.”

The visual writing sequence shows how each verbal transformation rule is encoded symbolically in the final equation.

1. Diagram
   Observation

   An orange horizontal line is drawn, then a curved shape above it, followed by a mirrored curved shape below it.

2. Audio
   Observation

   The instructor says that if something is on top, it would flip over and end up on the bottom because this is the x-axis.

1. orange horizontal line representing the x-axis

2. upper orange curve

3. lower orange curve

1. A curve above the line is drawn first.

2. A matching curve below the line is added to show the flipped position.

1. The horizontal line stays fixed as the mirror axis.

2. The sketch is used to represent sign change of output values.

The drawing visually supports the rule that reflection across the x-axis sends positive-height portions of the graph to negative-height portions, corresponding to multiplying the whole function by -1.

1. Diagram
   Observation

   Different parts of the board use blue, pink/magenta, orange, and green ink for prompts, intermediate terms, and final boxed answers.

2. Audio
   Observation

   The instructor mentions switching colors when discussing different transformation pieces.

1. Exact color assignment varies across moments and is not fully labeled on screen.

1. blue handwritten formulas

2. pink/magenta emphasis on vertical-shift discussion

3. orange emphasis for x-axis reflection work

4. green emphasis for y-axis reflection work

1. The board accumulates three worked examples from top to bottom.

2. Final answers are boxed in colored outlines.

1. The original prompts remain visible while the solutions are built underneath them.

The color coding helps separate horizontal-shift ideas, vertical-shift ideas, and the two different reflection rules.

1. Audio
   Observation

   The speaker says that when thinking of moving left from the origin, one naturally thinks of negative 4, “but that’s not what this is,” and then emphasizes that left uses plus inside the function.

Because moving left on a number line corresponds to negative values, students may expect a left translation to be written with a minus sign.

For horizontal translations of $y=f(x)$, a shift left is represented by adding inside the argument, as in $f(x+4)$, not by subtracting there.

1. Audio
   Observation

   The instructor explicitly warns that when thinking of right, one may think plus, but the needed sign is the opposite.

Learners may expect a rightward horizontal translation to be written with +x.

The clip states that horizontal movement goes inside the parentheses and the sign is opposite the intuitive direction: right uses x-h, left uses $x+h$.

1. Audio
   Observation

   The instructor distinguishes x-axis reflection from y-axis reflection by where the negative sign is placed.

2. Formula
   Observation

   Part (B) uses -f(x-5), while part (C) uses $f(-x-5)$.

Learners may confuse which reflection requires a negative outside the function and which requires a negative on x inside the function.

Reflection across the x-axis puts the negative in front of the entire function, while reflection across the y-axis puts the negative only on x inside the argument.

1. Audio
   Observation

   The speaker first defines the parent function and then immediately lists transformations that can be applied to it.

2. Diagram
   Observation

   The board places “Parent function” above the list of transformation types.

The notion of a parent function is needed before discussing transformations, because transformations are described as changes made to the graph of that simplest base function.

1. Audio
   Observation

   After listing transformations generally, the speaker narrows to the specific rule for horizontal shifts in Example A.

2. Formula
   Observation

   The general category “Shifts left, right” becomes the concrete expression $f(x+4)$ in the worked example.

The broad category of transformations includes horizontal translations, and the clip then specifies the rule for encoding a leftward shift inside the function argument.

1. Audio
   Observation

   The general list includes up/down shifts, and the worked example applies the down-shift rule explicitly.

2. Formula
   Observation

   The general category becomes the concrete appended term $-3$ in $f(x+4)-3$.

Vertical translations are one member of the transformation family introduced earlier, and the clip demonstrates the downward case by appending $-3$ outside the function.

1. Formula
   Observation

   Example A uses the rule directly by writing $f(x+4)$ for a left shift.

Example A applies the horizontal-shift rule to turn the verbal instruction “4 units left” into the symbolic term $x+4$ inside the function.

1. Formula
   Observation

   Example A uses the rule directly by appending $-3$ for a downward shift.

Example A applies the vertical-shift rule to turn the verbal instruction “3 down” into the symbolic term $-3$ outside the function.

1. Audio
   Observation

   The instructor contrasts inside-the-parentheses changes for left/right with outside changes for up/down.

2. Formula
   Observation

   Part (A) combines $f(x+4)$ with -3 outside.

Horizontal translations modify the input inside f( ), while vertical translations modify the output outside f( ).

1. Formula
   Observation

   The worked examples compare -f(x-5) and $f(-x-5)$.

2. Audio
   Observation

   The instructor explicitly says these are almost exactly the same except for where the negative sign goes.

Both are reflections, but the x-axis reflection negates the whole function value, whereas the y-axis reflection negates only the input variable.

1. Formula
   Observation

   Part (B) is built from a right shift followed by an x-axis reflection.

Example (B) applies the horizontal translation rule to obtain $f(x-5)$.

1. Formula
   Observation

   Part (B) ends as -f(x-5).

Example (B) applies the x-axis reflection rule by placing a negative sign in front of the whole function.

1. Formula
   Observation

   Part (C) is built from a right shift followed by a y-axis reflection.

Example (C) applies the horizontal translation rule to obtain $f(x-5)$.

1. Formula
   Observation

   Part (C) ends as $f(-x-5)$.

Example (C) applies the y-axis reflection rule by changing x to -x inside the argument.

1. Audio
   Observation

   The speaker defines parent function and gives $x^2$ as the example.

1. Parent function

1. Formula
   Observation

   The board lists shifts, reflections, and vertical stretches/compressions.

1. Types of function transformations introduced

1. Audio
   Observation

   The speaker explicitly contrasts the intuitive “negative 4” idea with the actual rule of using plus for a left shift.

1. Horizontal translation rule inside the function argument
2. Confusing a leftward shift with a minus sign

1. Formula
   Observation

   The worked answer for part A is shown as $y=f(x+4)-3$.

1. Example A: translate 4 units left, 3 down
2. Horizontal translation rule inside the function argument
3. Vertical translation rule outside the function

1. Caption evidence
   Observation

   Parts B and C are visible as prompts, but the clip ends before their solutions are written.

1. Example B prompt: translate 5 units right, reflect across the x-axis
2. Example C prompt: translate 5 units right, reflect across the y-axis

1. Audio
   Observation

   The instructor repeatedly says the sign inside the parentheses is opposite the direction word.

1. Horizontal translation rule for $f(x)$
2. Horizontal shifts use the opposite sign inside the argument
3. Example (B): translate 5 units right, reflect across the x-axis
4. Example (C): translate 5 units right, reflect across the y-axis

1. Audio
   Observation

   The instructor explains the top-to-bottom flip and places the negative in front of the whole function.

1. Reflection across the x-axis
2. Reflection across the x-axis places the negative sign outside the function
3. Example (B): translate 5 units right, reflect across the x-axis

1. Audio
   Observation

   The instructor says only the x becomes negative for reflection across the y-axis.

1. Reflection across the y-axis
2. Reflection across the y-axis places the negative sign on x
3. Example (C): translate 5 units right, reflect across the y-axis

1. Formula
   Observation

   The board shows combined transformations such as $f(x+4)-3$, -f(x-5), and $f(-x-5)$.

1. Example (A): translate 4 units left, 3 down
2. Example (B): translate 5 units right, reflect across the x-axis
3. Example (C): translate 5 units right, reflect across the y-axis
4. Horizontal translation rule for $f(x)$
5. Vertical translation rule for $f(x)$

1. Audio
   Observation

   The instructor notes that the last two problems are almost exactly the same except for the reflection type.

2. Formula
   Observation

   The final answers differ as -f(x-5) versus $f(-x-5)$.

1. Reflection across the x-axis
2. Reflection across the y-axis
3. Example (B): translate 5 units right, reflect across the x-axis
4. Example (C): translate 5 units right, reflect across the y-axis