What is the difference between $y = -f(x)$ and $y = f(-x)$ in terms of the axis of reflection?

$y = -f(x)$ is a reflection in the $x$-axis (vertical flip) because it negates the output values. $y = f(-x)$ is a reflection in the $y$-axis (horizontal flip) because it negates the input values.

How do the coordinates of a point $(a, b)$ change when the function is transformed to $y = -f(x)$?

The $x$-coordinate remains the same, while the $y$-coordinate changes sign. The new point is $(a, -b)$.

What happens to a point $(a, b)$ under the transformation $y = f(-x)$?

The $y$-coordinate remains the same, while the $x$-coordinate changes sign. The new point is $(-a, b)$.

What is an 'invariant point' in the context of a reflection?

An invariant point is a point on the graph that does not move during the transformation. This occurs when the point lies exactly on the axis of reflection (e.g., $x$-intercepts for $x$-axis reflections).

If a function has a vertical asymptote at $x = 3$, where is the asymptote after a reflection in the $y$-axis?

The asymptote moves to $x = -3$. This is because a $y$-axis reflection negates all $x$-values, including the position of vertical boundaries.

Why are the $x$-intercepts invariant during a reflection in the $x$-axis?

At an $x$-intercept, the $y$-value is $0$. Since the transformation $y = -f(x)$ negates the $y$-value, and $-0 = 0$, the coordinate remains unchanged.

What is a common error when reflecting a function like $f(x) = x^2 + 2x$ in the $y$-axis?

A common error is only negating the first $x$ term. You must replace *every* $x$ with $(-x)$, resulting in $f(-x) = (-x)^2 + 2(-x) = x^2 - 2x$.

Does a reflection in the $x$-axis change the domain or the range of a function?

It typically changes the range because the $y$-values are negated (e.g., $[0, \infty)$ becomes $(-\infty, 0]$). The domain usually remains the same as $x$-values are unaffected.

How does a reflection in the $y$-axis affect a horizontal asymptote $y = 5$?

It does not affect it. A $y$-axis reflection only changes $x$-coordinates; since a horizontal asymptote is defined by a constant $y$-value, its position remains $y = 5$.

When reflecting $y = f(x)$ to $y = -f(x)$, what happens to the local maximum points?

Local maximum points become local minimum points. Because the graph is flipped vertically, the highest points on the original curve become the lowest points on the reflected curve.

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Reflections

Reflections of Functions

Summary

Reflections are geometric transformations that flip a function's graph over a specific line of symmetry, known as the axis of reflection. In function notation, these are represented by a negative sign placed either outside the function, affecting vertical orientation, or inside the function, affecting horizontal orientation.

1. Definition & Core Concepts

A reflection is a transformation that maps every point of a graph to a new position on the opposite side of a fixed line, called the axis of reflection.

The resulting image is congruent to the original (pre-image) but has a reversed orientation, much like a mirror image.

In the context of coordinate geometry, reflections typically occur across the $x$ -axis or the $y$ -axis, altering the sign of one coordinate while leaving the other unchanged.

A coordinate plane showing a function f(x) in red and its reflection -f(x) in dashed blue across the x-axis.

2. Reflection in the x-axis: $y = -f(x)$

3. Reflection in the y-axis: $y = f(-x)$

4. Invariant Points

5. Key Distinctions

6. Exam Strategy & Tips

Reflections of Functions

Summary

1. Definition & Core Concepts

A reflection is a transformation that maps every point of a graph to a new position on the opposite side of a fixed line, called the axis of reflection.

The resulting image is congruent to the original (pre-image) but has a reversed orientation, much like a mirror image.

In the context of coordinate geometry, reflections typically occur across the $x$ -axis or the $y$ -axis, altering the sign of one coordinate while leaving the other unchanged.

A coordinate plane showing a function f(x) in red and its reflection -f(x) in dashed blue across the x-axis.

2. Reflection in the x-axis: $y = -f(x)$

When a negative sign is applied to the entire function ( $y = -f(x)$ ), the transformation is a vertical reflection across the $x$ -axis.

Every point $(x, y)$ on the original graph is mapped to the point $(x, -y)$ , meaning the $y$ -coordinates flip their signs while $x$ -coordinates remain constant.

This transformation effectively turns the graph 'upside down' relative to the horizontal axis.

3. Reflection in the y-axis: $y = f(-x)$

When the input variable is negated ( $y = f(-x)$ ), the transformation is a horizontal reflection across the $y$ -axis.

Every point $(x, y)$ on the original graph is mapped to the point $(-x, y)$ , meaning the $x$ -coordinates flip their signs while $y$ -coordinates remain constant.

This transformation creates a 'left-to-right' flip of the graph across the vertical axis.

4. Invariant Points

Invariant points are specific coordinates on a graph that do not change position after a transformation has been applied.

In an $x$ -axis reflection ( $y = -f(x)$ ), any point where the graph intersects the $x$ -axis (where $y=0$ ) is invariant because $-0 = 0$ .

In a $y$ -axis reflection ( $y = f(-x)$ ), any point where the graph intersects the $y$ -axis (where $x=0$ ) is invariant because $-0 = 0$ .

5. Key Distinctions

It is vital to distinguish between 'outside' and 'inside' negations to determine the direction of the flip.

Transformation	Notation	Axis of Reflection	Coordinate Change	Effect
Vertical Reflection	$y = -f(x)$	$x$ -axis	$(x, y) \to (x, -y)$	Upside down flip
Horizontal Reflection	$y = f(-x)$	$y$ -axis	$(x, y) \to (-x, y)$	Left-to-right flip

6. Exam Strategy & Tips

Check the Sign Placement: Always identify if the negative sign is affecting the output (vertical) or the input (horizontal) before sketching.
Track Key Points: When sketching, explicitly calculate the new coordinates for intercepts, turning points, and endpoints to ensure accuracy.
Asymptote Behavior: Remember that horizontal asymptotes ( $y=k$ ) are affected by $x$ -axis reflections, while vertical asymptotes ( $x=h$ ) are affected by $y$ -axis reflections.
Sanity Check: If you reflect a graph across the $x$ -axis, a point that was originally at $(2, 5)$ must end up at $(2, -5)$ . If your sketch doesn't show this, re-evaluate the axis of reflection.