Reflections Are Mirror Transformations Of Graphs
- A reflection is often the quickest way to generate a new graph because you do not need to recalculate lots of points, you "mirror" what you already know.
- In coordinate geometry, the "mirror line" is an axis.
- Reflecting in an axis changes the sign of one coordinate while keeping the other the same.
Analogy
- Think of the x-axis as the surface of a perfectly still lake.
- The graph above the lake has an identical "upside-down" image below the surface.
- That is what reflecting in the x-axis does.
Reflecting In The X-Axis Changes The Sign Of All y-Values
- When you reflect a point $(x,y)$ in the x-axis, its horizontal position does not change, but its vertical position flips. $$ (x,y) \mapsto (x,-y) $$
- So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=-f(x) $$
- This matches an important generalization:
The graph of $y=-f(x)$ is a reflection of the graph of $y=f(x)$ in the x-axis.
How To Sketch $y=-f(x)$ From A Known Graph
- Keep each $x$-coordinate the same.
- Multiply each $y$-value by $-1$.
- Plot the mirrored points and join with the same shape.
Example
- Suppose $f(x)=3x+2$.
- Reflecting in the x-axis gives $$y=-f(x)=-(3x+2)=-3x-2$$
- Check with intercepts: the original line crosses the y-axis at $(0,2)$, the reflection crosses at $(0,-2)$.
Common Mistake
- A reflection in the x-axis is not a left-right flip.
- Students often confuse $-f(x)$ with $f(-x)$.
- The negative sign outside the function changes y-values.
Example
- If $f(x)=(x-2)^2+3$, then $$-f(x)=-(x-2)^2-3.$$
- Notice what happens to key features:
- The vertex $(2,3)$ becomes $(2,-3)$.
- A parabola that opened upward now opens downward.
Reflecting In The Y-Axis Changes The Sign Of All x-Values
- When you reflect a point $(x,y)$ in the y-axis, its vertical position does not change, but its horizontal position flips. $$ (x,y) \mapsto (-x,y) $$
- So, for a whole function $y=f(x)$, the reflected graph has equation $$ y=f(-x) $$
- This matches another key generalization:
The graph of $y=f(-x)$ is a reflection of the graph of $y=f(x)$ in the y-axis.
How To Sketch $y=f(-x)$
- Keep each $y$-coordinate the same.
- Multiply each $x$-value by $-1$.
- Plot and join.
Example
- Let $f(x)=3x+2$.
- Then $$f(-x)=3(-x)+2=-3x+2$$
- The y-intercept stays $(0,2)$ because points on the y-axis do not move when reflecting in the y-axis.
Tip
To decide quickly:
- $y=-f(x)$ keeps $x$ the same and flips up/down.
- $y=f(-x)$ keeps $y$ the same and flips left/right.
Example
- Starting with $f(x)=(x-2)^2+3$, reflect in the y-axis: $$f(-x)=(-x-2)^2+3$$
- Because $(-x-2)^2=(x+2)^2$, we get $$f(-x)=(x+2)^2+3$$
- Geometrically, the vertex $(2,3)$ moves to $(-2,3)$.
How Reflections Affect Key Graph Features
Reflections are easiest when you track features (vertex, intercepts, endpoints) rather than lots of random points.
Intercepts
- x-intercepts (where $y=0$):
- For $y=-f(x)$, the x-intercepts stay the same (because $-0=0$).
- For $y=f(-x)$, the x-intercepts change sign: if $(a,0)$ is an intercept, it becomes $(-a,0)$.
- y-intercept (where $x=0$):
- For $y=-f(x)$, the y-intercept changes sign.
- For $y=f(-x)$, the y-intercept stays the same.
Domain And Range
- Reflecting in the x-axis keeps the domain the same, but changes the range by multiplying by $-1$.
- Reflecting in the y-axis keeps the range the same, but changes the domain by multiplying by $-1$.
Note
- When a function has a restricted domain (for example, a square-root curve), a y-axis reflection can force the domain to become negative.
- Always check the allowed x-values after replacing $x$ with $-x$.
Combining Reflections With Other Transformations
In practice, you often meet expressions like $$y=a\,f(bx+c)+d$$
where reflections can be "hidden" inside constants.
Reflection With A Vertical Dilation
Multiplying by $a$ changes y-values.
- If $a<0$, you get a reflection in the x-axis.
- If $|a|>1$, there is also a vertical stretch (dilation).
- If $0<|a|<1$, there is also a vertical compression.
Example
$y=-2f(x)$ is a reflection in the x-axis and a vertical dilation by scale factor $2$.
Reflection Inside The Input
- Replacing $x$ with $-x$ reflects in the y-axis. More generally, expressions like $$f(-x+5)$$ combine transformations.
- A helpful rewrite is to factor out the negative: $$-x+5=-(x-5)$$
- So $f(-x+5)=f(-(x-5))$ suggests:
- translate right by $5$ to get $f(x-5)$
- reflect in the y-axis (because of the negative)
Exam technique
- When the negative is inside the brackets, rewrite it by factoring: $$-x+5=-(x-5)$$
- This makes the reflection and translation easier to spot.
Same Graph, Different Transformations
Sometimes two different-looking transformations produce the same resulting graph.
Example
If $f(x)=2x^2$:
- Vertical dilation: $g(x)=4f(x)=8x^2$.
- Horizontal dilation: $h(x)=f(2x)=2(2x)^2=8x^2$.
So $g(x)=h(x)$, meaning the graphs are identical even though the transformation descriptions are different.
Common Mistake
Translation or reflection?
Consider $y=(x-2)^2$ and $y=(x+2)^2$.
- Each is a translation of $y=x^2$ (right by 2 and left by 2).
- They are also reflections of each other in the y-axis, because if $f(x)=(x-2)^2$ then $$f(-x)=(-x-2)^2=(x+2)^2$$
So two students can both be correct, depending on what they take as the "original" graph.
Self review
- What is the difference between $y=-f(x)$ and $y=f(-x)$?
- A point $(4,-1)$ is reflected in the y-axis. What are its new coordinates?
- If a graph crosses the x-axis at $x=3$, where will the reflected graph $y=f(-x)$ cross?
