Once you know the shape of a parent function, you rarely need to start from scratch to model new data.
A cooling drink, a tide, and a repeating signal each share a shape with a standard function, just shifted or stretched to fit the situation.
Transformations are the moves, slides, flips and stretches, that turn one known graph into the exact curve your data needs.
Learning the rules once lets you read a modified equation and picture its graph at a glance, and go back the other way just as fast.
Analogy
Think of the parent graph as a photo lying on a table.
You can slide it, flip it over, or stretch it like elastic without ever redrawing it.
Each of those physical moves is one of the transformations in this section.
Translations Slide The Graph Without Changing Its Shape
Vertical Translation f(x) + b
Adding a constant to the whole function, $f(x)+b$, raises every output by $b$.
Every point $(x,\,f(x))$ becomes $(x,\,f(x)+b)$, so the graph slides up by $b$ (or down when $b<0$).
The input $x$ is untouched, so nothing moves left or right.
Horizontal Translation f(x − a)
Replacing $x$ with $x-a$, giving $f(x-a)$, slides the graph horizontally.
Here is why the direction feels backwards:
the value $f(x-a)$ repeats whatever $f$ did $a$ units of input earlier,
so every feature appears $a$ units later, which moves the graph right by $a$ when $a>0$.
Common Mistake
Horizontal shifts go opposite to the sign inside the bracket: $f(x-a)$ moves right for positive $a$.
Vertical shifts follow the sign as written: $f(x)+b$ moves up for positive $b$.
Combining The Two
A vertical and a horizontal translation act on different axes, so they never interfere and can be done in either order.
Example question
The graph of $y=f(x)$ is transformed to $y=f(x-3)+2$. Describe the transformation and give the image of the point $(1,\,4)$.
Solution
Inside the bracket: $x-3$ shifts the graph right by 3.
Outside: $+2$ shifts it up by 2.
The point $(1,\,4)$ moves to $(1+3,\,4+2)=(4,\,6)$.
Reflections Flip The Graph Over An Axis
Reflection In The x-axis: −f(x)
Negatingthe whole output, $-f(x)$, flips the graph over the $x$-axis.
Each point $(x,\,f(x))$ becomes $(x,\,-f(x))$, so positive outputs become negative and vice versa, while points on the $x$-axis stay put.
Reflection In The y-axis: f(−x)
Negating the input, $f(-x)$, flips the graph over the $y$-axis.
The output that used to sit at $-x$ now appears at $x$, so left and right swap, while points on the $y$-axis stay put.
Example question
Given $f(x)=x^2+2x$, write $-f(x)$ and $f(-x)$ in expanded form.
Solution
$$-f(x) = -(x^2+2x) = -x^2-2x$$
$$f(-x) = (-x)^2 + 2(-x) = x^2-2x$$
They are different, because reflecting in the $x$-axis and in the $y$-axis are different moves.
Note
Negating the output flips vertically ($x$-axis); negating the input flips horizontally ($y$-axis).
The same reflections and stretches appear as matrix transformations of points in AHL 3.9.
Stretches Scale The Graph
Vertical Stretch p·f(x)
Multiplying the whole function by $p>0$, written $p\,f(x)$, stretches the graph vertically by scale factor $p$.
Each point $(x,\,f(x))$ becomes $(x,\,p\,f(x))$, so distances from the $x$-axis multiply by $p$ while points on the $x$-axis do not move.
If $p>1$ the graph stretches taller; if $0<p<1$ it is compressed toward the $x$-axis.
Horizontal Stretch f(qx)
Multiplying the input by $q>0$, written $f(qx)$, stretches the graph horizontally by scale factor $\dfrac{1}{q}$.
Here is why the factor is the reciprocal:
the transformed graph reaches at input $x$ the value $f$ had at $qx$,
so a feature at input $u$ now sits at $x=\dfrac{u}{q}$, meaning distances from the $y$-axis multiply by $\dfrac{1}{q}$.
So $q>1$ squeezes the graph toward the $y$-axis, and $0<q<1$ stretches it away.
Common Mistake
A horizontal stretch $f(qx)$ has scale factor $\dfrac{1}{q}$, not $q$.
This reciprocal is the single most-missed detail in transformation questions, so commit it to memory.
Example question
Describe the transformation from $y=f(x)$ to $y=f(2x)$, and give the image of the point $(6,\,5)$.
Solution
The input is multiplied by $2$, a horizontal stretch of scale factor $\tfrac{1}{2}$ (a squeeze toward the $y$-axis).
Outputs are unchanged, so the $y$-coordinate stays the same.
The point $(6,\,5)$ moves to $\left(\tfrac{6}{2},\,5\right)=(3,\,5)$.
Composite Transformations: Order Can Change The Result
Real models usually combine several transformations, and when two act in the same direction the order matters.
A safe reading: vertical transformations act on the whole output in the order written from the inside out, and horizontal transformations must be untangled by factoring the input first.
Example
For $y=2f(x)+3$, stretch vertically by $2$ first, then translate up $3$; doing it the other way gives a different graph.
For $y=f(2x-4)$, factor the bracket as $f\big(2(x-2)\big)$: read a horizontal stretch of factor $\tfrac{1}{2}$ together with a translation right $2$.
Example question
Describe the sequence of transformations taking $y=f(x)$ to $y=3f(x-2)-1$, and give the image of a general point $(a,\,b)$ on $f$.
Solution
Inside the bracket $x-2$: translate right by 2.
Factor of $3$ outside: vertical stretch, scale factor 3.
Then $-1$: translate down by 1.
The point $(a,\,b)$ maps to $(a+2,\,3b-1)$.
Exam technique
The formula booklet contains no transformation rules, so all six must be memorised.
Vertical transformations act on the outputs and follow the sign as written; horizontal transformations act on the inputs and behave in the opposite way.
Always factor a coefficient out of the bracket, $f(qx-d)=f\big(q(x-\tfrac{d}{q})\big)$, before reading off a horizontal shift.
Check a combined transformation by tracking one clear point, such as a vertex or intercept, through each step, or by graphing on your GDC.
All The Transformations At A Glance
Every transformation in this subtopic either acts on the output (vertical, as written) or on the input (horizontal, in reverse).
Transformation
Equation
Effect
Vertical translation
f(x) + b
up b (down if b < 0)
Horizontal translation
f(x − a)
right a (left if a < 0)
Reflection in x-axis
−f(x)
flip vertically
Reflection in y-axis
f(−x)
flip horizontally
Vertical stretch
p·f(x)
scale factor p from x-axis
Horizontal stretch
f(qx)
scale factor 1/q from y-axis
Active recall
What is the difference in effect between $f(x)+b$ and $f(x-a)$?
Why does $f(x-a)$ move the graph to the right when $a>0$?
State the effect of $-f(x)$ and of $f(-x)$.
A horizontal stretch $f(qx)$ has what scale factor, and why is it the reciprocal of $q$?
For $y=2f(x)+3$, which transformation is applied first, and why does the order matter here?
