International Baccalaureate · DP 1: Analysis and Approaches SL

Transformations of Functions

⌚ ~59 min ▢ 7 blocks ✓ 36 exercises ➣ Prerequisites : Quadratic Functions, Functions

I Translations, Stretches and Reflections

I.1 Translations

Recappreviously studied — click to reread

Function transformations allow us to take a basic “parent” function and modify it to create a new, related function. By applying a sequence of transformations, we can shift, stretch, compress, or reflect the graph of the parent function.
Understanding these transformations is essential, as it allows us to predict the graph of a complex function based on a simpler one and to model real-world phenomena by adjusting a basic function to fit observed data. The transformations are vertical and horizontal translations, stretches and compressions, and reflections.

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Definition — Vertical Translation by Adding a Constant

A vertical translation shifts the graph of a function up or down. The transformation is defined by:$$g(x) = f(x) + k$$ This transformation maps a point $(x, y)$ on the graph of $f$ to a new point $(x, y+k)$ on the graph of $g$.

If $k>0$, the graph is translated $\boldsymbol{k}$ units up.
If $k<0$, the graph is translated $\boldsymbol{|k|}$ units down.

Definition — Horizontal Translation

A horizontal translation shifts the graph of a function left or right. The transformation is defined by:$$g(x) = f(x-h)$$ This transformation maps a point $(x, y)$ on the graph of $f$ to a new point $(x+h, y)$ on the graph of $g$.

If $h>0$, the graph is translated $\boldsymbol{h}$ units to the right.
If $h<0$, the graph is translated $\boldsymbol{|h|}$ units to the left.

Example

From the graph of $f(x) = x^2$, plot $g(x) = f(x) + 2$ and $h(x) = f(x - 3)$.

$g(x) = f(x) + 2$: every output is raised by $2$ — a vertical translation $2$ units up. The vertex $(0, 0)$ moves to $(0, 2)$.
$h(x) = f(x - 3)$: the input $x = 3$ now plays the role $0$ used to play — a horizontal translation $3$ units to the right. The vertex $(0,0)$ moves to $(3, 0)$.

Translation by a Vector

The vertical and the horizontal translations combine into one. The translation by the vector $\begin{pmatrix} h \\ k \end{pmatrix}$ moves each point $(x, y)$ of the graph of $f$ to the point $(X, Y) = (x + h, y + k)$. Since $x = X - h$ and $Y = f(x) + k$, the image points satisfy $Y = f(X - h) + k$; writing $(x, y)$ again for the coordinates, the image of the graph of $y = f(x)$ is the graph of $$y = f(x - h) + k.$$ For example, the translation by the vector $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ maps the graph of $y = x^2$ onto the graph of $y = (x - 3)^2 - 2$, and its vertex $(0, 0)$ to $(3, -2)$.

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Skills to practice

Translating Graphs

I.2 Stretches

Recappreviously studied — click to reread

Definition — Vertical Stretch

A vertical stretch stretches or compresses the graph of a function vertically. It is defined, for a number $a \neq 0$, by:$$g(x) = a \cdot f(x)$$ This transformation maps a point $(x, y)$ to $(x, ay)$.

If $|a|>1$, the graph is stretched vertically by a factor of $|a|$.
If $0 < |a| < 1$, the graph is compressed vertically by a factor of $|a|$.
If $a < 0$, the stretch or compression is followed by a reflection in the $x$-axis ($a = -1$ gives the reflection alone).

Definition — Horizontal Stretch

A horizontal stretch stretches or compresses the graph of a function horizontally. It is defined, for a number $b \neq 0$, by:$$g(x) = f(bx)$$ This transformation maps a point $(x, y)$ to a new point $(\frac{x}{b}, y)$.

If $|b|>1$, the graph is compressed horizontally by a factor of $\frac{1}{|b|}$.
If $0 < |b| < 1$, the graph is stretched horizontally by a factor of $\frac{1}{|b|}$.
If $b < 0$, the stretch or compression is followed by a reflection in the $y$-axis ($b = -1$ gives the reflection alone).

Example

From the graph of $f(x) = x^2$, plot $a(x) = 2f(x)$ and $b(x) = f(2x)$.

$a(x) = 2f(x) = 2x^2$: every output is doubled — a vertical stretch by factor $2$. The point $(1, 1)$ moves to $(1, 2)$.
$b(x) = f(2x) = 4x^2$: the input is consumed twice as fast — a horizontal compression with scale factor $\frac12$: every $x$-coordinate is halved. The point $(1, 1)$ moves to $\left(\frac12, 1\right)$.

The two results differ: doubling the output is not the same as doubling the input.

Stretches and Scale Factors

The number by which a stretch multiplies the distances to the axis that stays fixed is its scale factor. For $a > 0$, the graph of $y = a f(x)$ is the image of the graph of $y = f(x)$ under the vertical stretch with scale factor $a$: each point $(x, y)$ goes to $(x, ay)$. For $b > 0$, the graph of $y = f(bx)$ is its image under the horizontal stretch with scale factor $\dfrac{1}{b}$: each point $(x, y)$ goes to $\left(\dfrac{x}{b}, y\right)$. A scale factor greater than $1$ moves the points away from the axis that stays fixed (the $x$-axis for a vertical stretch, the $y$-axis for a horizontal one), a scale factor between $0$ and $1$ brings them closer to it: the graph of $y = f(2x)$ is the image of the graph of $f$ under the horizontal stretch with scale factor $\dfrac{1}{2}$, a compression.

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Skills to practice

Stretching Graphs

I.3 Reflections

Recappreviously studied — click to reread

Definition — Reflection in the x-axis

A reflection in the x-axis flips the graph of a function vertically. It is a special case of vertical stretch where $a=-1$. It is defined by:$$g(x) = -f(x)$$ This transformation maps a point $(x, y)$ to $(x, -y)$.

Definition — Reflection in the y-axis

A reflection in the y-axis flips the graph of a function horizontally. It is a special case of horizontal stretch where $b=-1$. It is defined by:$$g(x) = f(-x)$$ This transformation maps a point $(x, y)$ to $(-x, y)$.

Example

From the graph of $f(x) = \sqrt{x}$, plot $r(x) = -f(x)$ and $s(x) = f(-x)$.

$r(x) = -\sqrt{x}$: every output changes sign — a reflection in the $x$-axis. The curve flips below the axis; the point $(4, 2)$ moves to $(4, -2)$.
$s(x) = \sqrt{-x}$: every input changes sign — a reflection in the $y$-axis. The curve flips to the left half-plane (defined for $x \le 0$); the point $(4, 2)$ moves to $(-4, 2)$.

Skills to practice

Reflecting Graphs

II Composite Transformations

II.1 Combining Transformations

Recappreviously studied — click to reread

When multiple transformations are applied to a function, the order in which they are performed is crucial. The standard form for a transformed function is:$$ g(x) = a \cdot f(b(x-c)) + d $$ To graph this function from the parent function $f(x)$, apply the transformations in the following order:

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Method — Order of Transformations

Horizontal Transformations (inside the brackets):
- Apply the horizontal stretch/compression by the factor $\frac{1}{|b|}$ ($b \neq 0$), followed by a reflection in the $y$-axis if $b < 0$.
- Apply the horizontal translation (shift) by $c$ units.
Vertical Transformations (outside the brackets):
- Apply the vertical stretch/compression by the factor $|a|$, followed by a reflection in the $x$-axis if $a < 0$.
- Apply the vertical translation (shift) by $d$ units.

Note: Always take out the coefficient $b$ as a factor of the term inside the function to correctly identify the horizontal shift $c$. For example, transform $f(2x-6)$ as $f(2(x-3))$. This shows a compression by $\frac{1}{2}$ followed by a shift of 3 units right, not 6.

Example

Describe the sequence of transformations that maps the graph of $f(x)=\sqrt{x}$ onto the graph of $g(x) = 3\sqrt{-x+2} - 4$.

Answer

First, we rewrite the function $g(x)$ in the standard form $a \cdot f(b(x-c)) + d$:$$ g(x) = 3\sqrt{-(x-2)} - 4 $$ Comparing this to $f(x)=\sqrt{x}$, we have $a=3$, $b=-1$, $c=2$, and $d=-4$. The transformations are applied as follows:

Horizontal Transformations on $f(x)=\sqrt{x}$:
- Reflection: Since $b=-1$, reflect the graph in the y-axis ($y = \sqrt{-x}$).
- Translation: Since $c=2$, translate the graph 2 units to the right ($y = \sqrt{-(x-2)}$).
Vertical Transformations on the result:
- Stretch: Since $a=3$, stretch the graph vertically by a factor of 3 ($y = 3\sqrt{-(x-2)}$).
- Translation: Since $d=-4$, translate the graph 4 units down ($y = 3\sqrt{-(x-2)} - 4$).

Skills to practice

Sketching Combined Transformations
Finding Equations from a Sequence of Transformations

II.2 Rational Functions as Images of $y \equal \frac{1}{x}$

Example

Describe a sequence of transformations that maps the graph of $y = \dfrac{1}{x}$ onto the graph of $y = \dfrac{2x + 1}{x - 1}$, and deduce the asymptotes of the second graph.

Answer

Write the numerator around the denominator: $2x + 1 = 2(x - 1) + 3$. With $f(x) = \dfrac{1}{x}$, for $x \neq 1$: $$\begin{aligned}[t] \frac{2x + 1}{x - 1} &= \frac{2(x - 1) + 3}{x - 1} \\ &= 2 + \frac{3}{x - 1} && \text{(split the fraction)} \\ &= 3f(x - 1) + 2. \end{aligned}$$ This is the standard form $a \cdot f(b(x - c)) + d$ with $a = 3$, $b = 1$, $c = 1$ and $d = 2$. In the order of transformations:

translate $1$ unit to the right: $y = \dfrac{1}{x - 1}$;
stretch vertically with scale factor $3$: $y = \dfrac{3}{x - 1}$;
translate $2$ units up: $y = \dfrac{3}{x - 1} + 2$.

The asymptotes $x = 0$ and $y = 0$ of $y = \dfrac{1}{x}$ follow the graph: the translation $1$ unit to the right moves them to $x = 1$ and $y = 0$; the vertical stretch leaves these two lines in place; the translation $2$ units up moves $y = 0$ to $y = 2$. The graph of $y = \dfrac{2x + 1}{x - 1}$ has the asymptotes $x = 1$ and $y = 2$. The point $(1, 1)$ of $y = \dfrac{1}{x}$ goes to $(2, 1)$, then $(2, 3)$, then $(2, 5)$; check: $\dfrac{2 \times 2 + 1}{2 - 1} = 5$.

Skills to practice

Relating Rational Functions to $y \equal \frac{1}{x}$

III Going further

Skills to practice

Reasoning and Solving Problems