Topic 1.12 Notes – Transformations of Functions

What Function Transformations Do

A transformation makes a new function $g$ from an original function $f$. Sometimes $f$ is a parent function like $x^2$ or $|x|$, but the rules work for any original function.

• The preimage is the set of original points on $f$.
• The image is the set of transformed points on $g$.

The rule that organizes everything is this:

• Outside the function changes outputs $\to$ vertical changes
• Inside the function changes inputs $\to$ horizontal changes

That is why horizontal changes feel backward. You are changing the input needed to get the same output.

The standard combined form is

$$g(x)=a\,f(b(x-H))+K$$

Here is what each part does:

• $a$ changes outputs
  • vertical stretch/compression by $|a|$
  • reflect over the $x$-axis if $a<0$
• $b$ changes inputs
  • horizontal scale factor is $\frac{1}{b}$
  • reflect over the $y$-axis if $b<0$
• $H$ shifts right $H$
• $K$ shifts up $K$

This graph gives you a quick preview of those ideas using the parent function $f(x)=x^2$. You can see an outside change move the graph up or stretch it vertically, and an inside change shift it horizontally.

Function transformations of $f(x)=x^2$

The Four Basic Transformations

Vertical translation

$$g(x)=f(x)+k$$

• $k>0$ moves the graph up
• $k<0$ moves it down
• Point mapping is $(x,y)\to(x,y+k)$

The shape stays the same. The range shifts because outputs changed.

Horizontal translation

$$g(x)=f(x+h) \quad \text{or} \quad g(x)=f(x-H)$$

This is the famous opposite-sign rule:

• $f(x+h)$ moves left $h$
• $f(x-H)$ moves right $H$

Point mapping:

• $(x,y)\to(x-h,y)$
• or $(x,y)\to(x+H,y)$

The domain shifts because inputs changed.

Vertical dilation and reflection

$$g(x)=af(x), \quad a\ne 0$$

• $|a|>1$ gives a vertical stretch
• $0<|a|<1$ gives a vertical compression
• $a<0$ reflects over the $x$-axis

Point mapping is $(x,y)\to(x,ay)$.

Horizontal dilation and reflection

$$g(x)=f(bx), \quad b\ne 0$$

This is where students lose points. The horizontal scale factor is $\frac{1}{b}$, not $b$.

• $|b|>1$ gives horizontal compression
• $0<|b|<1$ gives horizontal stretch
• $b<0$ reflects over the $y$-axis

Point mapping is $\left(x,y\right)\to\left(\frac{x}{b},y\right)$.

Building and Reading Combined Transformations

From words to equation, build horizontal changes inside and vertical changes outside:

1. Identify the original $f$
2. Write the inside as $b(x-H)$
3. Write the outside as $a(\ )+K$
4. Combine them

So the finished form is

$$g(x)=a f(b(x-H))+K$$

From equation to description, always factor the entire inside first.

• $f(2x+6)=f(2(x+3))$
  horizontal compression by $\frac12$, left 3
• $f(-4x+8)=f(-4(x-2))$
  horizontal compression by $\frac14$, reflection over $y$-axis, right 2

For any point $(u,v)$ on $f$, the full mapping is

$$(u,v)\to\left(H+\frac{u}{b},\,K+av\right)$$

This works for graphs, tables, intercepts, extrema, endpoints, and holes.

If a table has $u\mapsto v$, then the transformed row is

$$H+\frac{u}{b}\mapsto K+av$$

You must transform the input too. That is a common test trap.

Domain, Range, and Key Features After a Transformation

For $g(x)=a f(b(x-H))+K$:

• domain values move by $x=H+\frac{u}{b}$
• range values move by $y=K+av$

So:

• horizontal changes affect domain
• vertical changes affect range

Features that transform directly:

• points, endpoints, turning points, holes
  use the full point mapping
• vertical asymptote $x=c$ becomes $x=H+\frac{c}{b}$
• horizontal asymptote $y=L$ becomes $y=K+aL$

Two important cautions:

• Zeros only stay zeros when $K=0$
• The $y$-intercept comes from $g(0)=a f(-bH)+K$, not from transforming the old $y$-intercept

Common Mistakes and Exam Traps

• $f(x+4)$ means left 4
• $f(3x)$ means compression by $\frac13$
• Never read $bx+c$ term by term. Factor first.
• In tables, do not keep the same input and only change the output.
• Domain, range, and zeros do change under some transformations.
• Symmetry can hide reflections. For even or odd functions, the graph may look unchanged even when a reflection happened.