Functions and Graphs
2.6 Function Transformations
Why Transformations Matter
Transformations let us reuse graphs we already know. Instead of graphing every new function from scratch, we start with a parent function and describe what changed: shift, reflect, stretch, or compress.
Vertical Shifts
A vertical shift changes output values. Every $y$-value moves by the same amount.
Definition: Vertical Shift
If $g(x)=f(x)+k$, then the graph of $g$ is the graph of $f$ shifted vertically.
- Up by $k$ units when $k>0$.
- Down by $|k|$ units when $k<0$.
Example: Shift a line vertically
Let $f(x)=2x-1$ and $g(x)=f(x)+3$. Find a formula for $g$ and describe the shift.
Show Solution
Substitute $f(x)$ into the definition:
$$g(x)=f(x)+3=(2x-1)+3=2x+2.$$
The graph is shifted up 3 units.
Example: Shift a parabola down
Let $f(x)=x^2$ and $g(x)=f(x)-4$. What happens to the vertex?
Show Solution
Since we subtract 4 from every output, the graph moves down by 4 units.
Original vertex of $f(x)=x^2$ is $(0,0)$, so new vertex is
$$(0,-4).$$
A matching formula is
$$g(x)=x^2-4.$$
MyOpenMath: Practice: vertical shifts (placeholder)
Horizontal Shifts
Horizontal shifts can feel backwards at first, so it helps to check what input produces the same output.
Definition: Horizontal Shift
If $g(x)=f(x-h)$, then the graph of $g$ is the graph of $f$ shifted horizontally.
- Right by $h$ units when $h>0$.
- Left by $|h|$ units when $h<0$.
Example: Shift a cubic left
Let $f(x)=x^3+1$ and $g(x)=(x+3)^3+1$. Describe the horizontal shift.
Show Solution
Rewrite in function notation:
$$g(x)=f(x+3)=f\big(x-(-3)\big).$$
So $h=-3$, which means the graph shifts left 3 units.
Example: Mixed horizontal and vertical shift
Given $f(x)=\sqrt{x}$, describe the transformation in
$$g(x)=\sqrt{x-3}+1.$$
Show Solution
The expression has two changes:
- $x-3$ inside the root: shift right 3.
- $+1$ outside: shift up 1.
So the graph of $g$ is $f$ shifted right 3 and up 1.
MyOpenMath: Practice: horizontal and combined shifts (placeholder)
Reflections
A reflection flips a graph across an axis. This is a fast way to build new models from familiar ones.
Definition: Reflections of Functions
For a function $f$:
- $-f(x)$ reflects $f(x)$ across the $x$-axis.
- $f(-x)$ reflects $f(x)$ across the $y$-axis.
Example: Reflect a square-root function
Let $s(t)=\sqrt{t}$. Write formulas for its vertical and horizontal reflections.
Show Solution
Vertical reflection across the $x$-axis:
$$y=-s(t)=-\sqrt{t}.$$
Horizontal reflection across the $y$-axis:
$$y=s(-t)=\sqrt{-t}.$$
(Notice the horizontal reflection changes the domain to $t\le 0$.)
MyOpenMath: Practice: reflections (placeholder)
Stretches and Compressions
These transformations change how wide or tall a graph appears.
Fact: Vertical and horizontal scale rules
If $a>0$:
- $af(x)$ scales outputs (vertical scale).
- $a>1$: vertical stretch
- $0<a<1$: vertical compression
- $f(ax)$ scales inputs (horizontal scale).
- $a>1$: horizontal compression
- $0<a<1$: horizontal stretch
Example: Vertical scale
Compare $f(x)=x^2$ and $g(x)=2x^2$.
Show Solution
Because $g(x)=2f(x)$, every output doubles. The parabola becomes narrower, which is a vertical stretch by factor 2.
Example: Horizontal scale
Compare $f(x)=x^2$ and
$$g(x)=\left(\tfrac{1}{2}x\right)^2=f\left(\tfrac{1}{2}x\right).$$
Show Solution
Here $a=\tfrac{1}{2}$ in $f(ax)$, so $0<a<1$. That means a horizontal stretch by factor $2$.
Equivalent simplification:
$$g(x)=\frac{x^2}{4},$$
which also looks wider than $x^2$.
MyOpenMath: Practice: stretches and compressions (placeholder)
At this point, you can read most transformed equations quickly by scanning inside first (horizontal changes), then outside (vertical changes).