Transformations — Shifts, Reflections, and Stretches

Function transformations describe how the graph of a base function changes when certain values are added, subtracted, or multiplied. The general form of a transformed function is:

f(x) \rightarrow a\,f(b(x - h)) + k

Each parameter affects the graph in a specific way:

\begin{array}{c|l} \text{Parameter} & \text{Effect on Graph} \\ \hline h & \text{Horizontal shift: } +h \text{ right, } -h \text{ left} \\ k & \text{Vertical shift: } +k \text{ up, } -k \text{ down} \\ a & \text{Vertical stretch/shrink: } |a|>1 \text{ stretches, } |a|<1 \text{ shrinks} \\ a<0 & \text{Reflection over the } x\text{-axis} \\ b & \text{Horizontal stretch/shrink or reflection if negative} \end{array}

The graph below shows a base function

f(x) = x^2

alongside two transformations of it:

f(x - 3)

, a horizontal shift, and

-f(x)

, a reflection over the

x

-axis. Seeing all three together makes it clear that a transformation moves or flips the whole shape without changing it.

SAT questions often present transformations in function notation, e.g. “If

g(x) = f(x - 3) + 2

, how does the graph of

f

move?”

f(x - 3)
-f(x)
f(x) = x²

Core Skills

Identify transformations from function equations.
Describe shifts, reflections, and stretches verbally or graphically.
Write new function equations after a given transformation.
Match equations to transformed graphs.
Understand the difference between horizontal and vertical changes.

Example 1: Horizontal and Vertical Shifts

Let

f(x) = x^2.

g(x) = f(x - 2) + 3 = (x - 2)^2 + 3

This means the parabola moves **right 2 units** and **up 3 units.** New vertex: from

(0,0)

(2,3)

Example 2: Reflection

f(x) = x^2 \quad \Rightarrow \quad g(x) = -f(x) = -x^2

The negative sign reflects the graph across the **x-axis** — opening downward.

Example 3: Vertical Stretch

f(x) = x^2, \quad g(x) = 3f(x) = 3x^2

The graph is stretched vertically by a factor of 3 — it becomes narrower.

Example 4: Horizontal Stretch

f(x) = x^2, \quad g(x) = f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^2 = \frac{1}{4}x^2

The graph is stretched horizontally by a factor of 2 — points move farther from the y-axis.

Example 5: Combined Transformation

f(x) = |x|

, find and describe:

g(x) = -2f(x + 1) + 3 = -2|x + 1| + 3

Transformations:

Shift left 1 (inside $x + 1$ )
Reflect across x-axis (negative sign)
Vertical stretch by 2
Shift up 3

Example 6: SAT Context Example

f(x)

represents the height of a ball over time, then

f(x - 2)

represents the same motion **2 seconds later**, and

f(x) + 3

means the ball starts **3 meters higher.**

Key Takeaways

$f(x - h)$ shifts right, $f(x + h)$ shifts left.
$f(x) + k$ moves up, $f(x) - k$ moves down.
Multiply by $a$ to stretch/shrink vertically; negative $a$ reflects.
Multiply inside argument by $b$ to stretch/shrink horizontally; negative $b$ reflects over y-axis.
Combine multiple transformations carefully, applying inside changes first.