Transformations (Comprehensive)

Transformations let you build new functions from known ones by shifting, stretching, compressing, and reflecting. In Algebra 2, you learned the basics. Here we formalize the rules, handle multiple transformations at once, and develop the skill of writing equations from transformed graphs — all without a calculator.

The Master Transformation Formula

Every transformation of a parent function $f(x)$ can be written in the form:

$y = a \cdot f(b(x - h)) + k$

Each parameter controls a specific change:

Important sign conventions:

• $h > 0$ shifts right (the minus sign in the formula already handles it)
• $k > 0$ shifts up
• $a < 0$ reflects over the $x$-axis
• $b < 0$ reflects over the $y$-axis

Individual Transformations in Detail

Vertical Shifts: $y = f(x) + k$

Adding $k$ to the output shifts the graph vertically.

• $k > 0$: shift up $k$ units
• $k < 0$: shift down $|k|$ units

Every point $(x, y)$ on $f$ becomes $(x, y + k)$.

Horizontal Shifts: $y = f(x - h)$

Replacing $x$ with $(x - h)$ shifts the graph horizontally.

• $h > 0$: shift right $h$ units
• $h < 0$: shift left $|h|$ units

Every point $(x, y)$ on $f$ becomes $(x + h, y)$. The shift is opposite the sign inside the function.

Vertical Stretch and Compression: $y = a \cdot f(x)$

Multiplying the output by $a$ scales vertically.

• $|a| > 1$: vertical stretch by factor $|a|$
• $0 < |a| < 1$: vertical compression by factor $|a|$

Every point $(x, y)$ on $f$ becomes $(x, ay)$.

Horizontal Stretch and Compression: $y = f(bx)$

Multiplying the input by $b$ scales horizontally.

• $|b| > 1$: horizontal compression by factor $\frac{1}{|b|}$
• $0 < |b| < 1$: horizontal stretch by factor $\frac{1}{|b|}$

Every point $(x, y)$ on $f$ becomes $\left(\frac{x}{b}, y\right)$. The effect is the reciprocal of $b$ — this is counterintuitive and one of the most common mistakes in college algebra.

Reflections

• $y = -f(x)$: reflection over the $x$-axis (negate all $y$-values)
• $y = f(-x)$: reflection over the $y$-axis (negate all $x$-values)

Order of Transformations

When multiple transformations combine, the order matters. Starting from $y = a \cdot f(b(x - h)) + k$, apply transformations in this sequence:

1. Horizontal operations (inside the function) — in the order: factor out $b$, then shift by $h$
2. Vertical operations (outside the function) — in the order: stretch/reflect by $a$, then shift by $k$

Think of it as “inside first, then outside” — or equivalently, “horizontal first, then vertical.”

Worked Example 1: Identifying Transformations

Describe all transformations applied to $f(x) = x^2$ to produce $g(x) = -2(x + 3)^2 + 5$.

Rewrite to match the formula: $g(x) = -2 \cdot f(x - (-3)) + 5$.

• $a = -2$: vertical stretch by factor $2$ and reflection over the $x$-axis
• $h = -3$: horizontal shift left $3$ units
• $k = 5$: vertical shift up $5$ units

Sequence: Start with $y = x^2$. Shift left 3. Stretch vertically by 2. Reflect over $x$-axis. Shift up 5.

Worked Example 2: Graphing with Combined Transformations

Graph $h(x) = \frac{1}{2}\sqrt{-(x - 4)} + 1$ starting from $f(x) = \sqrt{x}$.

Rewrite: $h(x) = \frac{1}{2} \cdot f(-(x - 4)) + 1 = \frac{1}{2} \cdot f(-x + 4) + 1$.

Identify: $a = \frac{1}{2}$, $b = -1$ (reflection over $y$-axis), $h = 4$, $k = 1$.

Key points of $f(x) = \sqrt{x}$: $(0, 0)$, $(1, 1)$, $(4, 2)$, $(9, 3)$.

Apply transformations to each point $(x, y) \to \left(\frac{x}{-1} + 4,\; \frac{1}{2}y + 1\right) = (4 - x,\; \frac{y}{2} + 1)$:

• $(0, 0) \to (4, 1)$
• $(1, 1) \to (3, 1.5)$
• $(4, 2) \to (0, 2)$
• $(9, 3) \to (-5, 2.5)$

The transformed graph starts at $(4, 1)$ and extends to the left.

SVG: Transformation Visualization

The gray dashed curve is the parent function $y = x^2$. The green curve shifts right 2 and up 1. The blue curve reflects over the $x$-axis, compresses vertically by half, and shifts up 4.

Writing Equations from Transformed Graphs

Given a graph that you recognize as a transformed parent function, work backward:

1. Identify the parent function — look at the basic shape (V-shape = absolute value, U-shape = quadratic, etc.)
2. Locate the key point — vertex for quadratics, center for square roots, etc.
3. Determine shifts from the key point’s new position
4. Determine stretches/reflections from the shape’s width or orientation

Worked Example 3: Writing an Equation

A parabola opens downward with vertex at $(-1, 6)$ and passes through $(1, 2)$. Write its equation.

Since it opens downward with vertex $(-1, 6)$: $y = a(x + 1)^2 + 6$ with $a < 0$.

Substitute $(1, 2)$: $2 = a(1 + 1)^2 + 6 = 4a + 6$, so $4a = -4$ and $a = -1$.

Equation: $y = -(x + 1)^2 + 6$.

Worked Example 4: Combining Multiple Transformations

Start with $f(x) = |x|$. Apply: reflect over $y$-axis, stretch horizontally by 3, shift left 2, stretch vertically by 4, shift down 1.

Step 1: Reflect over $y$-axis: $f(-x) = |-x| = |x|$ (absolute value absorbs the reflection).

Step 2: Horizontal stretch by 3 means $b = \frac{1}{3}$: $f\!\left(\frac{x}{3}\right) = \left|\frac{x}{3}\right|$.

Step 3: Shift left 2: $f\!\left(\frac{x + 2}{3}\right) = \left|\frac{x + 2}{3}\right|$.

Step 4: Vertical stretch by 4: $4f\!\left(\frac{x + 2}{3}\right) = 4\left|\frac{x + 2}{3}\right| = \frac{4}{3}|x + 2|$.

Step 5: Shift down 1: $\frac{4}{3}|x + 2| - 1$.

Final equation: $g(x) = \frac{4}{3}|x + 2| - 1$.

Real-World Application: Signal Processing

In engineering, a signal $s(t)$ might be transformed as $g(t) = A \cdot s(b(t - t_0)) + D$, where:

• $A$ is the amplification factor (vertical stretch)
• $b$ is the time scaling (horizontal compression)
• $t_0$ is the time delay (horizontal shift)
• $D$ is the DC offset (vertical shift)

If a sensor outputs a base signal $s(t) = t^2$ for $0 \leq t \leq 1$, and the processed signal is delayed by 0.5 seconds, amplified by 3, and offset by $-2$:

$g(t) = 3(t - 0.5)^2 - 2$

At $t = 1$: $g(1) = 3(0.5)^2 - 2 = 3(0.25) - 2 = 0.75 - 2 = -1.25$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The master formula $y = a \cdot f(b(x - h)) + k$ encodes all possible transformations
• Horizontal transformations work opposite to intuition: minus shifts right, larger $b$ compresses
• Order of operations: apply horizontal changes (inside) first, then vertical changes (outside)
• Vertical stretch/compression affects $y$-values by factor $|a|$; horizontal by factor $\frac{1}{|b|}$
• To write an equation from a graph: identify the parent, locate the key point, then determine stretch and reflection
• Combined transformations require careful tracking of each step in sequence
• Engineering applications include signal processing, coordinate transformations, and scale modeling

Return to College Algebra for more topics in this section.