Transformations for MATH 122

Understanding Function Transformations

When you learn a "parent function" like $f(x) = x^2$, you can create infinitely many related functions by shifting, stretching, or flipping it. These modifications are called transformations.

Instead of memorizing the shape of every function, you can start with a basic shape and transform it. This makes graphing much faster and helps you understand how equations relate to their graphs.

The Four Main Transformations

Vertical Shifts: $f(x) + k$

Adding a constant outside the function shifts the graph vertically.

• $f(x) + k$ shifts up by $k$ units (when $k > 0$)
• $f(x) - k$ shifts down by $k$ units

Notice how the shape stays exactly the same — only the position changes. The vertex of $x^2$ moves from $(0,0)$ to $(0,3)$ or $(0,-2)$.

Horizontal Shifts: $f(x - h)$

Adding a constant inside the function (to $x$) shifts the graph horizontally.

⚠️ This one is counterintuitive! The shift is opposite to what you might expect.

• $f(x - h)$ shifts right by $h$ units (when $h > 0$)
• $f(x + h)$ shifts left by $h$ units

Why is it backwards? Think about it: $\frac{1}{x+2}$ has its vertical asymptote where $x + 2 = 0$, which is $x = -2$. So adding 2 inside shifts left. Similarly, $\frac{1}{x-3}$ shifts right to $x = 3$.

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

Multiplying the function by a constant $a$ stretches or compresses it vertically.

• $|a| > 1$: Stretch (taller)
• $0 < |a| < 1$: Compression (shorter)

Notice how $\sin(x)$ normally oscillates between $-1$ and $1$. With $2\sin(x)$, the amplitude doubles (oscillates between $-2$ and $2$). With $\frac{1}{2}\sin(x)$, the amplitude is halved.

Reflections

Reflection across the x-axis: $-f(x)$

Negating the output flips the graph upside down (over the x-axis).

Reflection across the y-axis: $f(-x)$

Negating the input flips the graph horizontally (mirror image over the y-axis).

Starting with $\ln(x)$ (defined for $x > 0$):

• $\ln(-x)$ reflects over the y-axis (now defined for $x < 0$)
• $-\ln(x)$ reflects over the x-axis (flipped upside down)
• $-\ln(-x)$ reflects over both axes (rotated 180°)

Example 3: Graphing with Transformations

Sketch $g(x) = f(x+1) + 2$ by transforming $f(x) = \sqrt{x}$.

Show Solution

Starting from $f(x) = \sqrt{x}$:

1. Shift right 1: The starting point moves from $(0, 0)$ to $(1, 0)$
2. Shift up 2: The starting point moves from $(1, 0)$ to $(1, 2)$

Key points on $f(x) = \sqrt{x}$: $(0,0)$, $(1,1)$, $(4,2)$

After transformation: $(1, 2)$, $(2, 3)$, $(5, 4)$

Order of Transformations

When multiple transformations are combined, apply them in this order:

1. Horizontal shifts (inside the function)
2. Horizontal stretches/reflections (inside the function)
3. Vertical stretches/reflections (outside the function)
4. Vertical shifts (outside the function)

Memory trick: Work from the inside out — transformations closer to $x$ happen first.

The General Form

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

Common Mistakes and Misunderstandings

❌ Mistake: Getting horizontal shifts backwards

Wrong: $f(x - 3)$ shifts left 3 units

Why it's wrong: The shift is opposite to the sign inside. Think: "What value of $x$ makes the inside equal zero?"

Correct: $f(x - 3)$ shifts right 3 units. $f(x + 3)$ shifts left.

❌ Mistake: Confusing stretch direction

Wrong: $2f(x)$ makes the graph wider

Why it's wrong: Multiplying by 2 makes every y-value twice as large, so the graph gets taller (narrower for parabolas).

Correct: $2f(x)$ is a vertical stretch — the graph gets taller/narrower. $\frac{1}{2}f(x)$ compresses it.

❌ Mistake: Applying transformations in the wrong order

Wrong: For $2(x-3)^2 + 1$, shifting up 1 first, then shifting right 3

Why it's wrong: You must follow the correct order: horizontal shifts, then stretches, then vertical shifts.

Correct: Shift right 3, then stretch by 2, then shift up 1.