Introduction to Function Transformations (Rigid)

TL;DR

You can move a function's graph around without changing its shape or size. We'll focus on shifts (up, down, left, right) and reflections (flipping). Understanding these "rigid" transformations makes graphing easier and helps you predict how changes to an equation affect its visual representation.

1. The Mental Model

Imagine you have a drawing on a transparent sheet. Rigid transformations are like sliding that sheet around on a table (shifts) or flipping it over (reflections). The drawing itself doesn't stretch, shrink, or become distorted.

2. The Core Material

When you transform a function, you're changing its equation, which in turn changes its graph. Rigid transformations involve moving the graph without altering its basic shape. These are super useful because once you know the basic shape of a function (like $y=x^2$ or $y=|x|$), you can easily sketch many related functions.

Shifting Up and Down (Vertical Shifts)

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Adding or subtracting a number outside the main function operation moves the whole graph up or down.

• $y = f(x) + c$: Shifts the graph of $f(x)$ up by $c$ units.
• $y = f(x) - c$: Shifts the graph of $f(x)$ down by $c$ units.

Think of it this way: for every $x$, the new $y$-value is just the old $y$-value plus or minus $c$.

Shifting Left and Right (Horizontal Shifts)

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Adding or subtracting a number inside the main function operation (affecting $x$ directly) moves the graph left or right. This often feels counter-intuitive!

• $y = f(x - c)$: Shifts the graph of $f(x)$ right by $c$ units.
• $y = f(x + c)$: Shifts the graph of $f(x)$ left by $c$ units.

Why is it opposite? Imagine $y = (x-2)^2$. To get the same $y$-value as $x=0$ in $y=x^2$ (which is $0^2=0$), you now need $x=2$ for $(2-2)^2=0$. So, the 'original' $x=0$ point effectively moved to $x=2$.

Reflecting Across Axes

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Reflections flip the graph over an axis.

• $y = -f(x)$: Reflects the graph of $f(x)$ across the x-axis. This changes the sign of all the $y$-values.
• $y = f(-x)$: Reflects the graph of $f(x)$ across the y-axis. This changes the sign of all the $x$-values before the function acts on them.

Here's how these transformations relate:

graph TD
    A["Starting Function y = f(x)"] --> B["Add/Subtract 'c' outside: y = f(x) ± c"]
    B --> B1["+c: Shift Up"]
    B --> B2["-c: Shift Down"]

    A --> C["Add/Subtract 'c' inside: y = f(x ± c)"]
    C --> C1["-c: Shift Right"]
    C --> C2["+c: Shift Left"]

    A --> D["Negate outside: y = -f(x)"]
    D --> D1["Reflect across X-axis"]

    A --> E["Negate inside: y = f(-x)"]
    E --> E1["Reflect across Y-axis"]

3. Worked Example

Let's start with the basic function $f(x) = x^2$. We want to graph $g(x) = -(x+3)^2 + 1$.

1. Identify the base function: $f(x) = x^2$. This is a parabola opening upwards, with its vertex at $(0,0)$.
2. Inside transformation: We have $(x+3)$. According to our rules, $f(x+c)$ shifts left. So, $y=(x+3)^2$ shifts the parabola left by 3 units. The vertex is now at $(-3,0)$.
3. Outside negation: We have $-(x+3)^2$. The negative sign is outside the squared term. This reflects the graph across the x-axis. So, the parabola now opens downwards, with its vertex still at $(-3,0)$.
4. Outside addition: Finally, we have $-(x+3)^2 + 1$. The $+1$ is outside the function. This shifts the graph up by 1 unit. The vertex is now at $(-3,1)$.

So, the graph of $g(x) = -(x+3)^2 + 1$ is a parabola that opens downwards, with its vertex at $(-3,1)$, shifted left 3 units and up 1 unit from the basic $y=x^2$ graph.

4. Key Takeaways

• Vertical shifts (up/down) happen when you add/subtract a constant outside the function, affecting the $y$-value directly.
• Horizontal shifts (left/right) happen when you add/subtract a constant inside the function, affecting the $x$-value (remember it's counter-intuitive: $x-c$ is right, $x+c$ is left).
• Reflections across the x-axis occur when you negate the entire function: $y = -f(x)$.
• Reflections across the y-axis occur when you negate the input variable: $y = f(-x)$.
• Rigid transformations don't change the shape or size of the graph, only its position or orientation.
• You can apply multiple transformations in sequence to get the final graph.

Common Mistakes to Avoid:
- Mixing up horizontal and vertical shifts; check if the change is inside or outside the function.
- Getting horizontal shifts backwards ($x-c$ means right, not left).
- Confusing reflections over the x-axis with reflections over the y-axis.
- Applying transformations in the wrong order, especially when non-rigid transformations are involved (though we haven't covered those yet).

5. Now Try It

Starting with the graph of $f(x) = |x|$, sketch the graph of $g(x) = -|x-2| - 3$. Identify the base function's "vertex" (the sharp corner), then track its movement and orientation through each transformation step.
Success looks like: a V-shaped graph pointing downwards, with its corner at $(2, -3)$.