Lesson 3: Transforming Linear Functions

Property

A transformation changes a function to create a new function.
A transformation of a function $f(x)$ produces a new function, $g(x)$, by applying specific operations to the input or output of $f(x)$.
These changes alter the graph of the original function by moving it, stretching it, compressing it, or reflecting it.

Examples

• Given the parent function $f(x) = x$, a transformation can create the new function $g(x) = f(x) + 3$, which simplifies to $g(x) = x + 3$.
• Given the function $f(x) = 2x - 1$, a transformation can create the new function $g(x) = 4 \cdot f(x)$, which simplifies to $g(x) = 4(2x - 1) = 8x - 4$.
• Given the function $f(x) = x$, a transformation can create the new function $g(x) = f(x-5)$, which simplifies to $g(x) = x-5$.

Explanation

A transformation is a general process that alters a function and its graph. Each point $(x, y)$ on the graph of the original function corresponds to a new point on the graph of the transformed function. The types of transformations include translations (shifts), dilations (stretches or compressions), and reflections. Understanding transformations allows you to predict how changes to a function''s equation will affect its graph.

Property

To compare an original function $f(x)$ with its transformation $g(x)$, create a table with columns for $x$, $f(x)$, and $g(x)$ using the same input values.
Calculate outputs systematically to identify patterns in how the transformation affects each point.

Examples

Property

Compared to the graph of $y = x$, the graph of $y = x + c$

• is shifted upward by $c$ units if $c > 0$.
• is shifted downward by $|c|$ units if $c < 0$.

Property

The graph of $f(x) = m(x - h) + b$ shifts the graph of $f(x) = mx + b$ horizontally $h$ units.

• If $h > 0$, shift the line horizontally right $h$ units.
• If $h < 0$, shift the line horizontally left $|h|$ units.

Examples

Property

For a linear function $y = mx + b$, scaling transformations affect the function differently depending on whether you multiply the input or output:

• Vertical scaling: $y = a(mx + b) = amx + ab$ stretches/compresses vertically by factor $|a|$
• Horizontal scaling: $y = m(cx) + b = mcx + b$ stretches/compresses horizontally by factor $\frac{1}{|c|}$