Lesson 1: Parent Functions and Transformations

Property

A parent function is the simplest function of a family of functions that preserves the definition (or shape) of the entire family.

• Constant: $f(x) = c$
• Linear: $f(x) = x$
• Absolute Value: $f(x) = |x|$
• Quadratic: $f(x) = x^2$

Examples

• The constant function $f(x) = 3$ is a horizontal line passing through $(0, 3)$.
• The linear parent function $f(x) = x$ is a straight line passing through the origin with a slope of 1.
• The absolute value parent function $f(x) = |x|$ forms a V-shape with its vertex at the origin.
• The quadratic parent function $f(x) = x^2$ forms a U-shaped parabola with its vertex at the origin.

Explanation

This skill summarizes the fundamental parent functions. Each parent function has a unique shape and set of properties. The constant function is a horizontal line, the linear function is a diagonal line, the absolute value function is V-shaped, and the quadratic function is a U-shaped parabola. Understanding these basic forms is the first step to learning how transformations like shifts, stretches, and reflections affect their graphs.

Property

A function $g(x) = a \cdot f(x - h) + k$ represents a transformation of a parent function $f(x)$.

• Translations (Shifts): The values of $h$ and $k$ shift the graph horizontally and vertically.
• Reflections: A negative value for $a$ reflects the graph across the x-axis.
• Stretches/Shrinks: The absolute value of $a$ vertically stretches or shrinks the graph.

Examples

• Translation: The graph of $g(x) = (x - 3)^2 + 5$ is the graph of $f(x) = x^2$ shifted 3 units right and 5 units up.
• Reflection: The graph of $g(x) = -|x|$ is the graph of $f(x) = |x|$ reflected across the x-axis.
• Vertical Stretch/Shrink: The graph of $g(x) = 2x^2$ is a vertical stretch of $f(x) = x^2$, while $g(x) = \frac{1}{2}x^2$ is a vertical shrink.

Explanation

Transformations alter the graph of a parent function in predictable ways. A translation is a rigid transformation that slides the graph horizontally or vertically without changing its shape or orientation. A reflection is another rigid transformation that flips the graph over a line, such as the x-axis. Stretches and shrinks are non-rigid transformations that change the size of the graph by multiplying the output values by a constant factor, making it appear narrower or wider.