Overview
This lecture introduces parent functions and explains how to apply translations and reflections as basic transformations to their graphs.
Parent Functions
- Parent functions are the simplest forms in a family of functions.
- The constant parent function is ( f(x) = 1 ); its graph is a horizontal line.
- The linear parent function is ( f(x) = x ); its graph is a straight line with constant slope.
- The absolute value parent function is ( f(x) = |x| ); its graph forms a V-shape and is always non-negative.
- The quadratic parent function is ( f(x) = x^2 ); its graph is a U-shaped parabola and always non-negative.
Transformations Overview
- Transformations alter the position or orientation of parent functions on the coordinate plane.
- Three main types: translations (shifts), reflections (flips), and stretches/shrinks (covered later).
Translations (Shifts)
- Vertical translation: ( g(x) = f(x) + k ) shifts the graph up by ( k ); ( g(x) = f(x) - k ) shifts down by ( k ).
- Example: ( f(x) = x^2 ); ( f(x) + 2 ) shifts up 2 units, ( f(x) - 3 ) shifts down 3 units.
- Horizontal translation: ( g(x) = f(x - h) ) shifts right by ( h ); ( g(x) = f(x + h) ) shifts left by ( h ).
- Note: For horizontal shifts, the sign is opposite the direction.
Reflections (Flips)
- Reflection over x-axis: ( g(x) = -f(x) ); flips the graph vertically.
- Example: ( f(x) = x^2 ) vs. ( g(x) = -x^2 ); the parabola opens downward.
- Reflection over y-axis: ( g(x) = f(-x) ); flips the graph horizontally.
- Example: ( f(x) = x^3 ); ( g(x) = (-x)^3 ) is the cubic reflected over the y-axis.
Key Terms & Definitions
- Parent Function — the simplest form of a function in a family.
- Transformation — operation that changes the position or orientation of a graph.
- Translation — shifting the entire graph horizontally or vertically.
- Reflection — flipping the graph over a specific axis.
- Vertical Shift — moving the graph up or down.
- Horizontal Shift — moving the graph left or right.
- Slope — rate of change of y with respect to x.
Action Items / Next Steps
- Review graphs of parent functions and identify their properties.
- Practice applying vertical and horizontal shifts to parent functions.
- Try sketching reflections over both axes for basic functions.