Overview
This lecture covers how to graphically transform functions using vertical and horizontal shifts, stretches, shrinks, and reflections, mainly using the example of quadratic functions.
Basic Function Transformations
- The graph of ( f(x) = x^2 ) is a parabola with vertex at the origin opening upwards.
- Transformations let us modify basic graphs more easily than plotting every point.
Vertical and Horizontal Shifts
- Adding or subtracting a number outside the function, ( f(x) + c ), moves the graph vertically by ( c ) units (up if positive, down if negative).
- Adding or subtracting inside the function, ( f(x + c) ), shifts the graph horizontally (left if ( c > 0 ), right if ( c < 0 )).
Stretches and Shrinks
- Multiplying the function by a constant, ( a f(x) ), vertically stretches if ( |a| > 1 ) and shrinks if ( 0 < |a| < 1 ).
- Multiplying the input by a constant, ( f(a x) ), horizontally shrinks for ( |a| > 1 ) and stretches for ( 0 < |a| < 1 ).
Reflections
- Multiplying the function by (-1), ( -f(x) ), reflects it across the x-axis.
- Multiplying input by (-1), ( f(-x) ), reflects it across the y-axis.
Combining Transformations
- Apply each transformation step-by-step: order does not matter if they’re all applied to the same parent function.
- Example: ( -2(x - 3)^2 + 4 ) reflects the graph downwards, vertically stretches, shifts right 3, then shifts up 4.
Applying Transformations to Any Function
- These rules apply to any parent function (e.g., absolute value, cube, square root), not just quadratics.
Key Terms & Definitions
- Vertical Shift — Moving the graph up or down by adding or subtracting a constant outside the function.
- Horizontal Shift — Moving the graph left or right by adding or subtracting inside the function’s input.
- Vertical Stretch/Shrink — Changing the graph’s steepness by multiplying the function by a constant.
- Horizontal Stretch/Shrink — Changing the graph’s width by multiplying the input by a constant.
- Reflection — Flipping the graph across the x-axis (( -f(x) )) or y-axis (( f(-x) )).
Action Items / Next Steps
- Review the summary table of transformations.
- Practice graphing functions using each transformation type without making a table for every point.
- Prepare for upcoming lessons on cases where transformations are not sufficient.