Overview
This lecture covers the main types of graph transformations, including translations, reflections, and how to apply them to various functions and exam questions.
Graph Translations (Vertical & Horizontal)
- The graph of y = x^2 is a parabola with points symmetric about the y-axis.
- Adding a constant to a function (y = x^2 + a) shifts the graph up by 'a' units.
- Subtracting a constant (y = x^2 - a) shifts the graph down by 'a' units.
- Adding a constant inside the function (y = (x + a)^2) shifts the graph left by 'a' units.
- Subtracting a constant inside (y = (x - a)^2) shifts the graph right by 'a' units.
Function Notation and General Translation Rules
- y = f(x) + a translates f(x) up by 'a' units; y = f(x) - a translates down by 'a' units.
- y = f(x + a) translates f(x) left by 'a' units; y = f(x - a) translates right by 'a' units.
Exam Question Strategies
- Mark key points on the original graph, then apply the translation to each point.
- For horizontal shifts, adjust the x-coordinate; for vertical, adjust the y-coordinate.
Reflections of Graphs
- y = -f(x) reflects the graph in the x-axis (output sign changes).
- y = f(-x) reflects the graph in the y-axis (input sign changes).
Multiple Transformations and Their Order
- Apply horizontal shifts before vertical for y = f(x + a) + b.
- For y = -f(x) + a, do the reflection in the x-axis before shifting up/down.
- For y = f(-(x + a)), do translation first, then reflect in y-axis.
Applying Transformations to Points and Equations
- Transformations applied to a single point: adjust x/y coordinates based on type.
- For y = f(x - a) + b, point (x, y) maps to (x + a, y + b).
- For reflections, change the sign of the relevant coordinate.
- To find new equations after transformation, substitute x with (x ± a) or -x as required, then expand.
Key Terms & Definitions
- Translation — shifting a graph horizontally or vertically without changing its shape.
- Reflection — flipping a graph across an axis (x-axis or y-axis).
- Function notation (f(x)) — general way to express a function.
- Input/Output — x is the input, f(x) (or y) is the output.
Action Items / Next Steps
- Practice by sketching transformed graphs using given translations/reflections.
- Complete assigned exam booklets or questions on graph transformations.
- Review how to expand expressions like (x + a)^3 for transformation problems.