Common Core: High School - Functions

This subject covers the essential concepts and applications of functions in high school mathematics, preparing students for advanced mathematical reasoning.

Understanding FunctionsFunction Notation and EvaluationGraphing Functions
Transformations of FunctionsInverse and Composite FunctionsFunction Models and Families
Functions in FinanceFunctions in Science and EngineeringFunctions in Everyday Life
Identifying Function Types on TestsChecking Your Work with the Vertical Line Test
Understanding Functi...Function Notation an...Graphing FunctionsTransformations of F...Inverse and Composit...Function Models and ...Functions in FinanceFunctions in Science...Functions in Everyda...Identifying Function...Checking Your Work w...
Advanced Topics

Transformations of Functions

Changing the Shape and Position

Transformations let us move and reshape graphs without changing their fundamental nature. The main types are translations (shifts), reflections (flips), and dilations (stretches and compressions).

Types of Transformations

  • Translations: Move the graph up, down, left, or right.
  • Reflections: Flip the graph over the \( x \)-axis or \( y \)-axis.
  • Dilations: Stretch or shrink the graph vertically or horizontally.

General Formulas

  • Vertical shift: \( f(x) + k \)
  • Horizontal shift: \( f(x - h) \)
  • Reflection over \( x \)-axis: \( -f(x) \)
  • Vertical stretch/compression: \( a \cdot f(x) \)

Exploring Patterns

Experiment with different values for \( h \), \( k \), and \( a \) to see how the graph changes. Graphing calculators and apps make this fun and interactive!

Key Formula

\[f(x) = a \cdot f(x-h) + k\]

Examples

  • \( f(x) = (x-2)^2 + 3 \) shifts the parabola right 2 units and up 3 units.

  • \( g(x) = -2x \) reflects the line over the \( x \)-axis and stretches it.

In a Nutshell

Transformations change a function's graph by shifting, flipping, or stretching it.

Key Terms

Transformation
Changing a graph's position, orientation, or size.
Translation
Sliding a graph horizontally or vertically.
Reflection
Flipping a graph over a line.
Dilation
Stretching or compressing a graph.
Next
Inverse and Composite Functions