Functions Summary

Algebra 2 · Axiom Academy

SUMMARY Functions and Their Graphs Let's review the essential tools for analyzing, transforming, and combining functions to model complex relationships. Functions are the foundation of advanced algebra — powerful tools for modeling complex relationships. Transformations let you predict a graph's behavior without re-plotting it from scratch. Operations and composition let you build sophisticated models by combining simple functions. Inverse functions reveal the reversibility of a relationship — how to "undo" f . Piecewise functions extend your toolkit to real-world situations with changing rules. Together, these concepts form the essential toolkit for all future mathematical modeling. Core Concept Function Transformations Parent Functions: Starting points include linear, quadratic, cubic, square root, absolute value, exponential, and logarithmic functions. Vertical Shifts: f(x)+k moves the graph up ( ) or down ( ). Horizontal Shifts: f(x-h) moves the graph right ( ) or left ( ). Stretches and Compressions: stretches vertically if , compresses if . Negative values reflect over the x -axis. Core Concept Function Operations Addition: (f+g)(x)=f(x)+g(x) combines outputs at each input value. Subtraction: (f-g)(x)=f(x)-g(x) finds the difference between functions. Multiplication: multiplies function outputs. Division: , where . Domain excludes zeros of g(x) .

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