Graphs of Functions

The independent variable is the x-coordinate representing horizontal position on a graph. Substituting x-values into the function produces the dependent variable, the y-coordinate representing vertical position. Multiple y-values are calculated using different x-values to plot the function's line or curve.

The domain includes all possible input values a function can accept, while the range consists of the corresponding output values it produces. Together, they define the complete set of ordered pairs plotted on the graph, establishing the function's scope and behavior across all valid inputs.

Even power functions form U-shaped curves symmetric about the vertical y-axis. These graphs have non-negative y-values for both positive and negative x-values, and their outputs increase rapidly as the input moves away from zero, creating a characteristic parabolic appearance.

Odd power functions create S-shaped curves that pass through the origin and extend in opposite directions, showing symmetry about the origin. Unlike even power functions, odd power functions can produce both positive and negative y-values depending on the input sign and the function's specific form.

Asymptotes are lines that a graph approaches indefinitely but never intersects. Reciprocal functions form two distinct curves that approach the coordinate axes as asymptotes without touching them, creating a characteristic hyperbolic pattern with gaps along both the x and y axes.

Linear functions like y = x produce straight lines indicating a constant rate of change. Nonlinear functions display more complex behaviors, forming curves such as U-shapes, S-shapes, or hyperbolas that show variable rates of change across different input values.

Function graphs model motion and forces in physics and engineering, represent cost and revenue in economics, and describe growth and decay rates in biology. Understanding these visual representations enables accurate interpretation and prediction of system behavior across diverse scientific and analytical domains.