ENG 207A – Graphical Analysis, Linear Fitting & Regression

Vocabulary flashcards covering the essential terms and definitions from Lecture 4 on Units, Dimensions & Measurements—specifically the section on graphical analysis, types of graph relationships, and linear regression methods.

Independent Relationship

A graph where y remains constant regardless of x; represented by y = k (a horizontal line).

Direct Proportionality

Relationship in which y increases in constant ratio with x; expressed as y = kx (line through the origin).

Inverse Proportionality

Relationship in which y decreases as x increases so that xy = k; rearranged as y = k/x (rectangular hyperbola).

Square Relationship

Relationship where y is proportional to x²; y = kx² (vertical parabola opening upward).

Square-Root Relationship

Relationship where y is proportional to √x; y = k√x (horizontal parabola opening rightward).

Power Relationship

General proportionality of form y = kxⁿ, where n can be any real power (positive, negative, fractional, etc.).

Linear Relationship

Combination of direct change and constant offset; y = ax + b (non-horizontal straight line).

Quadratic Relationship

Polynomial of degree two, y = ax² + bx + c; produces a vertical parabola that may be shifted.

Polynomial Relationship

Any relation expressed as y = a + bx + cx² + dx³ + … + aₙxⁿ, combining multiple power terms.

Exponential Growth

Relation where y multiplies by a constant factor per x-interval; y = a·n^{bx}, yielding rapid rise.

Exponential Decay

Relation where y is divided by a constant factor per x-interval; y = a·n^{−bx}, producing rapid fall.

Exponential Approach

Curve of the form y = a(1 − n^{−bx}) + c that approaches a final value asymptotically.

Periodic Function

Any function repeating its values at regular intervals; typically written y = a sin(bx + c).

Constant of Proportionality (k)

The fixed numerical factor linking two variables in proportional relationships (slope for direct).

Slope (m)

Rate of change of y with respect to x for a straight line; m = (y₂ − y₁)/(x₂ − x₁).

Y-Intercept (b)

The value of y where a line crosses the y-axis (x = 0) in y = mx + b.

Linear Curve Equation

General linear form y = ax + b; when a = m it becomes y = mx + b.

Linear Regression

Statistical method that fits the best straight line to multiple data points by minimizing squared errors.

Coefficient of Determination (r²)

Measure (0–1) indicating the fraction of variance in y explained by the regression line.

Coefficient of Correlation (r)

Statistic (−1 to +1) showing strength and direction of linear association; r = ±√r².

Rectangular Hyperbola

Graph shape of an inverse proportionality (y = k/x).

Vertical Parabola

Parabolic curve symmetric about the y-axis, produced by relationships like y = kx².

Horizontal Parabola

Parabolic curve symmetric about the x-axis, from relations such as y = k√x.

Universality of Mathematics

Concept that mathematics applies broadly across disciplines because of its abstract nature (AAAS quote).

Data Cloud

Scatter of experimental points around an ideal mathematical curve due to measurement limitations.

Best-Fit Line

Straight line that most closely represents the trend of a scatter plot, often found via regression.

Scatter Plot

Graph showing individual data points (x,y) used to visualize relationships before fitting a curve.

Graphical Analysis

Process of interpreting data relationships by plotting and examining curve shapes and fits.

Product Constant

Condition in inverse proportionality where the product xy equals a fixed value k.

Ratio Constant

Condition in direct proportionality where the ratio y/x equals a fixed value k.