Graph Analysis and Linearization Reference Cheat Sheet

Key Facts

• The slope of a straight line is $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$.
• The equation of a linear graph is $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.
• The units of slope are the units of the vertical axis divided by the units of the horizontal axis, so $\text{slope units} = \frac{\text{units of }y}{\text{units of }x}$.
• A direct proportion has the form $y = kx$ and produces a straight line through the origin with slope $k$.
• An inverse proportion has the form $y = \frac{k}{x}$, so plotting $y$ versus $\frac{1}{x}$ should produce a straight line.
• A quadratic relationship has the form $y = kx^2$, so plotting $y$ versus $x^2$ should produce a straight line.
• A square-root relationship has the form $y = k\sqrt{x}$, so plotting $y$ versus $\sqrt{x}$ should produce a straight line.
• Percent difference between an experimental value and accepted value is $\%\text{ difference} = \left|\frac{\text{experimental} - \text{accepted}}{\text{accepted}}\right| \times 100\%$.

Vocabulary

Slope

Slope is the rate of change of the vertical variable with respect to the horizontal variable, calculated by $m = \frac{\Delta y}{\Delta x}$.

Intercept

An intercept is the point where a graph crosses an axis, such as the $y$-intercept $b$ in $y = mx + b$.

Best-fit line

A best-fit line is a straight line drawn to represent the overall trend of scattered data points.

Linearization

Linearization is the process of transforming variables so curved data can be graphed as a straight line.

Direct proportion

A direct proportion is a relationship where $y = kx$, meaning one variable increases by the same factor as the other.

Inverse proportion

An inverse proportion is a relationship where $y = \frac{k}{x}$, meaning one variable decreases as the other increases.