Slope and proportional relationships Knowledge-Based Study

Study notes

1. Understanding Proportional Relationships

   A proportional relationship represents a linear relationship between two quantities where the ratio of the dependent variable to the independent variable remains constant.

   When graphed on the coordinate plane, every proportional relationship possesses two distinct visual characteristics:

   • The graph is a non-vertical straight line.
   • The line passes directly through the origin, which is the point $(0, 0)$.

   Algebraically, a proportional relationship is written in the form:

   $y = kx$

   Here, $k$ is a non-zero constant known as the constant of proportionality. Since the line includes the origin, any point $(x, y)$ on the line (where $x \neq 0$) satisfies $k = \frac{y}{x}$.

   Connecting Slope to the Constant of Proportionality

   The slope of a line measures its steepness and direction, defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$:

   $m = \frac{y_2 - y_1}{x_2 - x_1}$

   In a proportional relationship, one of the known points is always the origin, $(0, 0)$. If we compute the slope between $(0, 0)$ and any other point $(x, y)$ on the line, we obtain:

   $m = \frac{y - 0}{x - 0} = \frac{y}{x}$

   Notice that this ratio is identical to the constant of proportionality, $k$. Therefore, in any proportional relationship:

   $\text{Slope } (m) = \text{Constant of Proportionality } (k) = \text{Unit Rate}$

   The unit rate describes the change in $y$ for every 1-unit increase in $x$, corresponding to the coordinates $(1, k)$ or $(1, m)$ on the graph.

   Interpreting Steepness and Rates

   Because slope reflects the unit rate, we can compare rates by analyzing the steepness of lines on the coordinate plane:

   • A steeper line corresponds to a greater slope and therefore a greater unit rate.
   • A less steep line corresponds to a smaller slope and therefore a smaller unit rate.

   Worked Example

   Consider two vehicles, Vehicle A and Vehicle B, moving at constant speeds. Let $x$ represent time in hours and $y$ represent distance in miles.

   • Vehicle A passes through $(2, 110)$. Its slope is $m_A = \frac{110 - 0}{2 - 0} = 55$ miles per hour.
   • Vehicle B passes through $(3, 195)$. Its slope is $m_B = \frac{195 - 0}{3 - 0} = 65$ miles per hour.

   Because $65 > 55$, the line representing Vehicle B is steeper than the line for Vehicle A, confirming that Vehicle B travels at a faster unit rate.

   Summary of Key Concepts

   Keep these fundamental properties in mind when working with slope in proportional relationships:

   • Linearity and Origin: If a straight line does not pass through $(0, 0)$, the relationship is linear but not proportional.
   • Equation Form: Proportional lines have the equation $y = mx$, with a $y$-intercept equal to $0$.
   • Consistency: The slope is constant everywhere along the line; computing $\frac{y_2 - y_1}{x_2 - x_1}$ between any two points yields the exact same value as $\frac{y}{x}$ for any individual point.
   • Meaning of Coordinates: The point $(0, 0)$ indicates that zero units of the independent variable yield zero units of the dependent variable, while $(1, m)$ explicitly identifies the unit rate.