Understanding Linear Equations and Rate of Change

Slope-Intercept Form

This is the most common form used for graphing lines because it clearly identifies the starting point and the movement of the line. It is expressed as: $y = mx + b$ In this equation, $m$ is the slope and $b$ is the y-intercept.

Point-Slope Form

If you are given a single coordinate pair $(x_1, y_1)$ and the slope $m$, the point-slope form is the most efficient way to write the equation: $y - y_1 = m(x - x_1)$ For example, if you have a slope of $-1/3$ and the point $(6, -2)$, the equation is written as: $y + 2 = -\frac{1}{3}(x - 6)$

Standard Form

Equations can also appear in standard form, such as $2x - 5y = 40$. While this form is useful for certain algebraic manipulations, you can convert it to slope-intercept form to easily identify the slope and intercepts.

Intercepts and Graphing

Intercepts are the points where a line crosses the axes of a coordinate plane.

• Y-intercept: The point where the line crosses the y-axis. At this point, $x = 0$.
• X-intercept: The point where the line crosses the x-axis. At this point, $y = 0$. This is also known as the zero of the function.

Identifying intercepts is a primary step in graphing. For the equation $y = 3x - 9$, you can find the intercepts to plot the line quickly.

When graphing, you can use the slope to find additional points. For instance, if you have a point and a slope of $-\frac{4}{3}$, you can move down 4 units and right 3 units to plot your next point.

Direct Variation and the Constant of Variation

A special type of linear relationship is direct variation, where $y$ varies directly with $x$. This relationship can be expressed as: $y = kx$ In this context, $k$ is known as the constant of variation. To find $k$, you can substitute a known coordinate pair into the equation. For example, if $x = 15$ when $y = 45$, then: $45 = k(15)$ $k = 3$ Once you have the constant, you can find any other value in the relationship.

Summary Table of Linear Forms