3-3 Slope-Intercept Form

Property

The slope of a line is given by

The symbol $\Delta$ (delta) is used in mathematics to denote change in.

Examples

• If the change in y, $\Delta y$, is 6 and the change in x, $\Delta x$, is 2, the slope is $m = \frac{\Delta y}{\Delta x} = \frac{6}{2} = 3$.
• A line moves 4 units down ($Δy = -4$) for every 10 units it moves to the right ($Δx = 10$). The slope is $m = \frac{-4}{10} = -\frac{2}{5}$.
• For a horizontal line, the y-coordinate never changes, so $\Delta y = 0$. This means the slope $m = \frac{0}{\Delta x} = 0$, no matter the change in x.

Explanation

This is the official shorthand for slope. The letter $m$ stands for slope, and the Greek letter delta ($\Delta$) is a compact way to write 'change in'. So, $m = \frac{\Delta y}{\Delta x}$ is just a neat way of writing slope equals change in y over change in x.

Property

When data follows a linear pattern, the first differences between consecutive $y$-values are constant.

For data points with evenly spaced $x$-values, if $\Delta y = y_{i+1} - y_i$ is the same for all consecutive pairs, then the data can be modeled by a linear function.

Property

A linear equation written in the form

is said to be in slope-intercept form. The coefficient $m$ is the slope of the graph, and $b$ is the $y$-intercept.

Examples

• The equation $y = 3x + 5$ is in slope-intercept form. The slope is $3$ and the $y$-intercept is $(0, 5)$.
• For $y = -2x - 1$, the slope is $-2$ and the $y$-intercept is $(0, -1)$.
• In the equation $y = \frac{1}{2}x + 4$, the slope is $\frac{1}{2}$ and the $y$-intercept is $(0, 4)$.

Explanation

This form is a recipe for drawing a line. The '$b$' tells you your starting point on the y-axis, and the '$m$' (slope) gives you directions on how steep to draw the line from there.