Slope-Intercept Form of Linear Equations

Slope-intercept form is defined as $y = mx + b$ .
$m$ represents the slope of the line.
$b$ represents the $y$ -intercept, which is the location where the line crosses the $y$ -axis.

Slope measures the steepness and direction of a line.
Formula: $\text{Slope } (m) = \frac{\text{Rise}}{\text{Run}} = \frac{\text{Change in } y}{\text{Change in } x}$
Positive slope: Rises from left to right (e.g., an increase of $4$ in $y$ over $6$ in $x$ gives $m = \frac{4}{6} = \frac{2}{3}$ ).
Negative slope: Falls from left to right (e.g., a decrease of $2$ in $y$ over $5$ in $x$ gives $m = -\frac{2}{5}$ ).

Step 1: Plot the $y$ -intercept ( $b$ ) on the vertical $y$ -axis.
Step 2: Use the slope ( $m$ ) to find additional points by performing the rise and run starting from the $y$ -intercept.
Equation $y = \frac{1}{4}x + 7$ : Plot $b = 7$ , move up $1$ unit and right $4$ units ( $m = \frac{1}{4}$ ).
Equation $g(x) = -2x - 4$ : Plot $b = -4$ , write $m = -2$ as $\frac{-2}{1}$ , then move down $2$ units and right $1$ unit.

Step 1: Identify the $y$ -intercept ( $b$ ) where the line intersects the vertical $y$ -axis.
Step 2: Calculate the slope ( $m$ ) by measuring the vertical change over horizontal change between points on the line.
Example 1: A line crossing at $b = 1$ that drops $4$ units for every $1$ unit right ( $m = -4$ ) has the equation $y = -4x + 1$ .
Example 2: A line crossing at $b = -3$ that rises $1$ unit for every $2$ units right ( $m = \frac{1}{2}$ ) has the equation $y = \frac{1}{2}x - 3$ .