Sure! Let's define the slope in each of these contexts:
1. Slope from a Table:
The slope from a table is calculated using the change in the dependent variable (often represented as \( y \)) divided by the change in the independent variable (often represented as \( x \)). If you have two points from the table, say \((x_1, y_1)\) and \((x_2, y_2)\), the slope (\(m\)) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This represents the rate of change in \( y \) with respect to \( x \) between the two points provided in the table.
2. Slope from a Graph:
The slope from a graph represents the steepness or incline of a line. It can be visually determined by selecting two points on the line and applying the same formula used for the slope from a table:
\[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, "rise" is the vertical change (the difference in \( y \)-coordinates) and "run" is the horizontal change (the difference in \( x \)-coordinates) between the two selected points. A positive slope indicates that the line rises as it moves to the right, while a negative slope indicates that it falls.
3. Slope from an Equation:
The slope can also be derived from the equation of a line, particularly in slope-intercept form, which is given as:
\[ y = mx + b \]
In this equation, \( m \) represents the slope of the line, and \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)). The slope indicates how much \( y \) changes for a unit change in \( x \). For example, if \( m = 2 \), for every 1 unit increase in \( x \), \( y \) will increase by 2 units.
These definitions provide a comprehensive understanding of how to determine and interpret the slope in various mathematical contexts.
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