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The y-intercept is the point where a line crosses the y-axis on a graph. It is an essential feature in the equation of a line, usually represented as the constant term in the linear equation. For example, in the equation \(y = mx + b\), the \(b\) represents the y-intercept. In our specific problem, this is \(2\), which means every line in this family of lines passes through the point \((0, 2)\).
Understanding the y-intercept helps visualize where the line will intersect the vertical axis, serving as a starting point for graphing. This concept is particularly useful in identifying parallel lines, as they share the same y-intercept but can have varying slopes.

The slope of a line describes its steepness and direction. Represented by \(m\) in the equation \(y = mx + b\), it is calculated by the rise over run or the change in \(y\) over the change in \(x\).
The slope tells you how much \(y\) increases or decreases as \(x\) increases by 1 unit. A positive slope means the line ascends to the right, while a negative slope descends to the right.
In our exercise, different lines in the family \(y = mx + 2\) will have different slopes, but all share the same y-intercept. For instance, if the slope is \(-1\), the line goes down 1 unit for every 1 unit it goes right, giving the equation \(y = -x + 2\). This is the equation we derived for the specific line with an angle of inclination of \(135^\circ\).

The angle of inclination of a line refers to the angle \(\theta\) that the line forms with the positive x-axis. Here, it's a crucial element because it directly relates to the slope through the tangent function. The formula \(m = \tan(\theta)\) helps find the slope from the angle.
In our problem, the given angle of inclination is \(135^\circ\). Using the formula, the slope \(m\) is \(-1\), consistent with \(\tan(135^\circ)\). This results in the line \(y = -x + 2\), which agrees with the equation of the family of lines provided initially.

A family of lines shares a common feature, here defined by the y-intercept, \(b=2\). All lines in this family have equations of the form \(y = mx + 2\), with \(m\) as the varying slope. Every line intersects the y-axis at (0, 2), emphasizing their shared trait.
Visually, sketching this family involves drawing multiple lines through the y-intercept, but at different angles dictated by their slopes.
The exercise specifically derived one member of this family using an angle of inclination \(135^\circ\), yielding the line \(y = -x + 2\). Graphical representation includes this line among others with different slopes, all showcasing the influence of the fixed y-intercept.