Solution: The y-intercept occurs when $ x = 0 $. Substitute: $ t = \frac{4(0) + 1}{0 - 2} = \frac{1}{-2} = -\frac{1}{2} $. The y-intercept is $ \boxed{\left(0, -\dfrac{1}{2}\right)} $.

["Understanding the Y-Intercept: Definition, Formula, and Example", "The y-intercept is a fundamental concept in algebra and graphing that represents the point where a line crosses the y-axis. At this critical point, the value of the independent variable ( x ) is zero, and the corresponding ( y )-value reveals how a linear function behaves when no input is applied.", "### What is the Y-Intercept?", "The y-intercept occurs at ( x = 0 ). For any linear equation expressed in slope-intercept form ( y = mx + b ), the y-intercept is directly identified by the constant term ( b ). This means that when ( x = 0 ), the output ( y ) equals ( b )—the position of the intercept on the graph.", "### How to Calculate the Y-Intercept Step-by-Step", "To find the y-intercept, simply substitute ( x = 0 ) into the equation and solve for ( y ):", "[\ny = m(0) + b = b\n]", "This simplifies cleanly to ( y = b ), which is the exact coordinate:\n[ \boxed{\left(0, b\right)} ]", "### Example: Finding the Y-Intercept Using Substitution", "Consider the equation:", "[\nt = \frac{4(0) + 1}{0 - 2}\n]", "Substitute ( x ) (or in this case ( t )) with 0:", "[\nt = \frac{4(0) + 1}{0 - 2} = \frac{0 + 1}{-2} = \frac{1}{-2} = -\frac{1}{2}\n]", "Since ( t ) represents the dependent variable along the y-axis, the y-intercept is:", "[\n\left(0, -\frac{1}{2}\right)\n]", "This point tells us that when input ( t = 0 ) (in scaled units), the output ( y ) is ( -\frac{1}{2} ).", "### Why Is the Y-Intercept Important?", "Understanding the y-intercept helps interpret real-world relationships modeled by linear functions. It shows the starting value or baseline when no external influence (measured by ( x )) is present. Whether in finance, physics, or data analysis, the y-intercept offers a clear snapshot of initial conditions.", "### Summary", "- The y-intercept occurs at ( x = 0 )\n- To find it, substitute ( x = 0 ) and solve for ( y )\n- In the case ( t = \frac{4(0) + 1}{0 - 2} ), the y-intercept is ( \boxed{\left(0, -\dfrac{1}{2}\right)} )\n- It is a key feature in graphing and interpreting linear equations", "Mastering y-intercepts equips you with a vital tool for analyzing and visualizing linear relationships across mathematics and applied sciences."]