graph $y = \\frac{3}{2}x - \\frac{1}{2}$.

Explanation:

Step1: Identify the slope and y-intercept

The equation is in slope - intercept form (y = mx + b), where (m) is the slope and (b) is the y - intercept. For (y=\frac{3}{2}x-\frac{1}{2}), the slope (m = \frac{3}{2}) and the y - intercept (b=-\frac{1}{2}).

Step2: Plot the y - intercept

The y - intercept is (b =-\frac{1}{2}), so we plot the point ((0,-\frac{1}{2})) on the y - axis.

Step3: Use the slope to find another point

The slope (m=\frac{3}{2}=\frac{\text{rise}}{\text{run}}). From the point ((0,-\frac{1}{2})), we rise 3 units (upwards) and run 2 units (to the right). So we move from ((0,-\frac{1}{2})) up 3 units to (y=-\frac{1}{2}+ 3=\frac{5}{2}) and right 2 units to (x = 0 + 2=2). So the new point is ((2,\frac{5}{2})) or ((2,2.5)). We can also go down 3 units and left 2 units from the y - intercept. From ((0,-\frac{1}{2})), down 3 units: (y=-\frac{1}{2}-3=-\frac{7}{2}), left 2 units: (x = 0-2=-2), so the point is ((-2,-\frac{7}{2})) or ((-2, - 3.5)).

Step4: Draw the line

Draw a straight line through the two (or more) points we have plotted (e.g., ((0,-\frac{1}{2})) and ((2,\frac{5}{2}))).

Answer:

To graph (y=\frac{3}{2}x-\frac{1}{2}), plot the y - intercept ((0,-\frac{1}{2})) and use the slope (\frac{3}{2}) to find another point (e.g., ((2,\frac{5}{2}))) and draw a line through these points. The line has a positive slope, crosses the y - axis at ((0,-\frac{1}{2})) and passes through points like ((2,2.5)) and ((-2,-3.5)) when using the slope to find additional points.