Question

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.