Final answer:
To graph the equation 6y - 9x - 12 = 0, solve for y to get y = \( \frac{3}{2} \)x + 2, plot the y-intercept (0,2), and use the slope to find another point. Draw a straight line through these points. For the point-slope form, use the y-intercept to write y - 2 = \( \frac{3}{2} \)(x - 0).
Step-by-step explanation:
To graph the linear function 6y - 9x - 12 = 0 and write it in point-slope form, we must first solve for y in terms of x to find the slope-intercept form, which is y = mx + b where m is the slope and b is the y-intercept.
Rearranging the equation, we get:
- Add 9x to both sides: 6y = 9x + 12
- Divide by 6: y = \( \frac{3}{2} \)x + 2
The slope of the line (m) is \( \frac{3}{2} \) and the y-intercept (b) is 2. Plotting the y-intercept at (0, 2) on the y-axis, and using the slope, we rise 3 units and run 2 units to find another point on the line. Connect these points with a straight line to graph the function.
To write the equation in point-slope form, we choose a point on the line, which could be the y-intercept (0, 2). The point-slope form is:
y - y1 = m(x - x1)
Plugging in our point and slope we get:
y - 2 = \( \frac{3}{2} \)(x - 0)
This is the point-slope form of the equation, representing the same line as the original function but in a different format. It shows how a change in x-value corresponds to a change in y-value based on the slope of the line.