Final answer:
To create graphs of linear relationships, use the slope-intercept form y = b + mx, choosing various y-intercepts. For slopes 1/5, 3/5, 6/5, different y-intercepts provide equations like y = 2 + (1/5)x, y = 4 + (3/5)x, and y = 6 + (6/5)x. Plot these on the same graph to see the impact of these parameters on the line's position and steepness.
Step-by-step explanation:
To graph linear relationships with different slopes and y-intercepts, we use the slope-intercept form of a straight line, which is:
y = b + mx
Where b is the y-intercept and m is the slope. The slopes provided are 1/5, 3/5, and 6/5. For each slope, we can choose different y-intercepts, say b1, b2, and b3, making each equation unique. For instance:
- For a slope of 1/5, let's choose a y-intercept of 2: y = 2 + (1/5)x
- For a slope of 3/5, let's select a y-intercept of 4: y = 4 + (3/5)x
- For a slope of 6/5, we'll use a y-intercept of 6: y = 6 + (6/5)x
Each of these equations can be graphed on the same coordinate plane, with x on the horizontal axis and y on the vertical axis. All lines will have different starting points on the y-axis (their y-intercepts). The slope determines how steep the line is and the direction it tilts; positive slopes tilt upwards as you move right, and negative slopes tilt downwards.
The process of plotting each line involves finding two or more points for each line by plugging in values for x and solving for y. Then connect the points to form the straight lines. As a comparison with your reference figure, a line with a slope of 3 would rise 3 units for every 1 unit it moves to the right.
By graphing the lines, one can visualize how changes in the slope or y-intercept affect the position and steepness of the line.