Final answer:
To plot the bug's positions with lines and find the equations for those lines, use the given equations and their points on the coordinate axes. Connect the points on each line to form the required triangle. Then, write the equation for the line connecting the two given points.
Step-by-step explanation:
To plot the bug's positions on the coordinate axes with lines H_1 and H_2, we can use the given equations and their corresponding points. Let's start with part a. The equation for line H_1 is y = 2x - 3 and for line H_2, it is y = -x + 4. To find the points on these lines, we can choose any x-coordinate and substitute it into the equation to get the corresponding y-coordinate.
For example, for line H_1, when x = 0, y = 2(0) - 3 = -3, so one point on line H_1 is (0, -3). Repeat this process to find more points on both lines. Then, connect the points on the two lines with a line. Finally, for the line connecting them, the equation is given as y = x - 1.
Similarly, for parts b, c, and d, apply the same process. The equations for lines H_1 and H_2 are provided, along with their corresponding points. Find additional points on each line by substituting different x-coordinates into the equations. Connect the points from H_1 and H_2 with lines to form a triangle. Finally, write the equation for the line connecting the two given points. For example, for part b, the equation for the line connecting the two points is y = 3x + 2.