We draw the graph of y=4/x, by getting and plotting some points: x|y --|-- ±1|±4 ±2|±2 ±3|±1.3 ±4|±1Now the lines y=mx+4 will all have y intercept (0,4), but will have different slopes. Here are some lines that have the equation y=mx+4 where m has different values Some of those lines intersect the graph in 0 point, 1 points or 2 points. The brownish-green line does not intersect it at all The green and horizontal intersect the graph in 1 point The purple and the light blue lines intersect the curve in two points each. If we solve the system of equations: by substitution, we get What determines whether the line y=mx+4 intersects the curve 0, 1, or 2 times, is whether this quadratic equation has 0,1, or 2 real solutions. And what determines whether a quadratic equation has 0, 1, or 2 real solutions is the DISCRIMINANT . a=m, b=4, c=-4 so the discriminant = = = Case 1: If the discriminant is negative, there will be NO real solutions. So we set the discriminant less than 0 So when m is less than -4, there will be NO solutions. So we set the discriminant less than 0 So when m is less than -1, there will be NO solutions. and therefore the line will NOT intersect the curve at all. Case 2: If the discriminant is EQUAL to zero, there will be ONE real solution, and therefore the line will intersect the curve ONE time. So we set the discriminant EQUAL to 0 So when m is equal to -1, there will be ONE real solutions. and therefore the line will intersect the curve ONE time. Case 3: If the discriminant is positive, there will be TWO real solutions. So we set the discriminant GREATER than 0 So when m is greater than -1, there will be TWO solutions, and therefore the line will intersect the curve at TWICE. Answer: If the slope m < -1, the line y=mx+4 will intersect the curve 0 times. If the slope m = -1, the line y=mx+4 will intersect the curve 1 time. If the slope m > -1, the line y=mx+4 will intersect the curve 2 times. Edwin