At what points on the graph of y=x2 does the tangent line pass through (3, -7)?

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Jaden Petersen · Accepted Answer

The points are (-1, 1) and (7, 49).Method: Find the general form of the equation of a line tangent to the graph ofy=x2. Then find the particular points that satisfy the condition: the tangent line passes through (3, -7).I will continue to use x and y as variables.Consider a particular value of x, calls it a.The point an the graph at x=a has coordinates (a,a2) (The y-coordinate must satisfy y=x2 in order to be on the graph.)The slope of the tangent to the graph is determined by differentiating: y&#039;=2x, so at the point (a, a2) the slope of the tangent is m=2a.Use your favorite tehnique to find the equation of the line through (a, a2) with slope 2a.The tangent line has equation: y=2ax−a2.(One way to find the line: start with y−a2=2a(x−a), so y−a2=2ax−2a2.Add a2 to both sides to get y=2ax−a2.)We have been asked to make the point (3, -7) lie on the line. So we need,(−7)=2a(3)−a2. Now, solve for a−7=6a−a2 if and only if a2−6x−7=0.Solve by factoring: (a+1)(a-7)=0, which requires a=-1 or a=7.The points we are looking for, then, are (-1, 1) and (7, 49).You can now check the answers by verifying that the point (3, -7) lies on the lines:y=-2x-1 (the tangent when a=-1),and also on y=14x-49 (the tangent when a=7).

At what points on the graph of y=x^{2} does the

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