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

Jace Baxter 2022-02-11 Answered
At what points on the graph of y=x2 does the tangent line pass through (3, -7)?
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Expert Answer

Jaden Petersen
Answered 2022-02-12 Author has 15 answers
The points are (-1, 1) and (7, 49).
Method: Find the general form of the equation of a line tangent to the graph of
y=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'=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=2axa2.
(One way to find the line: start with ya2=2a(xa), so ya2=2ax2a2.
Add a2 to both sides to get y=2axa2.)
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=6aa2 if and only if a26x7=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).
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