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Tangent Lines
A line tangent to a parabola must intersect the parabola at exactly one point, as shown in the accompanying figure. Of course, it is impossible to draw a graph that really looks like there is one point of intersection. However, if we find a tangent line algebraically, then we can he sure that it intersects only once.
a) Consider y = x2 and the point (3, 9). If y – 9 = m(x – 3) is tangent to the parabola at (3, 9), then the system
y = x2
y – 9 = m(x – 3)
must have exactly one solution. For what value of m docs the system have exactly one solution?
b) Graph the parabola and the tangent line found in part (a) on a graphing calculator and then use the intersect feature of the calculator to find the point of intersection.
c) There is another line that intersects y = x2 at (3, 9) and only at (3, 9). What is it? Why did it not appear in part (a)?
d) Use the same reasoning as used in part (a) to find the slope of the tangent line to y = x2 at (x1, y1).
e) Use the result of part (d) to find the equation of the tangent line to y = x2 at (–2.5, 6.25).
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