Ideas in Geometry/Instructive examples/Section 1.2

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Section 1.2 Problem # 11

Find the line tangent to y = x^2 + 12x - 4 @ x=0

Well to start off, first you need to know that a tangent line to a parabola is a line that passes through the parabola in such a way that parabola(x)=line(x) has a double root. So thus, a tangent line is a line that passes through a curve such that curve(x)=line(x) has a double root.

Formula: p(x)-l(x)=(x-c)^2

Work: Start off by plugging the equation in to the formula. (x^2 + 12x - 4) would be plugged in for p(x) and 0 would be c. So the equation would be...(x^2+12x-4)-l(x)=(x-0)^2

This then is reduced down to... (x^2+12x-4)-l(x)=x^2

Now you can add l(x) to both sides to try to get it alone. (x^2+12x-4)=x^2+l(x)

From here, you want to subtract x^2 from both sides. 12x-4=l(x)

Answer: 12x-4