Find the equations of the two tangent lines to the graph of f that pass through the indicated point

Question

Find the equations of the two tangent lines to the graph of f that pass through the indicated point

f(x) = 4x − x²

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line with positive slope, y =?

line with negative slope, y =?

Answer 1

f'(x) = 4 - 2x = slope = m

directly below the point (2,5) is the vertex of the parabola (2,4)

go down one more unit to (2,3), then left and right 1 unit to get the points (1,3) and (2,3)

connect each of those two points with (2,4) to get two tangent lines

y=2x +1 and y=-2x +9. are the two tangent lines

the 2nd tangent line has a slope which is the negative of the 1st line's slope

y=4x-x^2, slope = y' = 4-2x 1st line has slope y'=4-2(1) = 2, 2nd line has slope y'= 4-2(3) = -2

check the solutions

plug in (3,3) into y=-2x+9

3=-2(3)+-9= 9-6 = 3

plug (1,3) into y=2x+1

3 = 2(1) +1 = 2 + 1 = 3

plug (2,5) into y=2x+1

5 = 2(2) + 1 = 5

plug (2,5) into y= -2x+9

5=-2(2) + 9= 9-4 = 5

Answer 2

(5-3)/(2-3) = (y-3)(x-3) => -2=(y-3)/(x-3)

(5-3)/(2-1) = (y--3)(x-1) => 2=(y-3)/(x-3)