If the parabola y = x^2 + 2x + 7 and the line y = 3x + b intersect at only one point, what is the value of b?
+37146 y = x^2 + 2x + 7 is a bowl shaped parabola.....the only ways a line can intersect it in ONE point is if the line is VERTICAL (this one has a slope of 3) ...or if the line is sloped , it must be tangent to the parabola ( a third way would be for the line to be horizontal (slope = 0) and tangent at vertex)
When the line is tangent to the parabola, the parabola will have the same slope as the line ( 3)
find where the slope = 3
derivative of the parabola equation
2x + 2 is the slope at each point x ..... eqaute to 3
2x+2 = 3
x = 1/2 then y = 8.25
so the line has the point (1/2 , 8.25 )
plug these into the line equation
8.25 = 3(1/2) + b b = 6.75
