Expert: Ahmed Salami - 10/5/2010

Question
I need to find the y-intercept of the line y=2x+b so that the
line is tangent to the parabola y=x^2-6x+6? the answer is -10
but i do not know how to find it, please help!

Answer
Hi Diana,
The slope of the line y = 2x + b is always 2.
The slope function of the parabola y = x² - 6x + 6 is given by
dy/dx = 2x - 6
If the line is to be a tangent to the parabola, then they should have the same slope at the point of contact as well as the same y values.
Now, the slope of the parabola is equal to 2 when
2x - 6 = 2
2x = 8
x = 4
At this point the parabola has a y value of
y = 4² - 6(4) + 6 = 16 - 24 + 6 = -2
The line must also have this y value and so
y = 2x + b
-2 = 2(4) + b
-2 = 8 + b
b = -2 - 8 = -10

Regards