Question

Find an equation of the tangent to the curve f(x) = 2x2 - 2x + 1 that has slope 2

Answer 1

Answer:

D. y = 2x + 1

Step-by-step explanation:

The tangent to the curve has one point in common with the curve.

The slope-intercept form of an equation of a line:

$y=mx+b$

$m$ - slope

$b$ - y-intercept

We have the slope $m=2$.

$y=2x+b$

Therefore we have the system of equations:

$\left\{\begin{array}{ccc}y=2x^2-2x+1&(1)\\y=2x+b&(2)\end{array}\right$

substitute (1) to (2):

$2x^2-2x+1=2x+b$        subtract 2x from both sides

$2x^2-4x+1=b$         subtract b from both sides

$2x^2-4x+1-b=0$

Use the discriminant of a quadratic equation:

$ax^2+bx+c=0\to \Delta=b^2-4ac$

If Δ = 0, then we have one common point.

Calculate:

$2x^2-4x+1-b=0\\\\\Delta=(-4)^2-4\cdot2\cdot(1-b)=16-8+8b=8-8b$

$\Delta=0\iff8-8b=0$   subtract 8 from both sides

$-8b=-8$        divide both sides by (-8)

$b=1$

Finally:

$y=2x+1$