Question

Match each function to the equation of the tangent line to its graph at x = 2.

Answer 1

1) The function is:

$f(x) = {x}^{2} + 6x - 16$

At x=2,

$f(2) = {2}^{2} + 6(2) - 16 \\ f(2) = 4 + 12 - 16 = 0$

Find the first derivative,

$f'(x) = 2x + 6$

Find the slope at x=2

$f'(2) = 2 \times 2 + 6 = 10$

The equation of the tangent line is given by:

$y - f(2) = f'(2)(x - 2)$

$y - 0= 10(x - 2) \\ y = 10x - 20$

Therefore

$\boxed{f(x) = {x}^{2} - 6x + 16 \to \: y = 10x - 21}$

2) The given function is

$g(x) = {x}^{2} - 49x - 456$

$g(2) = {2}^{2} - 49 \times 2 - 456 = - 550$

$g'(x) = 2x - 49 \\ g'(2) = 2 \times 2 - 49 = - 45$

The equation of the tangent is

$y - g(2) = g'(2)(x - 2) \\ y + 550 = - 45(x - 2) \\ y + 550 = - 45x + 90 \\ y = - 45x + 90 - 550 \\ y = - 45x - 460$

$\boxed{g(x) = {x}^{2} - 45x - 456 \to \: y = - 45x - 460}$

3) The function is

$h(x) = - {x}^{2} - 7x + 44$

Now

$h(2) = - {2}^{2} - 7 \times 2 + 44 = 26$

The first derivative is

$h'(x) = - 2x - 7$

$h'(2) = - 2 \times 2 - 7 = - 14$

The equation of tangent at:

x=2

$y - h(2) = h'(2)(x - 2) \\ y - 26 = - 14(x - 2) \\ y - 26 = - 14x + 28 \\ y = - 14x + 28 + 26 \\ y = - 14x + 54$

$\boxed{h(x) = - {x}^{2} - 7x + 44 \to \: y = - 14x + 54}$

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