Answer
a. $y$ $=$ $\frac{3}{4}$$x$ $-$ $\frac{9}{2}$
b. $c$ $=$ $\frac{7}{8}$
c. $y$ $=$ $\frac{3}{4}$$x$ $-$ $\frac{817}{64}$
d. See image
Work Step by Step
(a) Slope
$m = \frac{y_{2} - y_{1}}{x_{2}-x_{1}} = \frac{0 - (-6)}{4 - (-2)} = \frac{3}{4}$
Secant line: $y - y_{1} = m(x -x_{1})$
$y - (-6) = \frac{3}{4}(x - (-2)$
$y = \frac{3}{4} - \frac{9}{2}$
(b)
$f'(x) = 2x - 1$
$f'(x) = m$
$2x - 1 = \frac{3}{4}$
$x = \frac{7}{8}$
$c = \frac{7}{8}$
(c)
$f(c) = f(\frac{7}{8}) = (\frac{7}{8})^{2} - (\frac{7}{8}) - 12 = -\frac{775}{64}$
Tangent point: $(\frac{7}{8}, -\frac{775}{64})$
Tangent line: $y - y_{1} = m(x -x_{1})$
$y - (-\frac{775}{64}) = \frac{3}{4}(x - (\frac{7}{8}))$
$y = \frac{3}{4}x - \frac{817}{64}$
