Step-by-step explanation:
First, find the zeroes of the parabola
[tex] {x}^{2} - x - 12[/tex]
[tex] {x}^{2} - 4x + 3x - 12[/tex]
[tex]x(x - 4) + 3(x - 4)[/tex]
[tex](x + 3)(x - 4) = 0[/tex]
[tex]x = - 3[/tex]
[tex]x - 4 = 0[/tex]
[tex]x = 4[/tex]
So the zeroes or where the curve crosses the x axis is at 4 and -3.
Now, we take the derivative of the function.
[tex] \frac{d}{dx} ( {x}^{2} - x - 12) = 2x - 1[/tex]
Plug in -3, and 4 into the derivative function
[tex]2( - 3) = - 7[/tex]
[tex]2(4) - 1 = 7[/tex]
So at x=-3, our slope of the tangent line is -7 and must pass through (-3,0). So we use point slope formula.
[tex]y - y _{1} = m(x - x _{1})[/tex]
[tex]y - 0 = - 7(x - ( - 3)[/tex]
[tex]y = - 7x - 21[/tex]
At x=4, our slope of tangent line is 7, and pass through (4,0) so
[tex]y - 0 = 7(x - 4)[/tex]
[tex]y = 7x - 28[/tex]
So the equations of tangent is
[tex]y = - 7x - 21[/tex]
[tex]y = 7x - 28[/tex]
