In most reduced form, what is the x-value where the tangent to the graph of y = 3x² - 4x + 1 has a slope of -12?

To find the x-value where the tangent to the graph of the function ( y = 3x^2 - 4x + 1 ) has a slope of -12, we first need to determine the derivative of the function. The derivative represents the slope of the tangent line at any point on the curve.

The function ( y = 3x^2 - 4x + 1 ) can be differentiated using basic rules of differentiation. The derivative is calculated as follows:

[

\frac{dy}{dx} = 6x - 4

]

To find where the slope is -12, we set the derivative equal to -12:

[

6x - 4 = -12

]

Now, we solve for ( x ):

1. Add 4 to both sides:

[

6x = -12 + 4

]

[

6x = -8

]

2. Divide both sides by 6:

[

x = \frac{-8}{6} = -\frac{4}{3}

]

This result means that the x-value where the tangent line to the curve has a slope