`y = (x-2)(x^2 + 3x), (1,-4)` (a) Find an equation of the tangent line to the graph of f at the given point, (b) use a graphing utility to graph the function and its tangent line at the point, and (c) use the derivative feature of a graphing utility to confirm your results.
(a) Take the derivative. Use the product rule.
Given the point `(1,-4)` , substitute `x=1` .
`y'(1)= (1-2)(2(1)+3)+(1^2+3(1)) = -1(5)+(4)=-1`
The slope at is -1.
Write the slope intercept form, substitute the point and the slope to find b.
The equation of the tangent line is: `y=-x-3`
(b) Graph: This is only the original function... The graph will not allow me to plot y=-x-3 to add on this graph, but this function should show the tangent line on the graph. I have attached another image to the answer.
(c) You can do this on your own. It's the dy/dx function.