Question

May someone please explain how to complete this problem? (See attached image)

Answer 1

a) The derivative of a function `f(x)` at `x = a` represents the slope of the tangent at `f(a)`. On the graph, when `x = 3`, `y= 4` so the correct answer is V.

b) The rate of change is essentially the slope, so the same concepts as in a) are to be used here. On the graph, when `x = -1`, `y = 1`, so the correct answer is II.

c) A horizontal tangent line will have a slope of `0`. Since the slope is the y-value, we are looking for the x-intercepts of the graph, which are at `x = -4` and `x = 0`. The answer is III.

d) Repeat the same process as in c). I'll leave this one to you ;).

e) To find the equation of a line, you need a point `(x_1, y_1)` and the slope. In the graph, when `x= -5`, `y = -2`, so `-2` is our slope. Our point is `(-5, 1)`. By point-slope form, we have:

`y - y_1 = m(x - x_1)`

`y - 1 = -2(x - (-5))`

`y -1 = -2(x + 5)`

The correct answer is II.

Hopefully this helps!