Question

match the graphs a - d of functions with graphs a - c of their derivatives. the graph of function a corresponds to the graph of derivative the graph of function b corresponds to the graph of derivative the graph of function c corresponds to the graph of derivative the graph of function d corresponds to the graph of derivative

Explanation:

Step1: Recall derivative - function relationship

The derivative of a function represents the slope of the tangent line to the function. If a function is increasing, its derivative is positive; if decreasing, negative; and at a horizontal - tangent point, the derivative is zero.

Step2: Analyze function a

Function a is an increasing linear function. The slope of a linear function is constant. So, its derivative is a positive constant. Graph C has a positive constant value, so the graph of function a corresponds to the graph of derivative C.

Step3: Analyze function b

Function b is a decreasing linear function. Its slope is negative and constant. So, its derivative is a negative constant. Graph A has a negative constant value, so the graph of function b corresponds to the graph of derivative A.

Step4: Analyze function c

Function c is a horizontal line. The slope of a horizontal line is 0. So, its derivative is 0. Graph B has a value of 0, so the graph of function c corresponds to the graph of derivative B.

Step5: Analyze function d

Function d is first decreasing (negative - slope part), then has a horizontal part (slope = 0), and then is increasing (positive - slope part). Its derivative should be negative first, then 0, and then positive. There is no such graph among A - C for function d.

Answer:

The graph of function a corresponds to the graph of derivative C.
The graph of function b corresponds to the graph of derivative A.
The graph of function c corresponds to the graph of derivative B.