Use the following graph of the function f(x) = 3x^4 − x^3 + 3x^2 + x − 3 to answer this question: what is the average rate of change from x = −1 to x = 0?

a. −6

b. −3

c. 3

d. 6

Step-by-step explanation:

therefore, the slope of the secant line between any two points on this function is 2x + h.  To find the specific rate of change between two given values of x, is a simple matter of substitution.  Let's say we are asked to find the average rate of change between the points x1 = 2 and x2 = 4.  Then in our general answer, we will replace x with x1 and h = x2 - x1.  Replacing these values in the formula yields  2(2) + (4 - 2) = 4 + 2 = 6.  Thus, the slope of the secant line connecting the two points of the function is 6.  Note that the answer is a positive number.  That means what?  That's right, you know!  The line is going uphill or increasing as you look at it from left to right.  Be careful that you put the values for determining h in the correct order.  You already know that slope can be positive, negative or zero.

Now using the same function as above, find the average rate of change between x1 = -1 and x2 = -3.  The answer is 2(-1) + ( -3 + 1) = -2 + -2 = -4.  This means that the secant line is going downhill or decreasing as you look at it from left to right.