Question

find the average rate of change of f(x)=3x-9x+1 from x=1 to x=3 and the equation of the secant line joining the points (1,f(1)) and (3,f(3))

Answer 1

The average rate of change of the function f(x) from x=1 to x=3 is -6, and the equation of the secant line for those points is y = -6x + 1.

Step-by-step explanation:

To find the average rate of change of the function f(x) = 3x - 9x + 1, we need to evaluate the function at the given points and then use the formula for the average rate of change:

Average Rate of Change = (f(b) - f(a))/(b - a).

First, we find f(1) and f(3):

• f(1) = 3(1) - 9(1) + 1 = -5
• f(3) = 3(3) - 9(3) + 1 = -17

Then, the average rate of change from x = 1 to x = 3 is:

(-17 - (-5))/(3 - 1) = -12/2 = -6.

To find the equation of the secant line, we need two points. We've already found:

• (1, f(1)) = (1, -5)
• (3, f(3)) = (3, -17)

The slope of the secant line is the average rate of change, which is -6. Now, we can use the point-slope form of a line y - y1 = m(x - x1) with one of our points, (1, -5), and our slope, -6:

y - (-5) = -6(x - 1)

y + 5 = -6x + 6

y = -6x + 1 (equation of the secant line).