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Average rate of change and the equation of a secant line

The rule: average rate of change is a secant slope

For a function $f$ on $[a, b]$ , the average rate of change is $(f (b) - f (a)) / (b - a)$ . This is exactly the slope of the line through $(a, f (a))$ and $(b, f (b))$ . Once the slope is known, use point-slope form to write the secant line.

Problem

Let $h (x) = x^{2} - 5 x$ . (a) Find the average rate of change from $x = 3$ to $x = 4$ . (b) Find the equation of the secant line through $(3, h (3))$ and $(4, h (4))$ .

Answer

The average rate of change is

2

, and the secant line is

y = 2 x - 12

The line equation was checked at both endpoints of the interval.

Step-by-step solution

Evaluate the endpoints

First calculate the two function values:

h (3) = 3^{2} - 5 (3) = 9 - 15 = - 6

h (4) = 4^{2} - 5 (4) = 16 - 20 = - 4

The two points on the graph are therefore $(3, - 6)$ and $(4, - 4)$ .

Find the average rate of change

Apply the difference quotient:

\frac{h ( 4 ) - h ( 3 )}{4 - 3} = \frac{- 4 - ( - 6 )}{1} = 2

So the average rate of change is $2$ .

Write the secant line

The secant has slope $2$ and passes through $(3, - 6)$ . Point-slope form gives

y - (- 6) = 2 (x - 3)

Simplify:

y + 6 = 2 x - 6

y = 2 x - 12

How to check yourself

Insert $x = 3$ and $x = 4$ into the line. It returns $- 6$ and $- 4$ , exactly the same endpoint values as the function. If either point fails, the slope or the algebraic simplification is wrong.

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