Average Rate of Change

Question

Let h(x)=7x²-7

A) Find the average rate of change from -3 to 7

B) Find an equation of the secant line containing (-3, h(-3)) and (7, h(7))

Answer 1

It's a very handy check in your solution process to make sure both points determine the same equation. If they don't, go back and check your work for a mistake.

Answer 2

A) The average rate of change m from -3 to 7 can be computed as ratio of increments:

m = Δy/Δx = (h(7)-h(-3))/(7-(-3)) = 28

B) m is also the slope of the secant line through (-3, h(-3)) and (7, h(7)). The most general equation of a line is:

y = mx + q

We know m = 28 from (A); also, both (-3, h(-3)) and (7, h(7)) must satisfy the equation. We can use that to determine q and, hence, the secant line equation uniquely.

h(-3) = 28 * (-3) + q. ⇒ q = 56 + 84 = 140

So the equation reads: y = 28x + 130. Note that (7, h(7)) also satisfies this equation, as it should.