Question

average rate of change for f(x) = x2 − 3x − 10 from x = −5 to x = 10.

Answer 1

The average rate of change is the slope of the secant line connecting the points on f(x) from x=-5 to x=10.

In other words, it is just the slope of the straight line connecting the points
$(-5,f(-5)) \: and \: (10,f(10))$

So we were given ,

$f(x) = {x}^{2} - 3x - 10$

At
$x = - 5$
$f( - 5) = {( -5 )}^{2} - 3( - 5) - 10$
$f( - 5) = 25 + 15- 10$
$f( - 5) = 30$

Also at
$x = 10$
$f( 10) = {( 10)}^{2} - 3( 10) - 10.$
$f( 10) = 100- 30 - 10$
$f( 10) = 60$

The average rate of change
$= \frac{f(10) - f( - 5)}{10 - - 5}$
$= \frac{60- 30}{15}$
$= \frac{30}{15}$
$= 2$