Determine the average rate of change for the function between the values given:
\[ f(x) = -x-x^2\text{ for }x=-5\text{ and }x=a\text{ where }a>-5 \]
Solution
To determine the average rate of change, it helps to first calculate the net change. Since \(a\gt -5\), the net change will be \(f(a) - f(-5)\):
\[\begin{{array}}{{rcl}} f(a) - f(-5) &=&\left(-a-a^2\right)-\left(-(-5)-(-5)^2\right)\\ f(a) - f(-5) &=&-a-a^2-(5-25)\\ f(a) - f(-5) &=&-a^2-a+20 \end{{array}}\]Next we determine the horizontal change: \(a - (-5) = a+5\). Finally, we calculate the average rate of change by dividing the net change by the horizontal change:
\[ \frac{-a^2-a+20}{a+5} \]This, it turns out, be simplified:
\[ \solve{ \dfrac{-a^2-a+20}{a+5}&=&-\dfrac{a^2+a-20}{a+5}\\&=&-\dfrac{(a+5)(a-4)}{a+5}\\&=&-(a-4)\\&=-a+4 } \]As a reminder, the average rate of change between two points on a function is the same thing as the slope between the line connecting those points. Using the point \((-5, -20)\) on the function and the average rate of change as the slope, we can use a program like Desmos to create a graph, with a slider for different \(a\) values to see how it works:
\[ \text{Equation of line: } y =(-a+4)\left(x+5\right)-20 \]