Be the Change

Learning Focus

Find and interpret average rates of change for quadratic functions.

Big idea

Average rate of change of $f$ from $x=a$ to $x=b$:

$\frac{f(b)-f(a)}{b-a}.$

For a linear function this is constant. For a quadratic it depends on the interval. For $f(x)=x^2$ from $a$ to $a+h$:

$\frac{(a+h)^2-a^2}{h}=\frac{2ah+h^2}{h}=2a+h.$

Worked example

$f(x)=x^2$ from $x=1$ to $x=4$:

$\frac{16-1}{3}=5.$

(Check with $2a+h=2(1)+3=5$.)

Takeaways

• Average rate of change is the slope of the secant line.
• Quadratics have non-constant rates of change.
• A rate of $0$ over an interval means the function returns to the same value (net change zero).