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# How do the average rates of change for each pair of functions compare over the given interval? f(x) = -2x², g(x) = -4x², -4 ≤ x ≤ -2

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How do the average rate of change for $f(x)=-2\ x^2$ and $g(x)=-4\ x^2$ over the interval $-4\leq x\leq -2$

\begin{align*} f(x) &=-2\ x^2 \\ \\ f(-2) &=-2\cdot(-2)^2=-2\cdot 4=-8 \\ \\ f(-4) &=-2\cdot(-4)^2=-2 \cdot16=-32 \\ \\ g(x) &=-4\ x^2 \\ \\ f(-2) &=-4\cdot (-2)^2=-4 \cdot4=-16 \\ \\ f(-4) &=-4\cdot (-4)^2=-4 \cdot16=-64 \end{align*}

$f(x): \dfrac{-8-(-32)}{-2-(-4)}=\dfrac{-8+32}{-2+4}=\dfrac{24}{2}=12$

$g(x): \dfrac{-16-(-64)}{-2-(-4)}=\dfrac{-16+64}{-2+4}=\dfrac{48}{2}=24$

The rate of change $g$ is twice the rate of change $f$.

The values of function $f$ increase by $12$ units and the values of function $g$ increase by $24$ units for each unit increase in $x$ over the interval $-4\leq x\leq -2$

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