Arrange in ascending order:

$\dfrac{13}{-28}, \dfrac{-23}{42}, \dfrac{-4}{7} \text{ and } \dfrac{-9}{14}$

First write each rational number with a positive denominator, then write each of them with the LCM of the denominators as the common denominator and arrange the numerators in increasing order.

$\dfrac{13}{-28} = \dfrac{13 \times (-1)}{-28 \times (-1)} = \dfrac{-13}{28}$

So, the given rational numbers are $\dfrac{-13}{28}, \dfrac{-23}{42}, \dfrac{-4}{7} \text{ and } \dfrac{-9}{14}$.

Now write each rational number with the LCM of the denominators as the common denominator.

LCM of 28, 42, 7 and 14 = 84

$\Rightarrow \dfrac{-13}{28} = \dfrac{(-13) \times 3}{28 \times 3} = \dfrac{-39}{84} \\[1em] \Rightarrow \dfrac{-23}{42} = \dfrac{(-23) \times 2}{42 \times 2} = \dfrac{-46}{84} \\[1em] \Rightarrow \dfrac{-4}{7} = \dfrac{(-4) \times 12}{7 \times 12} = \dfrac{-48}{84} \\[1em] \Rightarrow \dfrac{-9}{14} = \dfrac{(-9) \times 6}{14 \times 6} = \dfrac{-54}{84}$

As -54 < -48 < -46 < -39

so $\dfrac{-54}{84} \lt \dfrac{-48}{84} \lt \dfrac{-46}{84} \lt \dfrac{-39}{84}$

$\Rightarrow \dfrac{-9}{14} \lt \dfrac{-4}{7} \lt \dfrac{-23}{42} \lt \dfrac{-13}{28}$

Hence, the given rational numbers in ascending order are $\mathbf{\dfrac{-9}{14}, \dfrac{-4}{7}, \dfrac{-23}{42} \text{ and } \dfrac{13}{-28}}$.