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Quantum entanglement and correlations in the spin-1 Heisenberg chain with single-ion anisotropy are investigated using the quantum renormalization group method. Negativity and quantum discord (QD) are calculated with various anisotropy parameters $\bigtriangleup$ and single-ion anisotropy parameters $D$. We focus on the relations between two abovementioned physical quantities and on transitions between the Néel, Haldane, and Large-D phases. It is found that both negativity and QD exhibit step-like patterns in different phases as the size of the system increases. Interestingly, the single-ion anisotropy parameter $D$, which can be modulated using nuclear electric resonance (2020 \textit{Nature} \textbf{579} 205), plays an important role in tuning the quantum phase transition (QPT) of the system. Both the first partial derivative of the negativity and quantum discord with respect to $D$ or $\bigtriangleup$ exhibit nonanalytic behavior at the phase transition points, which corresponds directly to the divergence of the correlation length. The quantum correlation critical exponents derived from negativity and QD are equal, and are the reciprocal of the correlation length exponent at each critical point. This work extends the application of quantum entanglement and correlations as tools for depicting QPTs in spin-1 systems.