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Based on the graph, I think it means that you need a lot of free energy added to the reactants to get the reaction started, but once it reaches the peak of the graph (achieves activation energy), it releases a lot of energy, more than what it started out with in the first place (to get the products' free energy below the reactants). This indicates that it is an exergonic reaction.
Alternatively, when the products are at a higher point than the reactants then it is an exergonic reaction, but a large activation energy will mean that the reaction might be slower because the reaction requires more energy for the reaction to take place.
A catalyst weakens the bonds of the species, ensuring that less energy is required to obtain the necessary activation barrier energy. This is how the reaction rates of the forward and reverse reactions are increased.
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