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It is false that in an energy diagram for a chemical reaction, energy is plotted along the x-axis (the horizontal axis) and reaction progress is plotted along the y-axis (the vertical axis). The energy is actually plotted along the y-axis and the reaction along the x-axis.
In the case of an exothermic reaction, the reactants are at a higher energy level as compared to the products, as shown below in the energy diagram. In other words, the products are more stable than the reactants. Overall for the reaction is negative, i.e., energy is released in the form of heat.