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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.
The activation energy diagram is drawn as a hill because there is a large amount of energy needed to form the unstable transition state (viewed as the upward slope on the diagram). The difference between this peak energy and the rectants' energy is the activation energy.