Identify the translations.

The orange triangle, A, is the pre-image.

The pre-image moved ___ units horizontally to form the purple triangle, B.

The pre-image moved ___ units vertically to form the green triangle, C.

The pre-image moved 4 units vertically to form the green triangle, C.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.