6.1 VERTICAL AND HORIZONTAL SHIFTS 1

Vertical Shift: The Heating Schedule for an Office Building Example 1 To save money, an office building is kept warm only during business hours. At midnight (t = 0), the building’s temperature (H) is 50 ◦ F. This temperature is maintained until 4 am. Then the building begins to warm up so that by 8 am the temperature is 70 ◦ F. At 4 pm the building begins to cool. By 8 pm, the temperature is again 50 ◦ F. Suppose that the building’s superintendent decides to keep the building 5 ◦ F warmer than before. Graph of the original heating schedule Graph of the new heating schedule obtained by shifting original graph upward by 5 units. t (hours after midnight) t H( ◦ F) H = f(t) H = p(t), new schedule H = f(t), original schedule 2

Vertical Shift: The Heating Schedule for an Office Building Example 2 What is the relationship between the formula for f(t), the original heating schedule and p(t), the new heating schedule? Solution 3

Vertical Shift If g(x) is a function and k is a positive constant, then the graph of y = g(x) + k is the graph of y = g(x) shifted vertically upward by k units. y = g(x) − k is the graph of y = g(x) shifted vertically downward by k units. 4

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Vertical Shift: The Heating Schedule for an Office Building Example 4 The superintendent then changes the original heating schedule to start two hours earlier. The building now begins to warm at 2 am instead of 4 am, reaches 70 ◦ F at 6 am instead of 8 am, begins cooling off at 2 pm instead of 4 pm, and returns to 50 ◦ F at 6 pm instead of 8 pm. How are these changes reflected in the graph of the heating schedule? Graph of the new heating schedule obtained by shifting original graph left by 2 units. t (hours after midnight) H( ◦ F) H = q(t), new schedule H = f(t), original schedule 7

Vertical Shift: The Heating Schedule for an Office Building Example 5 In Example 4 the heating schedule was changed to 2 hours earlier, shifting the graph horizontally 2 units to the left. Find a formula for q, this new schedule, in terms of f, the original schedule. Solution 8

Horizontal Shift If g(x) is a function and k is a positive constant, then the graph of y = g(x + k) is the graph of y = g(x) shifted horizontally to the left by k units. y = g(x − k) is the graph of y = g(x) shifted horizontally to the right by k units. 9

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Inside Versus Outside Changes Since the horizontal shift in the heating schedule, q(t) = f(t + 2), involves a change to the input value, it is called an inside change to f. Similarly the vertical shift, p(t) = f(t) + 5, is called an outside change because it involves changes to the output value. 11

Combining Horizontal and Vertical Shifts Example 9 A graph of f(x) = x 2 is shown in blue. Define g by shifting the graph of f to the right 2 units and down 1 unit; the graph of g is shown in red. Find a formula for g in terms of f. Find a formula for g in terms of x. Solution y = f(x) y = g(x) g(x) = f(x − 2) − 1. Since f(x) = x 2, we have f(x − 2) = (x − 2) 2. Therefore, g(x) = (x − 2) 2 − 1. 12