Example 1 from Algebra and Trigonometry, 3.5 Transformation of Functions

Adding a Constant to a Function

To regulate temperature in a green building, airflow vents near the roof open and close throughout the day. Figure 3 shows the area of open vents $V$ (in square feet) throughout the day in hours after midnight, $t .$ During the summer, the facilities manager decides to try to better regulate temperature by increasing the amount of open vents by 20 square feet throughout the day and night. Sketch a graph of this new function.

A blue graph is shown on a set of t and v axes. The scale is minus four to plus twenty-four for t and minus four to three hundred for v. The graph lies along the t axis from the origin to eight, then rises as a straight line to ten, two hundred twenty. Then it is a horizontal straight line to seventeen, two hundred twenty. It then is a straight line to the t axis at nineteen. It is then a straight line along the t axis until t equals twenty four.

Figure 3

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
We can sketch a graph of this new function by adding 20 to each of the output values of the original function. This will have the effect of shifting the graph vertically up, as shown in Figure 4.

Figure 4

Notice that in Figure 4, for each input value, the output value has increased by 20, so if we call the new function $S (t),$ we could write

$S (t) = V (t) + 20$
This notation tells us that, for any value of $t, S (t)$ can be found by evaluating the function $V$ at the same input and then adding 20 to the result. This defines $S$ as a transformation of the function $V,$ in this case a vertical shift up 20 units. Notice that, with a vertical shift, the input values stay the same and only the output values change. See Table 1.

$t$ 0 8 10 17 19 24

$V (t)$ 0 0 220 220 0 0

$S (t)$ 20 20 240 240 20 20

Table 1

How did it go?

$t$	0	8	10	17	19	24
$V (t)$	0	0	220	220	0	0
$S (t)$	20	20	240	240	20	20

Solution

Solution