3.5 Transformation of functions  (Page 2/21)

The formula $g(x)=f(x)-3$ tells us that we can find the output values of $g$ by subtracting 3 from the output values of $f.$ For example:

Subtracting 3 from each $f\left(x\right)$ value, we can complete a table of values for $g\left(x\right)$ as shown in [link] .

We can set $V\left(t\right)$ to be the original program and $F\left(t\right)$ to be the revised program.

In the new graph, at each time, the airflow is the same as the original function $V$ was 2 hours later. For example, in the original function $V,$ the airflow starts to change at 8 a.m., whereas for the function $F,$ the airflow starts to change at 6 a.m. The comparable function values are $V(8)=F(6).$ See [link] . Notice also that the vents first opened to $220{\text{ ft}}^2$ at 10 a.m. under the original plan, while under the new plan the vents reach $220{\text{ ft}}^{\text{2}}$ at
8 a.m., so $V(10)=F(8).$

In both cases, we see that, because $F\left(t\right)$ starts 2 hours sooner, $h=-2.$ That means that the same output values are reached when $F(t)=V(t-\left(-2\right))=V\left(t+2\right).$