F-BF A Sum of Functions
Using the graphs below, sketch a graph of the function \(s(x) = f(x) + g(x)\).
Solutions
Solution: Graphical solution
Students can create the graph shown below by:
- visually estimating the distance between the graph of \(f\) and the \(x\)-axis at a particular integer value of \(x\), and
- plotting a point this distance above (or below, if the \(f(x)\) value is negative) the graph of \(g\).
Some students may want to use a strip of paper to mark a distance and then use the mark to help them plot the point.
Solution: Numerical solution
Students may also create a chart of approximate values of \(f(x)\) and \(g(x)\) at various \(x\)-values by estimating from the provided graphs. We then add a row of \(s(x)\) values by summing the two rows above. Finally, we plot points of the form \((x, s(x))\) to sketch the graph of \(y=s(x)\).
| \(x\) | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| \(f(x)\approx\) | -1 | -0.5 | 0 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3 |
| \(g(x)\approx\) | 0.2 | 0.4 | 0.8 | 2 | 4 | 2 | 0.8 | 0.4 | 0.2 |
| \(s(x)=f(x) + g(x)\approx\) | -0.8 | -0.1 | 0.8 | 2.5 | 5 | 3.5 | 2.8 | 2.9 | 3.2 |
Commentary
The intent of this problem is to have students think about how function addition works on a fundamental level, so formulas have been omitted on purpose. In the graph shown, \(g(x)=\frac{4}{x^2+1}\). The task may be extended by asking students to sketch the graph of \(d(x)=f(x)-g(x)\).
Although this problem does not ask students to "write a function that describes a relationship between two quantities", it can provide students with understandings preparatory for F.BF.1b. In addition, this task makes use of the reasoning required for F.BF.3.
Source: Hilton Russell