Mechanics and Techniques Problems

11.1. This problem deals with what happens to equations of functions and graphs of functions if you apply several different transformations, one after the other. Take y = f(x) = x² and write out each of the following functions. For each step, explain what happens to the graph in terms of how the particular change affects the appearance of the previous graph in the sequence.


Function	Written out	What happens to graph

y = f(x) = x²


y = f(x - h)


y = af(x - h)


y = af(x - h) + k

11.2. How would the results in problem 1 be different if we changed the order to y = a[f(x - h) + k]?


Function	Written out	What happens to graph

y = f(x) = x²


y = f(x - h)


y = f(x - h) + k


y = a[f(x - h) + k]

11.3. Now repeat 1 with a basic exponential function.


Function	Written out	What happens to graph

y = f(x) = exp(x)


y = f(x - h)


y = af(x - h)


y = af(x - h) + k

11.4. Now repeat 1 with a basic logarithmic function.


Function	Written out	What happens to graph

y = f(x) = ln(x)


y = f(x - h)


y = af(x - h)


y = af(x - h) + k

11.5. For each of the five graphs below

Select the best basic function to fit the data,
Select appropriate shifts (direction) and scaling (stretch or compress), and
Write down a possible equation for the graph.

Figure 11.18: Graphs for problem 5.

11.6. Consider the data shown below in both table and graphical format.


x	y

0	2.05
5	2.69
10	3.55
15	4.23
20	4.35
24	5.08

Figure 11.19: Graph of the data from problem 6.

Create a scatterplot of the data, and determine which order polynomial function (2 through 6) fits the data best. Record the results of your investigation in a table like the one shown below.


Order	Equation	R²

2


3


4


5


6

Use the parameters from your quadratic trendline (the order 2 polynomial) to manually calculate the S_e for that model. You may want to set up a spreadsheet like the one in figure 11.20.

Figure 11.20: Set up to calculate R² and S_e in problem 6.

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