Example 11 from Algebra and Trigonometry, 3.5 Transformation of Functions

Applying a Learning Model Equation

A common model for learning has an equation similar to $k (t) = - 2^{- t} + 1,$ where $k$ is the percentage of mastery that can be achieved after $t$ practice sessions. This is a transformation of the function $f (t) = 2^{t}$ shown in Figure 15. Sketch a graph of $k (t) .$

Figure 15

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.

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Solution
This equation combines three transformations into one equation.
- A horizontal reflection: $f (− t) = 2^{- t}$
- A vertical reflection: $- f (− t) = - 2^{- t}$
- A vertical shift: $- f (− t) + 1 = - 2^{- t} + 1$
We can sketch a graph by applying these transformations one at a time to the original function. Let us follow two points through each of the three transformations. We will choose the points (0, 1) and (1, 2).
1. First, we apply a horizontal reflection: (0, 1) (–1, 2).
2. Then, we apply a vertical reflection: (0, -1) (-1, –2)
3. Finally, we apply a vertical shift: (0, 0) (-1, -1)).
This means that the original points, (0,1) and (1,2) become (0,0) and (-1,-1) after we apply the transformations.
In Figure 16, the first graph results from a horizontal reflection. The second results from a vertical reflection. The third results from a vertical shift up 1 unit.

Figure 16

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Given the toolkit function $f (x) = x^{2},$ graph $g (x) = - f (x)$ and $h (x) = f (- x) .$ Take note of any surprising behavior for these functions.

How did it go?

Solution

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Solution

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