## Shifting Graphs

FIGURE 2.25 The graphs of h = f(y - c) and T2 = f{y) - d.

FIGURE 2.25 The graphs of h = f(y - c) and T2 = f{y) - d.

d, then y—d is the reduced taxable income, and so the tax liability is f{y—d). By shifting the graph of the original tax function d units to the right, we obtain the graph of T\ = f(y —d)? The graph of T^ = /(>•) — c is obtained by lowering the graph of T — /(>) by c units. The income y* that gives the same tax under the two different schemes is the value of >• satisfying the equation

1. Determine the domain on which each of the following equations defines y as a function of x:

This value of y is marked y* in the figure.

Problems a. y = x + 2 b. y = ±-/x c. y = xA d. yA = x e. x2 - y2 = 1 f. y = g. j3 = * h. + y3 = 1

2. The graph of the function / is given in Fig. 2.26.

FIGURE 2.26

FIGURE 2.26

x

2As an example: >• = x~ is a parabola, whereas y = (x - I)2 is a parabola obtained by shifting the first parabola 1 unit to the right.

a. Find /(—5), /(-3), /(-2), /(0), /(3), and /(4) by examining the graph.

b. Find the domain and the range of /.

3. Explain how to get the graphs of the four functions defined by [2.4] based on the graph of y = f{x).

4. Use the rules obtained in Problem 3 to sketch the graphs of the following: