## Calculus AB and Calculus BC

## CHAPTER 1 Functions

### B. SPECIAL FUNCTIONS

The *absolute-value* function *f* (*x*) = |*x*| and the *greatest-integer* function *g*(*x*) = [*x*] are sketched in Figure N1–3.

**FIGURE N1–3**

**EXAMPLE 8**

A function *f* is defined on the interval [−2, 2] and has the graph shown in Figure N1–4.

(a) Sketch the graph of *y* = |*f* (*x*)|.

(b) Sketch the graph of *y* = *f* (|*x*|).

(c) Sketch the graph of *y* = − *f* (*x*).

(d) Sketch the graph of *y* = *f* (−*x*).

**FIGURE N1–4**

**SOLUTIONS:** The graphs are shown in Figures N1–4a through N1–4d.

**FIGURE N1–4a**

**FIGURE N1–4b**

**FIGURE N1–4c**

**FIGURE N1–4d**

Note that graph (c) of *y* = − *f* (*x*) is the reflection of *y* = *f* (*x*) in the *x*-axis, whereas graph (d) of *y* = *f* (−*x*) is the reflection of *y* = *f* (*x*) in the *y*-axis. How do the graphs of |*f* (*x*)| and *f* (|*x*|) compare with the graph of *f* (*x*)?

**EXAMPLE 9**

Let *f* (*x*) = *x*^{3} − 3 *x*^{2} + 2. Graph the following functions on your calculator in the window [−3,3] × [−3,3]: (a) *y* = *f* (*x*); (b) *y* = |*f* (*x*)|; (c) *y* = *f* (|*x*|).

**SOLUTIONS:**

**(a)** *y* = *f* (*x*)

See Figure N1–5a.

**FIGURE N1–5a**

**(b)** *y* = |*f* (*x*)|

See Figure N1–5b.

**FIGURE N1–5b**

**(c)** *y* = *f* (|*x*|)

See Figure N1–5c.

**FIGURE N1–5c**

Note how the graphs for (b) and (c) compare with the graph for (a).