Mathematics · Ch 15 — Functions

Power Functions: Square and Cube Function

15.1.5.2

Power Functions: Square and Cube Function

3. Power functions. Form: $f(x)=ax^n$ , $n\in N$ (a multiple of the $n$ -th power of $x$ ).

(i) Square function. Example: $f(x)=x^2$ (Fig. 6.19). Domain: $R$ ; Range: $[0,\infty)$ .

Properties: (1) The graph of $f(x)=x^2$ is a parabola opening upward with vertex at the origin. (2) The graph is symmetric about the $y$ -axis. (3) The graph of any even power of $x$ looks similar to the square function, e.g. $x^4,x^6$ (verify!). (4) $(y-k)=(x-h)^2$ represents a parabola with vertex at $(h,k)$ . (5) If $-2\le x\le2$ then $0\le x^2\le4$ , and if $-3\le x\le2$ then $0\le x^2\le9$ (see the figure). …

Figure 1Fig. 6.19 — graph of the square function f(x)=x^2

What this figure shows. A symmetric upward-opening parabola with its lowest point (vertex) at the origin (0,0), passing through points such as (-2,4), (-1,1), (1,1), (2,4), mirror-symmetric about the y-axis. Illustrates domain R and range [0,infinity) — the curve never dips below the x-axis. …

Figure 2Fig. 6.20 — graph of the cube function f(x)=x^3

What this figure shows. An S-shaped curve passing through the origin, through (-1,-1) and (1,1) and (2,8) and (-2,-8), flattening briefly as it crosses the origin (an inflection point) and then rising steeply for larger |x|, extending from bottom-left to top-right. Illustrates domain R and range R, and that the cube function is symmetric under 180-degree rotation about the origin (odd symmetry), unlike the square functi …

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