# Quod Erat Demonstrandum

## 2011/04/12

### Transformation of graphs

Filed under: mathematics,NSS — johnmayhk @ 5:59 下午
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$y=x^3$ 的圖象，右移 2 個單位，得出新圖象。問該圖象的代數表達式。

If the point P(a,b) is rotated through $90^o$ anti-clockwise about O to the point Q. What are the coordinates of Q?

Explain why the graph of $y=f(-x)$ is obtained by reflecting the graph of $y=f(x)$ about the y-axis.

For any point $(a,b)$ on the graph of $y=f(x)$,

$b=f(a)$

$b=f(-(-a))$

Hence, $(-a,b)$ lies on the graph of $y=f(-x)$.

Also, $(-a,b)$ is obtained by reflecting $(a,b)$ along the y-axis.

Thus, the graph of $y=f(-x)$ is obtained by reflecting the graph of $y=f(x)$ along the y-axis.

Describe the way of transforming the graph of $y=x^3+x$ to the graph of $y=8x^3+2x$.

The transformation is contracting $\frac{1}{2}$ time along the x-axis.

「收縮 $\frac{1}{2}$ 倍」等同「放大 2 倍」嗎？（否）

「收縮 $\frac{1}{3}$ 倍」是指「原來的 $\frac{1}{3}$」還是「原來的乘以 $(1-\frac{1}{3})$」？

「幸而」，課程不要求諸如 $f(\frac{2x-3}{5})$ 之類的變換，否則後果堪虞。

1.設 $C_1, C_2$ 分別為 $y=\frac{6x^2-24x+27}{3x^2-6x+14}$$y=\frac{1}{2+3x^2}$ 的圖像，試描述經由甚麼幾何變換，可以把 $C_1$ 變成 $C_2$

2.把 $y=f(x)$ 的圖象沿 y 軸反射，得圖像 $C_1$；如果把 $y=f(x)$ 的圖象沿 x 軸右移 2011 個單位，得圖像 $C_2$。已知 $C_1$$C_2$ 的代數表達式是完全一樣，除了 $f(x)\equiv k$$k$ 是常數）外，試給出一個 $f(x)$ 的可能代數式。