题目

Problem

Figure 1 shows a sketch of the curve with equation $y=f(x)$ , where $x\in\mathbb{R}$ and $f(x)$ is a polynomial.

The curve passes through the origin and touches the $x$ -axis at the point $(3,0)$ .

There is a maximum turning point at $(1,2)$ and a minimum turning point at $(3,0)$ .

On separate diagrams, sketch the curve with equation

(i) $y=3f(2x)$

(3)

(ii) $y=f(-x)-1$

(3)

On each sketch, show clearly the coordinates of

题目中文翻译

图 1 给出了曲线 $y=f(x)$ 的草图，其中 $x\in\mathbb{R}$ ，且 $f(x)$ 为多项式。

该曲线经过原点，并在点 $(3,0)$ 处与 $x$ 轴相切。

曲线在 $(1,2)$ 处有一个极大值点，在 $(3,0)$ 处有一个极小值点。

在分别的图上，画出下列方程所表示曲线的草图：

(i) $y=3f(2x)$

(ii) $y=f(-x)-1$

在每幅草图上清楚标出下列点的坐标：

解答

For

y=3f(2x)

the graph of $y=f(x)$ is transformed as follows:

The original curve crosses the $y$ -axis at

(0,0)

This remains

(0,0)

The maximum turning point

(1,2)

becomes

\left(\frac12,6\right)

The minimum turning point

(3,0)

becomes

\left(\frac32,0\right)

For

y=f(-x)-1

the graph of $y=f(x)$ is reflected in the $y$ -axis, then translated down by $1$ .

The original $y$ -axis crossing

(0,0)

becomes

(0,-1)

The maximum turning point

(1,2)

first reflects to

(-1,2)

then translates down to

(-1,1)

The minimum turning point

(3,0)

first reflects to

(-3,0)

then translates down to

(-3,-1)