题目

Problem

Figure 4 shows a sketch of the graph of $y=g(x)$ , $-3\leqslant x\leqslant4$ and part of the line $l$ with equation $y=\dfrac12x$ .

Figure 4

The graph of $y=g(x)$ consists of three line segments, from $P(-3,4)$ to $Q(0,4)$ , from $Q(0,4)$ to $R(2,0)$ and from $R(2,0)$ to $S(4,10)$ .

The line $l$ intersects $y=g(x)$ at the points $A$ and $B$ as shown in Figure 4.

(a) Use algebra to find the $x$ coordinate of the point $A$ and the $x$ coordinate of the point $B$ . Show each step of your working and give your answers as exact fractions.

(6)

(b) Sketch the graph with equation

\begin{align*} y=\frac32g(x), \qquad -3\leqslant x\leqslant4 \end{align*}

On your sketch show the coordinates of the points to which $P$ , $Q$ , $R$ and $S$ are transformed.

(2)

解答

(a)

解法一

思路

展开

交点 $A$ 在 $QR$ 上，交点 $B$ 在 $RS$ 上。先分别求这两段直线的方程，再与 $y=\frac12x$ 联立。

答题过程

展开

For line segment $QR$ , using $Q(0,4)$ and $R(2,0)$ , the gradient is

\begin{align*} \frac{0-4}{2-0}=-2. \end{align*}

So the equation of $QR$ is

\begin{align*} y=-2x+4. \end{align*}

At $A$ ,

\begin{align*} \frac12x=&\,-2x+4\\ \frac52x=&\,4\\ x=&\,\frac85. \end{align*}

For line segment $RS$ , using $R(2,0)$ and $S(4,10)$ , the gradient is

\begin{align*} \frac{10-0}{4-2}=5. \end{align*}

So the equation of $RS$ is

\begin{align*} y=5x-10. \end{align*}

At $B$ ,

\begin{align*} \frac12x=&\,5x-10\\ \frac92x=&\,10\\ x=&\,\frac{20}{9}. \end{align*}

Therefore

\begin{align*} x_A=\frac85,\qquad x_B=\frac{20}{9}. \end{align*}

(b)

解法一

思路

展开

$y=\frac32g(x)$ 是竖直方向放大 $\frac32$ 倍，所以所有 $x$ 坐标不变，所有 $y$ 坐标乘以 $\frac32$ 。折线形状保持为三段直线。

答题过程

展开

Under

\begin{align*} y=\frac32g(x), \end{align*}

the $x$ coordinates stay the same and the $y$ coordinates are multiplied by $\dfrac32$ .

Thus

\begin{align*} P(-3,4)&\mapsto P'(-3,6),\\ Q(0,4)&\mapsto Q'(0,6),\\ R(2,0)&\mapsto R'(2,0),\\ S(4,10)&\mapsto S'(4,15). \end{align*}

The sketch should join these four transformed points with straight line segments.

A completed sketch is:

IAL 2025 Oct A Q8

题目

解答

(a)

解法一

思路

答题过程

(b)

解法一

思路

答题过程