变量 x 和 y 满足方程 $y=\frac{x^3}{5}-2x+2$ 。下表给出了 x 和 y 的一些对应值。

(a)
求 p 的值。

答案 (a) p = ...
(b)
在提供的网格上，画出当 $-3\le x\le4$ 时 $y=\frac{x^3}{5}-2x+2$ 的图像。
(c)
方程 $\frac{x^3}{5}-2x=3$ 只有一个解。

说明如何从你的图像中看出这一点。

答案 (c)

...
(d)
1. (i)
  在 (b) 的同一网格中，画出对应 $-1\le x\le3$ 的直线 $y=-2x+5$ 。
2. (ii)
  写出这条直线与曲线交点的 $x$ 坐标。
  
  答案 (d)(ii) $x=$ ...
3. (iii)
  此 $x$ 的值是方程 $x^3+Ax+B=0$ 的一个解。
  
  求 A 的值和 B 的值。
  
  答案 (d)(iii) A = ...
  
  B = ...

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完整解法

8(a)

1. Substitute $x=-3$ into the given equation and evaluate $y$.

For $x=-3$ , $y=p$ .

$p=\frac{(-3)^3}{5}-2(-3)+2$

$p=\frac{-27}{5}+6+2$

$p=-5.4+8=2.6$

答案

$p=2.6$

8(b)

1. Use the table values, including the value of $p$ found in part (a), as coordinates to be plotted.

The points to plot are

$(-3,2.6),\ (-2,4.4),\ (-1,3.8),\ (0,2),\ (1,0.2),\ (2,-0.4),\ (3,1.4),\ (4,6.8).$

2. Plot all the ordered pairs accurately on the given Cartesian grid.

On the grid, mark each point at its correct horizontal position $x$ and vertical position $y$ :

$(-3,2.6),\ (-2,4.4),\ (-1,3.8),\ (0,2),\ (1,0.2),\ (2,-0.4),\ (3,1.4),\ (4,6.8).$

3. Join the plotted points with a smooth curve over the required domain.

Draw a smooth continuous curve passing through the plotted points from $x=-3$ to $x=4$ .

Do not join the points with straight line segments.

答案

A smooth curve of $y=\dfrac{x^3}{5}-2x+2$ drawn through the points $(-3,2.6)$ , $(-2,4.4)$ , $(-1,3.8)$ , $(0,2)$ , $(1,0.2)$ , $(2,-0.4)$ , $(3,1.4)$ and $(4,6.8)$ for $-3\le x\le4$ .

8(c)

1. Rewrite the given equation so it matches the plotted curve equation.

The plotted curve is
$y=\dfrac{x^3}{5}-2x+2$ .

Given
$\dfrac{x^3}{5}-2x=3$ ,
add $2$ to both sides:
$\dfrac{x^3}{5}-2x+2=5.$
So the solution occurs where the curve has $y=5$ .

2. Use the graph to explain the number of solutions.

Draw or imagine the horizontal line $y=5$ on the graph.

From the graph, the line $y=5$ cuts the curve $y=\dfrac{x^3}{5}-2x+2$ at exactly one point. Therefore, there is only one value of $x$ satisfying the equation.

答案

$\dfrac{x^3}{5}-2x=3$ can be written as $\dfrac{x^3}{5}-2x+2=5$ . Hence it is represented on the graph by the horizontal line $y=5$ . This line intersects the curve only once, so the equation has only one solution.

8(d)(i)

1. Find two points on the line within the required domain.

For $y=-2x+5$ :

When $x=-1$ , $y=-2(-1)+5=7$ , giving $(-1,7)$ .

When $x=3$ , $y=-2(3)+5=-1$ , giving $(3,-1)$ .

2. Plot the two points and draw the required line segment.

Plot $(-1,7)$ and $(3,-1)$ on the same grid.

Join them with a straight line for $-1\le x\le 3$ .

答案

Draw the straight line segment of $y=-2x+5$ from $(-1,7)$ to $(3,-1)$ on the same grid.

8(d)(ii)

1. Identify the point where the straight line meets the curve and read its x-coordinate from the graph.

From the graph, the line $y=-2x+5$ intersects the curve $y=\dfrac{x^3}{5}-2x+2$ at about $x=2.5$ .

答案

$x \approx 2.5$

8(d)(iii)

1. Use the fact that the intersection point lies on both the curve and the line.

At the point of intersection,
$\frac{x^3}{5}-2x+2=-2x+5.$

2. Simplify the equation and rearrange it into cubic form.

$\frac{x^3}{5}-2x+2=-2x+5$
Cancel $-2x$ from both sides:
$\frac{x^3}{5}+2=5$
$\frac{x^3}{5}=3$
$x^3=15$
So
$x^3-15=0.$

3. Compare with the required form $x^3+Ax+B=0$.

Compare
$x^3-15=0$
with
$x^3+Ax+B=0.$
There is no $x$ term, so $A=0$ .
The constant term is $-15$ , so $B=-15$ .

答案

$A=0,\quad B=-15$

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变量 x 和 y 满足方程 y=x35−2x+2y=\frac{x^3}{5}-2x+2y=5x3​−2x+2。 下表给出了 x 和 y 的一些对应值。