变量 x 和 y 由方程 $y = \frac{x^3}{2} - 5x - 2$ 联系起来。表中给出了 x 和 y 的一些对应值。

(a)
求 k 的值。

答 k = ...
(b)
在第 19 页的网格上，画出 $-3 \le x \le 4$ 范围内 $y = \frac{x^3}{2} - 5x - 2$ 的图像。
(c)
方程 $\frac{x^3}{2} - 5x = 7$ 只有一个解。

解释如何从你的图像中推导出这一点。

答
(d)
通过画切线，估算曲线在 (1, -6.5) 处的斜率。

答斜率 = ...
(e)
1. (i)
  在同一个网格上，画出 $-1 \le x \le 4$ 范围内的直线 $y = 4 - x$ 。
2. (ii)
  写出这条直线与曲线交点的 x 坐标。
  
  答 x = ...
3. (iii)
  这个 x 的值是方程 $x^3 + Ax + B = 0$ 的一个解。
  
  求 A 的值和 B 的值。
  
  答 A = ... , B = ...

幂函数图像曲线斜率估算一次函数图像列方程解应用题有序数对与图像数系与四则运算直角坐标系一次函数斜率一次函数与二次函数用字母表示数 English

先不看答案，让 AI 诊断你卡在哪里？

它不会直接告诉你答案，而是问你卡在哪一步看看 AI 是怎么陪你一步步弄懂这道题的

完整解法

9(a)

1. Evaluate $y$ when $x=2$.

Since $k$ is the value of $y$ when $x=2$ ,
$k=\frac{2^3}{2}-5(2)-2=\frac{8}{2}-10-2=4-10-2=-8.$

答案

$k = -8$

9(b)

1. List the coordinates to be plotted from the table, using the value of $k$ found earlier.

From the table and $k=-8$ , plot the points
$(-3,-0.5),\ (-2,4),\ (-1,2.5),\ (0,-2),\ (1,-6.5),\ (2,-8),\ (3,-3.5),\ (4,10).$

2. Draw the curve accurately through the plotted points for the given range of $x$.

Join the plotted points with a smooth curve, not straight line segments, for
$-3\le x\le 4.$

答案

A smooth curve of $y=\frac{x^3}{2}-5x-2$ for $-3\le x\le 4$ passing through
$(-3,-0.5),(-2,4),(-1,2.5),(0,-2),(1,-6.5),(2,-8),(3,-3.5),(4,10)$ .

9(c)

1. Rewrite the equation in terms of the graph already drawn.

The graph drawn is
$y=\dfrac{x^3}{2}-5x-2$ .

Given
$\dfrac{x^3}{2}-5x=7$ ,
subtract $2$ from both sides:
$\dfrac{x^3}{2}-5x-2=5.$
So this corresponds to points on the curve where $y=5$ .

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

On the graph, the horizontal line $y=5$ cuts the curve $y=\dfrac{x^3}{2}-5x-2$ at only one point.

Therefore, there is only one value of $x$ satisfying the equation.

答案

Since $\dfrac{x^3}{2}-5x=7$ is equivalent to $y=5$ on the curve, draw/consider the horizontal line $y=5$ . It intersects the curve at only one point, so the equation has only one solution.

9(d)

1. Draw a tangent to the curve at $(1,-6.5)$.

On the graph, draw a straight line that just touches the curve at $(1,-6.5)$ and follows the direction of the curve at that point.

2. Find the gradient of the tangent using two points on the tangent.

Choose two well-spaced points on the tangent. For example, the tangent should pass approximately through $(0,-3)$ and $(2,-10)$ .

$\text{Gradient}=\frac{\text{change in }y}{\text{change in }x}$

$=\frac{-10-(-3)}{2-0}=\frac{-7}{2}=-3.5$

答案

Gradient $\approx -3.5$

9(e)(i)

1. Find suitable points on the line within the given range of $x$.

For $y=4-x$ :

When $x=-1$ ,
$y=4-(-1)=5.$

When $x=4$ ,
$y=4-4=0.$

So two points on the line are $(-1,5)$ and $(4,0)$ .

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

Plot $(-1,5)$ and $(4,0)$ on the existing grid.

Join these two points with a straight line segment for $-1\le x\le 4$ .

答案

Draw the straight line segment $y=4-x$ from $(-1,5)$ to $(4,0)$ on the same grid.

9(e)(ii)

1. Identify the intersection of the curve and the line on the graph and read its x-coordinate.

From the graph, the line $y=4-x$ intersects the curve $y=\dfrac{x^3}{2}-5x-2$ at about $x=3.4$ .

答案

$x \approx 3.4$

9(e)(iii)

1. Equate the curve and line equations at their intersection.

At the intersection,
$\frac{x^3}{2}-5x-2=4-x.$

2. Rearrange the equation into the form $x^3+Ax+B=0$ and compare coefficients.

$\frac{x^3}{2}-5x-2=4-x$
$\frac{x^3}{2}-4x-6=0$
Multiplying by $2$ ,
$x^3-8x-12=0.$
Comparing with $x^3+Ax+B=0$ ,
$A=-8,\quad B=-12.$

答案

$A=-8,\quad B=-12$

AI 分析，可能出错

用 AI 批改，圈出写错的那一步，并自动将你的薄弱点存入错题本，形成你的错因库。

变量 x 和 y 由方程 y=x32−5x−2y = \frac{x^3}{2} - 5x - 2y=2x3​−5x−2 联系起来。 表中给出了 x 和 y 的一些对应值。