下表是关于函数 $y = \dfrac{x^{3}}{4} - 2x + 1$ 的。

((a))
完成关于函数 $y = \dfrac{x^{3}}{4} - 2x + 1$ 的数值表。
1. x -3 -2 -1 0 1 2 3
  y 0.25 3 2.75 1 -0.75 1.75
((b))
在对面的网格上，画出 $-3 \le x \le 3$ 范围内的 $y = \dfrac{x^{3}}{4} - 2x + 1$ 函数图像。
((c))
说明你的图像如何展示出，对于某些 $k$ 值，方程 $\dfrac{x^{3}}{4} - 2x + 1 = k$ 只有唯一解。
((d))
方程 $x^{3} - 9x - 4 = 0$ 可以通过在网格上画一条合适的直线来求解。
1. ((i))
  求该直线的方程。
  
  答案 ___
2. ((ii))
  通过画出这条直线，求解方程 $x^{3} - 9x - 4 = 0$ 。
  
  答案 x = ___ 或 ___

x	-3	-2	-1	0	1	2	3
y	0.25	3	2.75	1	-0.75		1.75

幂函数图像一次函数与二次函数一次函数图像数系与四则运算有序数对与图像直角坐标系列方程解应用题一次式化简直线方程 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

7((a))

1. Evaluate the function at the missing table entry $x=2$.

For $x=2$ ,
$y=\frac{2^3}{4}-2(2)+1$
$=\frac{8}{4}-4+1$
$=2-4+1=-1$

答案

The missing value is $y=0$ when $x=2$ .

7((b))

1. Use the completed table to identify all coordinates to be plotted.

From the completed table,

$(-3,0.25),\ (-2,3),\ (-1,2.75),\ (0,1),\ (1,-0.75),\ (2,0),\ (3,1.75).$

2. Plot the points accurately on the given grid.

Plot each point on the Cartesian grid at its corresponding $x$ -value and $y$ -value. For example, plot $(2,0)$ on the $x$ -axis and $(1,-0.75)$ three-quarters of a unit below the $x$ -axis.

3. Join the plotted points with an appropriate smooth curve over the required interval.

Join the plotted points with a smooth curve from $x=-3$ to $x=3$ , not with straight line segments. The curve should pass close to all the plotted points and stop at $x=-3$ and $x=3$ .

答案

A smooth curve for $y=\dfrac{x^3}{4}-2x+1$ is drawn through the plotted points $(-3,0.25)$ , $(-2,3)$ , $(-1,2.75)$ , $(0,1)$ , $(1,-0.75)$ , $(2,0)$ and $(3,1.75)$ for $-3\le x\le 3$ .

7((c))

1. Relate the equation to intersections of the graph with a horizontal line.

The equation $\dfrac{x^3}{4}-2x+1=k$ is solved graphically by drawing the horizontal line $y=k$ and finding where it meets the curve $y=\dfrac{x^3}{4}-2x+1$ . For some values of $k$ , this horizontal line cuts the curve at only one point, so there is only one solution.

答案

For some values of $k$ , the horizontal line $y=k$ cuts the curve at only one point. Hence $\dfrac{x^3}{4}-2x+1=k$ has only one solution for those values of $k$ .

7((d))((i))

1. Express the given cubic equation in terms of the graphed function.

The graph already drawn is
$y=\dfrac{x^3}{4}-2x+1.$
From
$x^3-9x-4=0,$
divide throughout by $4$ :
$\dfrac{x^3}{4}-\dfrac{9x}{4}-1=0.$

2. Rearrange to obtain the equation of the straight line.

Rearrange so that $\dfrac{x^3}{4}-2x+1$ is on one side:
$\dfrac{x^3}{4}-\dfrac{9x}{4}-1=0$
$\dfrac{x^3}{4}-2x+1=\dfrac{x}{4}+2.$
Since the curve is $y=\dfrac{x^3}{4}-2x+1$ , the suitable straight line is
$y=\dfrac{x}{4}+2.$

答案

$y=\dfrac{x}{4}+2$

7((d))((ii))

1. Draw the straight line obtained in part (d)(i) on the same graph.

From part (d)(i), draw the line

$y=\frac{x}{4}+2.$

For example, it passes through convenient points such as $(0,2)$ and $(2,2.5)$ .

2. Read the $x$-coordinates where the line intersects the cubic curve.

The solutions of

$x^3-9x-4=0$

are given by the intersection points of the curve

$y=\frac{x^3}{4}-2x+1$

and the line

$y=\frac{x}{4}+2.$

From the graph, the two visible intersections occur at approximately

$x\approx -2.7 \quad \text{and} \quad x\approx -0.5.$

答案

$x \approx -2.7$ or $x \approx -0.5$

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。

下表是关于函数 y=x34−2x+1y = \dfrac{x^{3}}{4} - 2x + 1y=4x3​−2x+1 的。