已知 y=2x3−x2−10xy = 2x^3 - x^2 - 10x。

  1. (a)

    完成 y=2x3−x2−10xy = 2x^3 - x^2 - 10x 的数值表。

  2. (b)

    在对面的网格纸上，画出 y=2x3−x2−10xy = 2x^3 - x^2 - 10x 在 −2≤x≤2-2 \le x \le 2 范围内的图像。

  3. (c)

    方程 2x3−x2−5x=−82x^3 - x^2 - 5x = -8 只有一个解。

    1. (i)

      解释如何从你的图像中看出来。

      答案

    2. (ii)

      利用图像求方程 2x3−x2−5x=−82x^3 - x^2 - 5x = -8 的解。

      答案 ___

  4. (d)

    利用图像求极大值点的 x 坐标。

    答案 ___

一次函数与二次函数幂函数图像数系与四则运算二次函数性质直线方程有序数对与图像直角坐标系一次函数图像English
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完整解法

5(a)

1. Find the missing table value when $x=2$.

For x=2x=2,
y=2(2)3−(2)2−10(2)y=2(2)^3-(2)^2-10(2)
=16−4−20=16-4-20
=−8?=-8?
Check carefully:
2(2)3=16,2(2)^3=16, so
16−4−20=−8.16-4-20=-8.

答案

The missing value is y=−12y=-12.

5(b)

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

From the completed table,

(−2,0), (−1.5,6), (−1,7), (−0.5,4.5), (0,0), (1,−9), (1.5,−10.5), (2,−12).(-2,0),\ (-1.5,6),\ (-1,7),\ (-0.5,4.5),\ (0,0),\ (1,-9),\ (1.5,-10.5),\ (2,-12).

2. Plot the points accurately on the given grid.

Plot each coordinate on the axes, using the given vertical scale marked in units of 22 and the horizontal values from x=−2x=-2 to x=2x=2.

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

Draw a smooth continuous curve through the plotted points for

−2≤x≤2.-2\le x\le 2.

The curve should pass through the origin and continue down to approximately (2,−12)(2,-12) on the right.

答案

A smooth curve of y=2x3−x2−10xy=2x^3-x^2-10x drawn through the points (−2,0)(-2,0), (−1.5,6)(-1.5,6), (−1,7)(-1,7), (−0.5,4.5)(-0.5,4.5), (0,0)(0,0), (1,−9)(1,-9), (1.5,−10.5)(1.5,-10.5) and (2,−12)(2,-12) for −2≤x≤2-2\le x\le 2.

5(c)(i)

1. Relate the given equation to the graph already drawn.

The graph drawn is
y=2x3−x2−10xy=2x^3-x^2-10x.

Given
2x3−x2−5x=−82x^3-x^2-5x=-8,
write
(2x3−x2−10x)+5x=−8(2x^3-x^2-10x)+5x=-8.

So
y+5x=−8y+5x=-8.

2. Rewrite the equation as a straight line to compare with the curve.

From
y+5x=−8y+5x=-8,
we get
y=−5x−8y=-5x-8.

3. Explain why there is only one solution from the graph.

The solutions of the equation are the xx-coordinates where the curve
y=2x3−x2−10xy=2x^3-x^2-10x
meets the line
y=−5x−8y=-5x-8.

From the graph, the line meets the curve at only one point, so there is only one solution.

答案

2x3−x2−5x=−82x^3-x^2-5x=-8 can be written as y=−5x−8y=-5x-8. On the graph, the line y=−5x−8y=-5x-8 intersects the curve y=2x3−x2−10xy=2x^3-x^2-10x at only one point. Hence the equation has only one solution.

5(c)(ii)

1. Rewrite the equation in a form that can be used with the drawn curve.

The graph drawn is for
y=2x3−x2−10xy=2x^3-x^2-10x.

Given
2x3−x2−5x=−8,2x^3-x^2-5x=-8,
write
2x3−x2−10x=−5x−8.2x^3-x^2-10x=-5x-8.
So draw the straight line
y=−5x−8y=-5x-8
on the same graph and read the xx-coordinate where it meets the curve.

2. Read the solution from the point of intersection.

From the graph, the line y=−5x−8y=-5x-8 intersects the curve at approximately
x=−1.9.x=-1.9.

答案

x≈−1.9x \approx -1.9

5(d)

1. Identify the maximum point on the drawn curve and read its $x$-coordinate.

From the graph of y=2x3−x2−10xy=2x^3-x^2-10x, the maximum point is the highest turning point on the curve. Reading its horizontal position from the graph gives

x≈−1.1.x \approx -1.1.

答案

x≈−1.1x \approx -1.1

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同一知识点的其他题

  1. 1在下方空白处画出函数图像。 y=4xy = \frac{4}{x} y=−2x3y = -2x^{3} 由此，确定方程 4x+2x3=0\frac{4}{x} + 2x^{3} = 0 的解的个数。
  2. 2通过配方法或其他方法，在已知 y=x2−4x+6y = x^2 - 4x + 6 的情况下，用 yy 表示 xx。 答案 x = ___ 或 ___ 解方程 38=a6−4a3+638 = a^6 - 4a^3 + 6。 答案 a = ___ 或 ___
  3. 3若二次函数的图象开口向下，且经过(2,−3)(2,-3)点.符合条件的一个二次函数的解析式为 ___ ．
  4. 4在下方坐标系中画出 y=16−(x+3)2y = 16 - (x + 3)^2 的函数图象草图，并清晰地标出图象上截距和顶点的坐标。
  5. 5函数y=kx+b交x轴于（2，0），若其图象过二、四象限，则此函数关系式可以是 ．