Course: 10.1 赤道函数及其图

Section outline

Select section General

Collapse Expand
General

Collapse all Expand all
- Select activity 新闻通告
  
  新闻通告 Forum

二次曲线函数及其图表

Quadratic Functions and Their Graphs
::二次曲线函数及其图表

The are curved lines called parabolas . You don’t have to look hard to find parabolic shapes around you. Here are a few examples:
::曲线线被称为parapolas。您不必寻找周围的抛物线形状。以下是几个例子:

The path that a ball or a rocket takes through the air.
::球或火箭穿过空中的路径

Water flowing out of a drinking fountain.
::泉水从饮用喷泉流出

The shape of a satellite dish.
::卫星天线的形状

The shape of the mirror in car headlights or a flashlight.
::车头灯或手电筒里镜子的形状

The cables in a suspension bridge.
::悬吊桥上的电缆

Graphing Quadratic Functions
::刻度函数

Let’s see what a parabola looks like by graphing the simplest quadratic function , $\begin{aligned} y = x^{2} \end{aligned}$ .
::让我们通过绘制最简单的四方函数 y=x2 的图形来观察抛物线的长相。

We’ll graph this function by making a table of values. Since the graph will be curved, we need to plot a fair number of points to make it accurate.
::我们将通过绘制一个数值表来绘制这个函数。由于这个图表将被弯曲, 我们需要绘制数量相当的点, 才能使其准确化。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} (- 3)^{2} = 9 \end{aligned}$
–2	$\begin{aligned} (- 2)^{2} = 4 \end{aligned}$
–1	$\begin{aligned} (- 1)^{2} = 1 \end{aligned}$
0	$\begin{aligned} (0)^{2} = 0 \end{aligned}$
1	$\begin{aligned} (1)^{2} = 1 \end{aligned}$
2	$\begin{aligned} (2)^{2} = 4 \end{aligned}$
3	$\begin{aligned} (3)^{2} = 9 \end{aligned}$

Here are the points plotted on a coordinate graph:
::以下是坐标图上绘制的点数 :

To draw the parabola, draw a smooth curve through all the points. (Do not connect the points with straight lines).
::要绘制抛物线,请通过所有点绘制一个平滑的曲线。 (不要将点与直线连接)。

Let’s graph a few more examples.
::让我们再举几个例子。

Graphing Parabolas
::图示图

Graph the following parabolas.
::如下图所示。

a) $\begin{aligned} y = 2 x^{2} + 4 x + 1 \end{aligned}$
::a) y=2x2+4x+1

Make a table of values:
::绘制数值表:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = 2 x^{2} + 4 x + 1 \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} 2 (- 3)^{2} + 4 (- 3) + 1 = 7 \end{aligned}$
–2	$\begin{aligned} 2 (- 2)^{2} + 4 (- 2) + 1 = 1 \end{aligned}$
–1	$\begin{aligned} 2 (- 1)^{2} + 4 (- 1) + 1 = - 1 \end{aligned}$
0	$\begin{aligned} 2 (0)^{2} + 4 (0) + 1 = 1 \end{aligned}$
1	$\begin{aligned} 2 (1)^{2} + 4 (1) + 1 = 7 \end{aligned}$
2	$\begin{aligned} 2 (2)^{2} + 4 (2) + 1 = 17 \end{aligned}$
3	$\begin{aligned} 2 (3)^{2} + 4 (3) + 1 = 31 \end{aligned}$

Notice that the last two points have very large $\begin{aligned} y - \end{aligned}$ values. Since we don’t want to make our $\begin{aligned} y - \end{aligned}$ scale too big, we’ll just skip graphing those two points. But we’ll plot the remaining points and join them with a smooth curve.
::请注意后两点有非常大的y-价值。由于我们不想让我们的y-规模太大,我们将跳过这两点的图表。但我们将勾画其余的点并用一个平稳的曲线加入它们。

b) $\begin{aligned} y = - x^{2} + 3 \end{aligned}$
::b) yx2+3

Make a table of values:
::绘制数值表:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = - x^{2} + 3 \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} - (- 3)^{2} + 3 = - 6 \end{aligned}$
–2	$\begin{aligned} - (- 2)^{2} + 3 = - 1 \end{aligned}$
–1	$\begin{aligned} - (- 1)^{2} + 3 = 2 \end{aligned}$
0	$\begin{aligned} - (0)^{2} + 3 = 3 \end{aligned}$
1	$\begin{aligned} - (1)^{2} + 3 = 2 \end{aligned}$
2	$\begin{aligned} - (2)^{2} + 3 = - 1 \end{aligned}$
3	$\begin{aligned} - (3)^{2} + 3 = - 6 \end{aligned}$

Plot the points and join them with a smooth curve.
::绘制点数并用一个平滑的曲线加入它们。

Notice that this time we get an “upside down” parabola. That’s because our equation has a negative sign in front of the $\begin{aligned} x^{2} \end{aligned}$ term . The sign of the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ term determines whether the parabola turns up or down: the parabola turns up if it’s positive and down if it’s negative.
::请注意,这次我们得到了一个“上下向”抛物线。这是因为我们的方程式在 x2 术语之前有一个负符号。 x2 术语系数的标志决定了抛物线是上升还是下降:如果它是正的,则抛物线会上升,如果是负的,则下降。

c) $\begin{aligned} y = x^{2} - 8 x + 3 \end{aligned}$
::c) y=x2-8x+3

Make a table of values:
::绘制数值表:

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} - 8 x + 3 \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} (- 3)^{2} - 8 (- 3) + 3 = 36 \end{aligned}$
–2	$\begin{aligned} (- 2)^{2} - 8 (- 2) + 3 = 23 \end{aligned}$
–1	$\begin{aligned} (- 1)^{2} - 8 (- 1) + 3 = 12 \end{aligned}$
0	$\begin{aligned} (0)^{2} - 8 (0) + 3 = 3 \end{aligned}$
1	$\begin{aligned} (1)^{2} - 8 (1) + 3 = - 4 \end{aligned}$
2	$\begin{aligned} (2)^{2} - 8 (2) + 3 = - 9 \end{aligned}$
3	$\begin{aligned} (3)^{2} - 8 (3) + 3 = - 12 \end{aligned}$

Let’s not graph the first two points in the table since the values are so big. Plot the remaining points and join them with a smooth curve.
::让我们不要在表格中绘制前两个点的图表,因为数值如此之大。绘制剩余点,然后用一个光滑曲线加入它们。

Wait—this doesn’t look like a parabola. What’s going on here?
::等等,这看起来不像是抛物线,这是怎么回事?

Maybe if we graph more points, the curve will look more familiar. For negative values of $\begin{aligned} x \end{aligned}$ it looks like the values of $\begin{aligned} y \end{aligned}$ are just getting bigger and bigger, so let’s pick more positive values of $\begin{aligned} x \end{aligned}$ beyond $\begin{aligned} x = 3 \end{aligned}$ .
::也许如果我们绘制更多的点, 曲线会看起来更熟悉。对于 x 的负值来说, y 的值似乎正在变得越来越大, 所以让我们选择 x x 的正值大于 x = 3 。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} - 8 x + 3 \end{aligned}$
$\begin{aligned} - 1 \end{aligned}$	$\begin{aligned} (- 1)^{2} - 8 (- 1) + 3 = 12 \end{aligned}$
0	$\begin{aligned} (0)^{2} - 8 (0) + 3 = 3 \end{aligned}$
1	$\begin{aligned} (1)^{2} - 8 (1) + 3 = - 4 \end{aligned}$
2	$\begin{aligned} (2)^{2} - 8 (2) + 3 = - 9 \end{aligned}$
3	$\begin{aligned} (3)^{2} - 8 (3) + 3 = - 12 \end{aligned}$
4	$\begin{aligned} (4)^{2} - 8 (4) + 3 = - 13 \end{aligned}$
5	$\begin{aligned} (5)^{2} - 8 (5) + 3 = - 12 \end{aligned}$
6	$\begin{aligned} (6)^{2} - 8 (6) + 3 = - 9 \end{aligned}$
7	$\begin{aligned} (7)^{2} - 8 (7) + 3 = - 4 \end{aligned}$
8	$\begin{aligned} (8)^{2} - 8 (8) + 3 = 3 \end{aligned}$

Plot the points again and join them with a smooth curve.
::重新标出点数, 并用一个平滑的曲线加入它们。

Now we can see the familiar parabolic shape. And now we can see the drawback to graphing quadratics by making a table of values—if we don’t pick the right values, we won’t get to see the important parts of the graph.
::现在我们可以看到熟悉的抛物线形状。现在我们可以看到通过绘制一个数值表来绘制象方图的缺点 — — 如果我们不选择正确的数值,我们就无法看到图中的重要部分。

In the next couple of lessons, we’ll find out how to graph quadratic equations more efficiently—but first we need to learn more about the properties of parabolas.
::在接下来的几节课中, 我们会发现如何更高效地绘制二次方程, 但首先我们需要了解更多关于parapolas的特性。

Compare Graphs of Quadratic Functions
::比较二次函数的图形

The general form (or standard form ) of a quadratic function is:
::二次函数的一般形式(或标准形式)是:

\begin{aligned} y = a x^{2} + b x + c \end{aligned}

::y=ax2+bx+c y=ax2+bx+c

Here $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ are the coefficients. Remember, a coefficient is just a number (a constant term) that can go before a variable or appear alone.
::a, b 和 c 是系数。记住, 系数只是一个数字( 恒定值) , 可以在变量之前出现或单独出现。

Although the graph of a quadratic equation in standard form is always a parabola, the shape of the parabola depends on the values of the coefficients $\begin{aligned} a, b \end{aligned}$ and $\begin{aligned} c \end{aligned}$ . Let’s explore some of the ways the coefficients can affect the graph.
::虽然标准形式的二次方程图始终是抛物线,但抛物线的形状取决于a、b和c等系数的值。让我们探讨一下这些系数如何影响该图。

Dilation
::关系

Changing the value of $\begin{aligned} a \end{aligned}$ makes the graph “fatter” or “skinnier”. Let’s look at how graphs compare for different positive values of $\begin{aligned} a \end{aligned}$ .
::更改一个图的值使图“ 折叠” 或“ 皮肤” 。让我们看看图表如何比较一个图的不同正值。

Comparing Graphs
::比较图表

The plot on the left shows the graphs of $\begin{aligned} y = x^{2} \end{aligned}$ and $\begin{aligned} y = 3 x^{2} \end{aligned}$ . The plot on the right shows the graphs of $\begin{aligned} y = x^{2} \end{aligned}$ and $\begin{aligned} y = \frac{1}{3} x^{2} \end{aligned}$ .
::左边的图示显示 y=x2 和 y= 3x2 的图示。右边的图示显示 y=x2 和 y= 13x2 的图示。

Notice that the larger the value of $\begin{aligned} a \end{aligned}$ is, the skinnier the graph is – for example, in the first plot, the graph of $\begin{aligned} y = 3 x^{2} \end{aligned}$ is skinnier than the graph of $\begin{aligned} y = x^{2} \end{aligned}$ . Also, the smaller $\begin{aligned} a \end{aligned}$ is, the fatter the graph is – for example, in the second plot, the graph of $\begin{aligned} y = \frac{1}{3} x^{2} \end{aligned}$ is fatter than the graph of $\begin{aligned} y = x^{2} \end{aligned}$ . This might seem counterintuitive, but if you think about it, it should make sense. Let’s look at a table of values of these graphs and see if we can explain why this happens.
::注意一个值越大, 图形的皮肤越小, 例如在第一个图中, y= 3x2 的图比 y=x2 的图要浅。另外, 越小, 图越胖, 例如, 在第二个图中, y= 13x2 的图比 y=x2 的图要肥, 这看起来似乎是反直观的, 但如果你考虑它, 它应该是有道理的。让我们看看这些图的数值表, 看看我们能否解释为什么发生这种情况。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} \end{aligned}$	$\begin{aligned} y = 3 x^{2} \end{aligned}$	$\begin{aligned} y = \frac{1}{3} x^{2} \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} (- 3)^{2} = 9 \end{aligned}$	$\begin{aligned} 3 (- 3)^{2} = 27 \end{aligned}$	$\begin{aligned} \frac{(- 3)^{2}}{3} = 3 \end{aligned}$
–2	$\begin{aligned} (- 2)^{2} = 4 \end{aligned}$	$\begin{aligned} 3 (- 2)^{2} = 12 \end{aligned}$	$\begin{aligned} \frac{(- 2)^{2}}{3} = \frac{4}{3} \end{aligned}$
–1	$\begin{aligned} (- 1)^{2} = 1 \end{aligned}$	$\begin{aligned} 3 (- 1)^{2} = 3 \end{aligned}$	$\begin{aligned} \frac{(- 1)^{2}}{3} = \frac{1}{3} \end{aligned}$
0	$\begin{aligned} (0)^{2} = 0 \end{aligned}$	$\begin{aligned} 3 (0)^{2} = 0 \end{aligned}$	$\begin{aligned} \frac{(0)^{2}}{3} = 0 \end{aligned}$
1	$\begin{aligned} (1)^{2} = 1 \end{aligned}$	$\begin{aligned} 3 (1)^{2} = 3 \end{aligned}$	$\begin{aligned} \frac{(1)^{2}}{3} = \frac{1}{3} \end{aligned}$
2	$\begin{aligned} (2)^{2} = 4 \end{aligned}$	$\begin{aligned} 3 (2)^{2} = 12 \end{aligned}$	$\begin{aligned} \frac{(2)^{2}}{3} = \frac{4}{3} \end{aligned}$
3	$\begin{aligned} (3)^{2} = 9 \end{aligned}$	$\begin{aligned} 3 (3)^{2} = 27 \end{aligned}$	$\begin{aligned} \frac{(3)^{2}}{3} = 3 \end{aligned}$

From the table, you can see that the values of $\begin{aligned} y = 3 x^{2} \end{aligned}$ are bigger than the values of $\begin{aligned} y = x^{2} \end{aligned}$ . This is because each value of $\begin{aligned} y \end{aligned}$ gets multiplied by 3. As a result the parabola will be skinnier because it grows three times faster than $\begin{aligned} y = x^{2} \end{aligned}$ . On the other hand, you can see that the values of $\begin{aligned} y = \frac{1}{3} x^{2} \end{aligned}$ are smaller than the values of $\begin{aligned} y = x^{2} \end{aligned}$ , because each value of $\begin{aligned} y \end{aligned}$ gets divided by 3. As a result the parabola will be fatter because it grows at one third the rate of $\begin{aligned} y = x^{2} \end{aligned}$ .
::从表格中可以看到,y=3x2的值大于y=x2的值。这是因为y的每个值乘以3。这是因为, y的每个值乘以3 。因此, 抛物线的增速会比y=x2快三倍, 因为它的增速比y=x2快三倍。另一方面, 你可以看到, y=13x2的值小于y=x2的值, 因为y的每值除以3 。因此, 抛物线会变胖, 因为它的增速是y=x2的三分之一。

Orientation
::方向方向方向

As the value of $\begin{aligned} a \end{aligned}$ gets smaller and smaller, then, the parabola gets wider and flatter. What happens when $\begin{aligned} a \end{aligned}$ gets all the way down to zero? What happens when it’s negative?
::随着一个价值越来越小和越来越小,那么,抛物线会变得更宽和受宠若惊。当一个价值大到零时会怎样? 当它变成负时会怎样?

Well, when $\begin{aligned} a = 0 \end{aligned}$ , the $\begin{aligned} x^{2} \end{aligned}$ term drops out of the equation entirely, so the equation becomes linear and the graph is just a straight line. For example, we just saw what happens to $\begin{aligned} y = a x^{2} \end{aligned}$ when we change the value of $\begin{aligned} a \end{aligned}$ ; if we tried to graph $\begin{aligned} y = 0 x^{2} \end{aligned}$ , we would just be graphing $\begin{aligned} y = 0 \end{aligned}$ , which would be a horizontal line.
::当 a=0 时, x2 术语会完全从方程式中消失, 所以方程式会变成线性, 图形只是一条直线。例如, 我们刚刚看到当我们改变一个值时, y= ax2 会发生什么; 如果我们试图绘制 y= 0x2, 我们就会绘制 y=0 的图, 也就是水平线。

So as $\begin{aligned} a \end{aligned}$ gets smaller and smaller, the graph of $\begin{aligned} y = a x^{2} \end{aligned}$ gets flattened all the way out into a horizontal line. Then, when $\begin{aligned} a \end{aligned}$ becomes negative, the graph of $\begin{aligned} y = a x^{2} \end{aligned}$ starts to curve again, only it curves downward instead of upward. This fits with what you’ve already learned: the graph opens upward if $\begin{aligned} a \end{aligned}$ is positive and downward if $\begin{aligned} a \end{aligned}$ is negative.
::因此,当一个变小和变小时,y=ax2的图形就会被平整成一条水平线。然后,当一个为负时,y=ax2的图形开始再次曲线,只有它向下曲线而不是向上曲线。这符合你已经学到的东西:如果一个为正向上,如果是一个为负向下,这个图形会向上打开。

Comparing Two Equations
::比较两个等号

What do the graphs of $\begin{aligned} y = x^{2} \end{aligned}$ and $\begin{aligned} y = - x^{2} \end{aligned}$ look like?
::y=x2 和 yx2 的图形看起来像什么?

You can see that the parabola has the same shape in both graphs, but the graph of $\begin{aligned} y = x^{2} \end{aligned}$ is right-side-up and the graph of $\begin{aligned} y = - x^{2} \end{aligned}$ is upside-down.
::您可以看到, 抛物线在两个图形中都有相同的形状, 但 y=x2 的图形是右侧的, 而 yx2 的图形是上下向的。

Vertical Shifts
::垂直垂直移动

Changing the constant $\begin{aligned} c \end{aligned}$ just shifts the parabola up or down.
::改变常数C只是改变抛物线的上下移动。

Graphing Equations
::图形平方

What do the graphs of $\begin{aligned} y = x^{2}, y = x^{2} + 1, y = x^{2} - 1, y = x^{2} + 2, \end{aligned}$ and $\begin{aligned} y = x^{2} - 2 \end{aligned}$ look like?
::Y=x2,y=x2,y=x2+1,y=x2 -1,y=x2+2和y=x2-2的图象是什么?

You can see that when $\begin{aligned} c \end{aligned}$ is positive, the graph shifts up, and when $\begin{aligned} c \end{aligned}$ is negative the graph shifts down; in either case, it shifts by $\begin{aligned} | c | \end{aligned}$ units. In one of the later Concepts, we’ll learn about horizontal shift (i.e. moving to the right or to the left). Before we can do that, though, we need to learn how to rewrite quadratic equations in different forms - our objective for the next Concept.
::当 c 呈正数时, 图表会向上移动, 当 c 呈负数时, 图表会向下移动; 无论哪种情况, 它都会由 {c} 单位转移。在后来的概念之一中, 我们将会了解横向变化( 即向右或向左移动 ) 。但是, 在我们能够做到这一点之前, 我们需要学习如何以不同的形式重写二次方程, 也就是我们下一个概念的目标。

Example
::示例示例示例示例

Example 1
::例1

Graph the quadratic function, $\begin{aligned} y = - x^{2} + 2 \end{aligned}$ .
::构造二次函数, yx2+2。

$\begin{aligned} x \end{aligned}$	$\begin{aligned} y = x^{2} \end{aligned}$
$\begin{aligned} - 3 \end{aligned}$	$\begin{aligned} - (- 3)^{2} + 2 = - 7 \end{aligned}$
–2	$\begin{aligned} - (- 2)^{2} + 2 = - 2 \end{aligned}$
–1	$\begin{aligned} - (- 1)^{2} + 2 = 1 \end{aligned}$
0	$\begin{aligned} - (0)^{2} + 2 = 2 \end{aligned}$
1	$\begin{aligned} - (1)^{2} + 2 = 1 \end{aligned}$
2	$\begin{aligned} - (2)^{2} + 2 = - 2 \end{aligned}$
3	$\begin{aligned} - (3)^{2} + 2 = - 7 \end{aligned}$

Plot the points and connect them with a smooth curve:
::绘制点并用一个光滑曲线将其连接 :

Review
::回顾

For 1-5, does the graph of the parabola turn up or down?
::1 -5, 抛物线的图是上升还是下降?

$\begin{aligned} y = - 2 x^{2} - 2 x - 3 \end{aligned}$
::y2x2 - 2x - 3

$\begin{aligned} y = 3 x^{2} \end{aligned}$
::y=3x2 y=3x2

$\begin{aligned} y = 16 - 4 x^{2} \end{aligned}$
::y=16 - 4x2

$\begin{aligned} y = - 100 + 0.25 x^{2} \end{aligned}$
::y$100+0.25x2

$\begin{aligned} y = 3 x^{2} - 2 x - 4 x^{2} + 3 \end{aligned}$
::y= 3x2 - 2x - 4x2+3

For 6-10, which parabola is wider?
::6 -10,哪个抛物线宽?

$\begin{aligned} y = x^{2} \end{aligned}$ or $\begin{aligned} y = 4 x^{2} \end{aligned}$
::y=x2 或 y= 4x2

$\begin{aligned} y = 2 x^{2} + 4 \end{aligned}$ or $\begin{aligned} y = \frac{1}{2} x^{2} + 4 \end{aligned}$
::y=2x2+4或y=12x2+4

$\begin{aligned} y = - 2 x^{2} - 2 \end{aligned}$ or $\begin{aligned} y = - x^{2} - 2 \end{aligned}$
::y2x2 - 2 或yx2 - 2

$\begin{aligned} y = x^{2} + 3 x^{2} \end{aligned}$ or $\begin{aligned} y = x^{2} + 3 \end{aligned}$
::y=x2+3x2 或y=x2+3

$\begin{aligned} y = - x^{2} \end{aligned}$ or $\begin{aligned} y = \frac{1}{10} x^{2} \end{aligned}$
::yx2 或y= 110x2

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Section outline

General

二次曲线函数及其图表

Quadratic Functions and Their Graphs ::二次曲线函数及其图表

Graphing Quadratic Functions ::刻度函数

Graphing Parabolas ::图示图

Compare Graphs of Quadratic Functions ::比较二次函数的图形

Dilation ::关系

Comparing Graphs ::比较图表

Comparing Two Equations ::比较两个等号

Vertical Shifts ::垂直垂直移动

Graphing Equations ::图形平方

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Quadratic Functions and Their Graphs
::二次曲线函数及其图表

Graphing Quadratic Functions
::刻度函数

Graphing Parabolas
::图示图

Compare Graphs of Quadratic Functions
::比较二次函数的图形

Dilation
::关系

Comparing Graphs
::比较图表

Comparing Two Equations
::比较两个等号

Vertical Shifts
::垂直垂直移动

Graphing Equations
::图形平方

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)