Course: 1.13 伸展和反射转变

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伸展和反射转换
Understanding how changes in the equation of a function result in stretching and/or reflecting the graph of the function is a great way to take some of the mystery out of graphing more complicated equations. By recognizing the family to which a more complex equation belongs, and then identifying what changes have been made to the parent of that family, the graph of even quite detailed functions can be made much more understandable.
::理解函数方程式的变化如何导致函数的伸展和/或反映图解,这是将一些神秘之处从更复杂的方程式的图形化中解开的绝佳方法。通过识别一个更复杂的方程式所属的家庭,然后确定该家庭父子的改变,甚至相当详细的函数图可以更容易理解。

See if you can identify what parts of the equation: $\begin{aligned} y = - \frac{1}{5} x^{2} \end{aligned}$ represent either a stretch or a reflection of the parent function $\begin{aligned} y = x^{2} \end{aligned}$ before the examples in this section.
::查看您能否确定公式的哪个部分: y 15x2 代表父函数 y=x2 的伸缩或反射。

Stretching and Reflecting Transformations
::伸展和反射转换

Stretching and Compressing Graphs
::伸展和压缩图

If we multiply a function by a coefficient , the graph of the function will be stretched or compressed.
::如果我们将函数乘以系数,函数的图形将被拉伸或压缩。

Given a function f(x), we can formalize compressing and stretching the graph of f(x) as follows:
::根据 f( x) 函数, 我们可以正式化f( x) 图形的压缩和拉伸如下 :

A function g ( x ) represents a vertical stretch of f ( x ) if g ( x ) = cf ( x ) and c > 1.
::函数 g(x) 代表 f(x) 的垂直伸展, 如果 g(x) = cf(x) 和 c > 1 。

A function g ( x ) represents a vertical compression of f ( x ) if g ( x ) = cf ( x ) and 0 < c < 1.
::函数 g(x) 代表 f(x) 的垂直压缩, 如果 g(x) = cf(x) 和 0 < c < 1 。

A function h(x) represents a horizontal compression of f ( x ) if h ( x ) = f ( cx ) and c > 1.
::函数 h(x) 代表 f(x) 水平压缩, 如果 h(x) = f(cx) 和 c > 1 。

A function h ( x ) represents a horizontal stretch of f ( x ) if h ( x ) = f ( cx ) and 0 < c < 1.
::函数 h(x) 表示f(x) 的水平伸展, 如果h(x) = f(cx) 和 0 < c < 1 。

Notice that a vertical compression or a horizontal stretch occurs when the coefficient is a number between 0 and 1.
::注意当系数在0到1之间时会发生垂直压缩或水平拉伸。

Reflecting Graphs Over the y-axis and x-axis
::Y 轴和 x 轴的反射图

Consider the graphs of the functions y = x ² and y = - x ² , shown below.
::考虑以下函数 y = x2 和 y = -x2 的图形。

The graph of y = - x ² represents a reflection of y = x ² , over the x -axis. That is, every function value of y = - x ² is the negative of a function value of y = x ² . In general, g ( x ) = - f ( x ) has a graph that is the graph of f ( x ), reflected over the x -axis.
::y = -x2 的图形代表 x 轴的 Y = x2 的反射。也就是说, y = - x2 的每个函数值是 y = x2 的负值。一般而言, g(x) = - f(x) 的图形是 f(x) 的图, 反射到 x 轴。

Examples
::实例

Example 1
::例1

Earlier, you were asked a question about identifying transformations .
::早些时候,有人问过你一个有关识别变异的问题。

The function $\begin{aligned} y = - \frac{1}{5} x^{2} \end{aligned}$ is the result of transforming $\begin{aligned} y = x^{2} \end{aligned}$ by reflecting it over the x axis, because of the negative co-efficient on the x , and vertically compressing it (making it wider ), because the co-efficient on the x is a fraction between 0 and 1.
::y15x2 函数是将 y=x2 反射到 x 轴上而将其反射到 x 轴上的结果,因为 x 的负共增效作用,而垂直压缩(使其扩大),因为 x 的共增效作用在 0 和 1 之间是一个分数。

Example 2
::例2

Identify the graph of the function y = (3 x ) ² .
::标明函数 y = (3x) 2 的图形。

We have multiplied x by 3. This should affect the graph horizontally . However, if we simplify the equation, we get y = 9 x ² . Therefore the graph if this parabola will be taller/thinner than y = x ² . Multiplying x by a number greater than 1 creates a horizontal compression, which looks like a vertical stretch.
::我们已经将x乘以3 。这将水平影响图形。但是, 如果我们简化方程, 我们就会得到y = 9x2 。因此, 如果这个抛物线会比 y = x2 高/ 深, 图形会比 y = x2 高/ 高。乘以乘以乘以乘以大于 1 的乘以 x 会形成水平压缩, 看起来像是垂直拉伸。

Example 3
::例3

Identify the transformation described by y = ((1/2) x ) ² .
::识别y = ((1/2)x) 2 描述的变换。

If we simplify this equation, we get y = (1/4) x ² . Therefore multiplying x by a number between 0 and 1 creates a horizontal stretch, which looks like a vertical compression. That is, the parabola will be shorter/wider.
::如果我们简化这个方程, 我们就会得到 Y = (1/ 4) x2。因此乘以 x 乘以 0 和 1 之间的一个数字, 会产生一个水平拉伸, 看起来像是垂直压缩。也就是说, 抛物线会缩短/ 宽度。

Example 4
::例4

Sketch a graph of y = x ³ and y = - x ³ on the same axes.
::在相同的轴上绘制 y = x3 和 y = - x3 的图。

At first the two functions might look like two parabolas. If you graph by hand, or if you set your calculator to sequential mode (and not simultaneous), you can see that the graph of y = - x ³ is in fact a reflection of y = x ³ over the x-axis .
::首先,这两个函数可能看起来像两个parabolas。如果用手图,或者将计算器设置为顺序模式(而不是同步),您可以看到 y = -x3 的图形实际上是 X 轴上的 Y = x3 的反射。

However, if you look at the graph, you can see that it is a reflection over the y -axis as well. This is the case because in order to obtain a reflection over the y -axis, we negate x . In other words, h ( x ) = f (- x ) is a reflection of f ( x ) over the y -axis. For the function y = x ³ , h ( x ) = (- x ) ³ = (- x ) (- x ) (- x ) = - x ³ . This is the same function as the one we have already graphed.
::然而,如果您查看图表,您可以看到它也是y轴的反射。这是因为为了在y轴上取得反射, 我们否定 x。换句话说, h(x) = f(x) = f(x) 是y轴上的 f(x) 反射。对于函数 y = x3, h(x) = (-x) = (-x) 3 = (-x) (-x) (x) (x) (x) (x) = -x3) = -x3, 这和我们已经绘制的函数相同。

It is important to note that this is a special case. The graph of y = x ² is also a special case. If we want to reflect y = x ² over the y -axis, we will just get the same graph! This can be explained algebraically: y = (- x ) ² = (- x ) (- x ) = x ² .
::必须指出,这是一个特例。 y = x2 的图形也是一个特例。如果我们想要在 y 轴上反映 y = x2, 我们只能得到相同的图! 这可以解释代数 : y = (- x) 2 = (- x) (- x) (- x) = x2 。

Example 5
::例5

Graph the functions $\begin{aligned} y = \sqrt{x} \end{aligned}$ and $\begin{aligned} y = \sqrt{- x} \end{aligned}$ .
::函数 y=x 和 yx 的图形。

The equation $\begin{aligned} y = \sqrt{- x} \end{aligned}$ might look confusing because of the - x under the square root . It is important to keep in mind that - x means the opposite of x . Therefore the domain of this function is restricted to values ≤ 0. For example, if x = - 4, $\begin{aligned} y = \sqrt{- (- 4)} = \sqrt{4} = 2 \end{aligned}$ . It is this domain, which includes all real numbers not in the domain of $\begin{aligned} y = \sqrt{x} \end{aligned}$ plus zero, that gives us a graph that is a reflection over the y -axis.
::yx 等式可能会因为平方根下的 -x 表示混淆。重要的是要记住 - x 表示 x 的反面。因此, 此函数的域限为 + + 0 。例如, 如果 x = - 4, y( 4) = 4= 2, 则此域包含不在 y=x + 0 域内的所有真实数字, 就会给我们一个反射于 Y 轴的图形。

In sum, a graph represents a reflection over the x -axis if the function has been negated (i.e. the y has been negated if we think of y = f ( x )). The graph represents a reflection over the y -axis if the variable x has been negated.
::总而言之,如果函数被否定(即如果我们想到 y = f(x),y y 被否定),一个图代表 x 轴的反射。如果变量 x 被否定,该图代表 y 轴的反射。

Example 6
::例6

Identify the function and sketch the graph of $\begin{aligned} y = \sqrt{x} \end{aligned}$ reflected over both axes.
::识别函数并绘制两个轴上反射的 y=x 的图形的草图。

To reflect the graph of $\begin{aligned} y = \sqrt{x} \end{aligned}$ over both axes, the function must be negated both outside and inside the root : $\begin{aligned} y = - \sqrt{- x} \end{aligned}$ . The negation (negative) outside of the root has the effect of reflecting the graph vertically, and the negation inside of the root reflects the graph horizontally. The image below shows three versions:
::要在两个轴上反射 y=x 的图形,函数必须从根的外部和内部反弹:yx。根的外部否定(负)效果是垂直反射图,根的反射效果是水平反射图。下图显示三个版本:

BLUE: $\begin{aligned} y = \sqrt{x} \end{aligned}$
::BLUE: Y=x

GREEN: $\begin{aligned} y = - \sqrt{x} \end{aligned}$
::你们这群家伙

RED: $\begin{aligned} y = \sqrt{- x} \end{aligned}$
::红:你x

Review
::回顾

If a function is multiplied by a coefficient, what will happen to the graph of the function?
::如果函数乘以系数,函数的图形会怎样?

What does multiplying a functions by a number greater than one create?
::将函数乘以一个数大于一个数会创造什么?

What happens when we a function by a number between 0 and 1?
::当我们以 0 和 1 之间的数字计算函数时会怎样?

In order to obtain a reflection over the y axis what do we have to do to x?
::为了在Y轴上取得反射,我们要对x做些什么?

How do we obtain a reflection over the x-axis?
::我们如何在 X 轴上取得反射?

Write a function that will create a horizontal compression of the following: $\begin{aligned} f (x) = x^{2} + 3 \end{aligned}$
::写入一个函数, 以创建以下水平压缩: f( x)=x2+3

Write a function that will horizontally stretch the following: $\begin{aligned} f (x) = x^{2} - 6 \end{aligned}$
::写入将水平延伸以下的函数: f( x) =x2 - 6

Rewrite the function $\begin{aligned} f (x) = - \sqrt{x} \end{aligned}$ to get a reflection over the x-axis.
::重写函数 f(x) x 以获得 X 轴的反射。

Rewrite the function $\begin{aligned} f (x) = \sqrt{x} \end{aligned}$ to get a reflection over the y-axis.
::重写函数 f( x) =x 以获得 Y 轴的反射。

Graph each of the following using transformations. Identify the translations and reflections.
::使用变换方法绘制下列各图。标明翻译和反射。

$\begin{aligned} f (x) = | x | - 2 \end{aligned}$
::f(xx) x2

$\begin{aligned} h (x) = \sqrt{x + 3} \end{aligned}$
::h(x)=x+3

$\begin{aligned} g (x) = \frac{1}{x + 1} \end{aligned}$
::g(x)=1x+1

$\begin{aligned} f (x) = - 4 x^{3} \end{aligned}$
:xx)4x3

$\begin{aligned} h (x) = (x + 3)^{3} + 1 \end{aligned}$
::h(x) = (x+3) 3+1

$\begin{aligned} f (x) = \frac{1}{3} (x - 3)^{2} + 1 \end{aligned}$
:xx)=13(x-3)2+1

$\begin{aligned} f (x) = - 4 \sqrt{x + 1} - 2 \end{aligned}$
:xx) 4x+1-2

$\begin{aligned} f (x) = \frac{2}{3 (x - 2)} + \frac{1}{4} \end{aligned}$
:xx)=23(x-2)+14

Let $\begin{aligned} y = f (x) \end{aligned}$ be the function defined by the line segment connecting the points (-1, 4) and (2, 5). Graph each of the following transformations of $\begin{aligned} y = f (x) \end{aligned}$ .
::Let y=f(x) 是连接点( 1, 4) 和点(2, 5) 的线条段定义的函数。

$\begin{aligned} y = f (x) + 1 \end{aligned}$
::y=f(x)+1

$\begin{aligned} y = f (x + 2) \end{aligned}$
::y=f( x+2)

$\begin{aligned} y = f (- x) \end{aligned}$
::y=f( - x)

$\begin{aligned} y = f (x + 3) - 2 \end{aligned}$
::y=f( x+3)-2

Given the graph of $\begin{aligned} y = x, \end{aligned}$ sketch the graph of each of the following transformations.
::根据 y =x 的图形,绘制以下每个变形的图形。

$\begin{aligned} y = x + 3 \end{aligned}$
::y=x+3 y=x+3

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

$\begin{aligned} y = - x \end{aligned}$
::yx

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

伸展和反射转换

Stretching and Reflecting Transformations ::伸展和反射转换

Stretching and Compressing Graphs ::伸展和压缩图

Reflecting Graphs Over the y-axis and x-axis ::Y 轴和 x 轴的反射图

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Review ::回顾

Review (Answers) ::回顾(答复)

Stretching and Reflecting Transformations
::伸展和反射转换

Stretching and Compressing Graphs
::伸展和压缩图

Reflecting Graphs Over the y-axis and x-axis
::Y 轴和 x 轴的反射图

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Example 6
::例6

Review
::回顾

Review (Answers)
::回顾(答复)