Course: 1.12 纵向和横向转变

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垂直和水平转换

Horizontal and vertical transformations are two of the many ways to convert the basic parent functions in a function family into their more complex counterparts.
::横向和纵向转变是将功能家庭的基本母函数转换为更为复杂的对应功能的多种方式中的两种。

What vertical and/or horizontal shifts must be applied to the parent function of $\begin{aligned} y = x^{2} \end{aligned}$ in order to graph $\begin{aligned} g (x) = (x - 3)^{2} + 4 \end{aligned}$ ?
::为图g(x)=(x-3)2+4, Y=x2 的父函数必须应用什么垂直和(或)水平移动?

Vertical and Horizontal Transformations
::垂直和水平转换

Have you ever tried to draw a picture of a rabbit, or cat, or dog? Unless you are talented, even the most common animals can be a bit of a challenge to draw accurately (or even recognizably!). One trick that can help even the most "artistically challenged" to create a clearly recognizable basic sketch is demonstrated in nearly all "learn to draw" courses: start with basic shapes . By starting your sketch with simple circles, ellipses, rectangles, etc., the basic outline of the more complex figure is easily arrived at, then details can be added as necessary, but the figure is already recognizable for what it is.
::你有没有试过画兔子、猫、狗的画?除非你很有才华,否则,即使是最常见的动物也可能是准确画画(甚至可以辨认出来 ) 的一个挑战。几乎在所有的“精练画画”课程中都展示了一种能帮助甚至最“艺术挑战者”制作一个明显基本草图的技巧:从基本形状开始。通过以简单的圆形、椭圆、矩形等开始绘制你的草图,比较复杂的图的基本轮廓可以很容易地达成,然后可以在必要时添加细节,但数字已经可以识别出来。

The same trick works when graphing equations. By learning the basic shapes of different types of function graphs, and then adjusting the graphs with different types of transformations , even complex graphs can be sketched rather easily. This lesson will focus on two particular types of transformations: vertical shifts and horizontal shifts .
::当图形化方程式时,同样的把戏也可以用同样的把戏。通过学习不同类型函数图形的基本形状,然后用不同类型的变形调整图形,甚至复杂的图形也可以很容易地绘制。这个课程将侧重于两种特定的变形类型:垂直变形和水平变形。

We can express the application of vertical shifts this way:
::我们可以这样表达垂直转变的应用:

Formally: For any function f ( x ), the function g ( x ) = f ( x ) + c has a graph that is the same as f ( x ), shifted c units vertically . If c is positive, the graph is shifted up. If c is negative, the graph is shifted down.
::形式 : 对于任何 f(x) 函数, 函数 g(x) = f(x) + c 的图形与 f(x) 相同, 垂直移动 c 单位。如果 c 是正, 则向上移动。如果 c 是负, 则向下移动。

Informally: Adding a positive number after the x outside the " data-term="Parentheses" role="term" tabindex="0"> parentheses shifts the graph up, adding a negative (or subtracting) shifts the graph down.
::非正式: 在x 身外的括号中加上正数,将图向上移动,加上负(或减)将图向下移动。

We can express the application of horizontal shifts this way:
::我们可以这样表达横向转变的应用:

Formally: given a function f ( x ), and a constant a > 0, the function g ( x ) = f ( x - a ) represents a horizontal shift a units to the right from f ( x ). The function h ( x ) = f ( x + a ) represents a horizontal shift a units to the left.
::形式 : 给定函数 f( x) 和常数 > 0, 函数 g( x) = f( x) - a) 代表单位从 f( x) 向右水平移动单位。函数 h( x) = f( x) + a) 代表单位向左水平移动单位。

Informally: Adding a positive number after the x inside the parentheses shifts the graph left , adding a negative (or subtracting) shifts the graph right .
::非正式: 在xinide 后添加正数, 括号括号将图向左移动, 加上负( 减) 将图向右移动。

Examples
::实例

Example 1
::例1

Earlier, you were given a question about applying vertical and/or horizontal shifts to a parent function in order to graph a different function in the same function family.
::早些时候,您被问及对父函数应用垂直和(或)水平移动的问题,以便在同一函数族中绘制不同函数的图表。

What transformations must be applied to $\begin{aligned} y = x^{2} \end{aligned}$ , in order to graph $\begin{aligned} g (x) = (x - 3)^{2} + 4 \end{aligned}$ ?
::要图g(x)=(x-3)2+4,Y=x2必须应用什么变换?

The graph of $\begin{aligned} g (x) = (x - 3)^{2} + 4 \end{aligned}$ is the graph of $\begin{aligned} y = x^{2} \end{aligned}$ shifted 3 units to the right, and 4 units up.
::g(x) = (x-3) 2+4 的图形是 y=x2 向右移动 3 个单位, 向上移动 4 个单位。

Example 2
::例2

What must be done to the graph of y = x ² to convert it into the graphs of y = x ² - 3, and y = x ² + 4?
::要将 y = x2 的图形转换成 y = x2 - 3 和 y = x2 + 4 的图形,必须对 y = x2 = x2 + 4 的图形做些什么?

At first glance, it may seem that the graphs have different widths. For example, it might look like y = x ² + 4, the uppermost of the three parabolas, is thinner than the other two parabolas. However, this is not the case. The parabolas are congruent .
::乍一看,图表似乎有不同的宽度。例如,它可能看起来象是y = x2 + 4,即三个parabolas的顶部,比另外两个parabolas更薄。然而,情况并非如此。Parabolas是相同的。

If we shifted the graph of y = x ² up four units, we would have the exact same graph as y = x ² + 4. If we shifted y = x ² down three units, we would have the graph of y = x ² - 3.
::如果我们将 y = x2 的图形向四个单位移动, 我们将会拥有与 y = x2 + 4 相同的图形。如果我们向三个单位移动 y = x2, 我们将会拥有 y = x2 - 3 的图形。

Example 3
::例3

Identify the transformation(s) involved in converting the graph of f ( x ) = | x | into g ( x ) = | x - 3|.
::识别将 f(x) = {x} = {x} = = g(x) = {x - 3} 的图形转换的变换。

From the examples of vertical shifts above, you might think that the graph of g ( x ) is the graph of f(x), shifted 3 units to the left. However, this is not the case. The graph of g ( x ) is the graph of f ( x ), shifted 3 units to the right.
::从以上垂直移动的示例中,您可能会认为 g(x) 的图形是 f(x) 的图形, 将 3 个单位移到左边。但是, 情况并非如此。 g(x) 的图形是 f(x) 的图形, 将 3 个单位移到右边。

The direction of the shift makes sense if we look at specific function values.
::如果我们看看具体的功能值,转变的方向就有意义了。

x	g ( x ) = abs( x - 3)
0	3
1	2
2	1
3	0
4	1
5	2
6	3

From the table we can see that the vertex of the graph is the point (3, 0). The function values on either side of x = 3 are symmetric, and greater than 0.
::从表中可以看出,图形的顶点是点(3,0)。 x = 3 的两侧的函数值对称,大于0。

Example 4
::例4

What transformations must be applied to $\begin{aligned} y = x^{2} \end{aligned}$ , in order to graph $\begin{aligned} g (x) = (x + 2)^{2} - 2 \end{aligned}$ ?
::要图g(x)=(x+2)2-2,Y=x2必须应用什么变换?

The graph of $\begin{aligned} g (x) = (x + 2)^{2} - 2 \end{aligned}$ is the graph of $\begin{aligned} y = x^{2} \end{aligned}$ shifted 2 units to the left, and 2 units down.
::g(x) = (x+2) 2-2 的图形是 y=x2 向左移动 2 个单位, 向下移动 2 个单位。

Example 5
::例5

Use the parent function f(x) = x ² to graph f(x) = x ² + 3.
::使用父函数 f( x) = x2 到图形 f( x) = x2 + 3 。

The function f(x) = x ² is a parabola with the vertex at (0, 0).
::函数 f(x) = x2 是一个有顶点为 0, 0 的抛物线。

Adding outside the parenthesis shifts the graph vertically.
::括号外添加将图形垂直移动。

Therefore , f(x) = x ² + 3 will be a parabola with the vertex 3 units up.
::因此, f(x) = x2 + 3 将是一个有顶端 3 单元的抛物线。

Example 6
::例6

Use the parent function f(x) = |x| to graph f(x) = |x - 4|.
::使用父函数 f( x) =\\\\\\\\\\\\\\\\\\\\ x\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

The graph of the absolute value function family parent function f(x) = |x| is a large "V" with the vertex at the origin.
::绝对值函数的家庭父函数 f(x) =\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\"\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\F\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\

Adding or subtracting inside the parenthesis results in horizontal movement.
::括号内添加或减去导致水平移动。

Recall that the horizontal shift is right for negative numbers, and left for positive numbers.
::回顾横向变化对负数来说是正确的,对正数则是正确的。

Therefore f(x) = |x - 4| is a large "V" with the vertex 4 units to the right of the origin.
::因此 f(x) = x - 4 是一个大的“V” , 上面有源头右侧的顶部 4 个单位。

Review
::回顾

Graph the function $\begin{aligned} f (x) = 2 | x - 1 | - 3 \end{aligned}$ without a calculator.
::函数 f( x) = 2\\\\\\\\\\\\\\\\\\\\\\\3没有计算器的图形。

What is the vertex of the graph and how do you know?
::图形的顶点是什么? 你怎么知道的?

Does it open up or down and how do you know?
::开着还是下着呢? 你怎么知道的?

For the function: $\begin{aligned} f (x) = | x | + c \end{aligned}$ if c is positive, the graph shifts in what direction?
::对于函数 : f(x) x c 如果 c 是正, 图形会向哪个方向移动 ?

For the function: $\begin{aligned} f (x) = | x | + c \end{aligned}$ if c is negative, the graph shifts in what direction?
::对于函数 : f(x) xc 如果 c 是负的, 图形会向哪个方向移动 ?

The function $\begin{aligned} g (x) = | x - a | \end{aligned}$ represents a shift to the right or the left?
::函数 g(x)\\\\\\\\\a\\\\ 表示向右或向左的移动?

The function $\begin{aligned} h (x) = | x + a | \end{aligned}$ represents a shift to the right or the left?
::函数 h(x) x+a 表示向右或向左的移动 ?

If a graph is in the form $\begin{aligned} a \cdot f (x) \end{aligned}$ . What is the effect of changing the a ?
::如果图形以 af(x) 的形式出现。更改 a 的效果是什么 ?

Describe the transformation that has taken place for the parent function $\begin{aligned} f (x) = | x | \end{aligned}$ .
::描述父函数 f (x) 的变换情况。

$\begin{aligned} f (x) = | x | - 5 \end{aligned}$
::f(xx) x* 5

$\begin{aligned} f (x) = 5 | x + 7 | \end{aligned}$
:xx)=5x+7

Write an equation that reflects the transformation that has taken place for the parent function $\begin{aligned} g (x) = \frac{1}{x} \end{aligned}$ , for it to move in the following ways:
::写入反映父函数 g( x) = 1x 所发生变异的方程式, 以便以下列方式移动 :

Move two spaces up
::向上移动两个空格

Move four spaces to the right
::向右移动四个空格

Stretch it by 2 in the y-direction
::将之伸展在Y方向上,再伸展两下

Write an equation for each described transformation.
::为每个描述的变换写一个方程。

a V-shape shifted down 4 units.
::a V形状向下移动4个单元。

a V-shape shifted left 6 units
::a V-形状向左移动6个单元

a V-shape shifted right 2 units and up 1 unit.
::a V-形状向右移动2个单元,向上移动1个单元。

The following graphs are transformations of the parent function $\begin{aligned} f (x) = | x | \end{aligned}$ in the form of $\begin{aligned} f (x) = a | x - h | = k \end{aligned}$ . Graph or sketch each to observe the type of transformation.
::以下图表是父函数 f(x) 的转换, 以 f(x) = a* x- hk 的形式。图表或草图各用于观察转换类型。

$\begin{aligned} f (x) = | x | + 2 \end{aligned}$ . What happens to the graph when you add a number to the function? (i.e. f(x) + k).
::f(x) 2. 函数中添加数字时图形会怎样? (即 f(x) + k) 。)

$\begin{aligned} f (x) = | x | - 4 \end{aligned}$ . What happens to the graph when you subtract a number from the function? (i.e. f(x) - k).
::f(x) {x}} {x}} 。从函数中减去数字时, 图形会怎样? (即 f(x) - k) 。)

$\begin{aligned} f (x) = | x - 4 | \end{aligned}$ . What happens to the graph when you subtract a number in the function? (i.e. f(x - h)).
::f(x) x- 4}}。当您在函数中减去数字时, 图形会怎样? (即 f(x) - h) 。)

$\begin{aligned} f (x) = | x + 2 | \end{aligned}$ . What happens to the graph when you add a number in the function? (i.e. f(x + h)).
::f(x) x+2 。当您在函数中添加数字时, 图形会怎样? (即 f(x) + h) 。

Practice: Graph the following.
::实践:如下图所示。

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

$\begin{aligned} f (x) = \frac{5}{2} | x | \end{aligned}$
:xx)=52x

$\begin{aligned} f (x) = \frac{1}{2} | x | \end{aligned}$
:xx)=12x}(fx)=12x}(xx)=12x*}(xx)=12x*}(xx)=

$\begin{aligned} f (x) = \frac{2}{5} | x | \end{aligned}$
:xx)=25x}(fx)=25x*}(fx)=25x*}(fx)=25x*x*}(fx)=25

Let $\begin{aligned} f (x) = x^{2} \end{aligned}$ . Let $\begin{aligned} g (x) \end{aligned}$ be the function obtained by shifting the graph of $\begin{aligned} f (x) \end{aligned}$ two units to the right and then up three units. Find a formula for $\begin{aligned} g (x) \end{aligned}$ and then draw its graph
::letf(x) =x2. letg(x) 是将 f(x) 两个单位的图图向右移取的函数, 然后向上移三个单位。查找一个公式 forg( x) , 然后绘制它的图

Suppose H(t) gives the height of high tide in Hawaii(H) on a Tuesday, (t) of the year. Use shifts of the function H(t) to find formulas of each of the following functions:
::假设 H( t) 在当年的星期二, (t) 给予夏威夷( H) 高潮的高度。使用函数 H( t) 的移动来查找下列函数的公式 :

F(t), the height of high tide on Fiji on Tuesday (t), given that high tide in Fiji is always one foot higher than high tide in Hawaii.
::F(t),星期二(t)斐济高潮,因为斐济高潮总是比夏威夷高一英尺。

S(d), the height of high tide in Saint Thomas on Tuesday (t), given that high tide in Saint Thomas is the same height as the previous day's height in Hawaii.
::S(d),星期二(t)圣托马斯高潮,因为圣托马斯高潮与前一天夏威夷高潮相同。

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

垂直和水平转换

Vertical and Horizontal Transformations ::垂直和水平转换

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)