2.15 摘要:职能和图表

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  摘要:函数和图表

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  Chapter Summary
  ::章次摘要

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  In this chapter, we learned about:
  ::在本章中,我们了解到:

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    Parent functions for the following function families: identity function, quadratic function, cubic function, square root function, reciprocal function, exponential function, logarithm function, sine function, absolute value function, and logistic function. 
    ::下列函数系的父函数:身份功能、二次函数、立方函数、平根函数、对等函数、指数函数、对数函数、正弦函数、绝对值函数和后勤功能。

    
    ::下列函数系的父函数:身份功能、二次函数、立方函数、平根函数、对等函数、指数函数、对数函数、正弦函数、绝对值函数和后勤功能。
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    These parent functions can be transformed using:
    ::这些父函数可通过以下方式转换:

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    • Vertical shift:  $\begin{array}{r}y=f(x)+d\end{array}$  shift up,  $\begin{array}{r}y=f(x)-d\end{array}$  shift down
      ::垂直移动: y=f( x)+d 向上移动, y=f( x)-d 向下移动
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    • Horizontal shift:  $\begin{array}{r}y=f(x+c)\end{array}$  shift left,  $\begin{array}{r}y=f(x-c)\end{array}$  shift right
      ::水平移动: y=f( x+c) 向左, y=f( x- c) 向右
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    • Vertical stretch or shrink:  $\begin{array}{r}y=a\cdot f(x)\end{array}$
      ::垂直伸展或缩缩: y=af(x)
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    • Horizontal stretch or shrink:  $\begin{array}{r}y=f(ax)\end{array}$
      ::水平拉伸或缩缩: y=f( 轴)
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    • Reflection:  $\begin{array}{r}y=-f(x)\end{array}$  over the  $\begin{array}{r}x\end{array}$ -axis, $\begin{array}{r}y=f(-x)\end{array}$  over the $\begin{array}{r}y\end{array}$ -axis 
      ::反射 : x 轴的y f(x), y 轴的y=f(x)
    
    ::这些父函数可以使用下列方式转换: 垂直移动 : y=f( x)+d 向上移动, y=f( x) - d 向下向下向向上移动 : y=f( x+c) 向左移动 , y=f( x- c) 向右移动 垂直拉伸或缩缩 : y=a*f( x) 水平拉伸或缩缩缩 : y=f( 轴) 反射 : y=f( x) 轴, y=f( x) 向上
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  • Domain is the set of  inputs, and range i s the set of  outputs for a function.
    ::域是投入的一组,范围是函数的一组产出。
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  • The zeros or roots of a function are the $\begin{array}{r}x\end{array}$ -values where a function crosses the  $\begin{array}{r}x\end{array}$ -axis.
    ::函数的零或根是函数交叉 x 轴的 x 值。
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  • Maximums and minimums are the highest points and the lowest points, respectively, of a function. 
    ::最大值和最低值分别为函数的最高值和最低值。
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  • A function i s  increasing   if  $\begin{array}{r}f\left(b\right)\le f\left(c\right)\end{array}$  for any  $\begin{array}{r}b\text{ and }c\end{array}$  in the interval   whenever  $\begin{array}{r}b<c\end{array}$ , and a  function  is de creasing   if  $\begin{array}{r}f\left(b\right)\ge f\left(c\right)\end{array}$   whenever  $\begin{array}{r}b<c\end{array}$ .
    ::a 函数正在增加,如果 f(b)\f(c) 用于 b <c 时的任何 b 和 c 间隔;如果 f(b)\f(c) 时的一个函数正在减少,f(b)\f(c) 时的一个函数正在减少。
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  • A function is even if  $\begin{array}{r}f(-x)=f(x),\end{array}$  and is odd if  $\begin{array}{r}f(-x)=-f(x)\end{array}$ .   
    ::函数即使 f(- x) =f( x) ,即使 f(- x) 也为函数,如果 f(- x) =f( x) 则为函数,如果 f(- x) =f( x) 则为函数,如果 f(- x) =f( x) 则为函数奇特。
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  • A vertical  asymptote is a vertical line that visually shows where  the function is not defined. A horizontal asymptote  is a horizontal line  that a function  approaches  as  $\begin{array}{r}x\end{array}$  approaches positive or negative infinity.
    ::垂直无线线是指直观显示函数未定义位置的垂直线。水平无线线是指当 x 接近正或负无线时,函数会接近正或负无线的水平线。
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  • There are two  types of  discontinuities :  removable and non-removable. There are two types of non-removable discontinuities: jump and  infinite .
    ::有两种不连续:可移动和不可移动。有两种不可移动的不连续:跳跃和无限。
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  • A function has an inverse only if it is a one-to-one function. To algebraically solve for the inverse function, switch the variables in the function and solve for  $\begin{array}{r}y\end{array}$  in terms of  $\begin{array}{r}x\end{array}$ .
    ::函数只有一对一函数才具有反向函数。要对反函数进行代数解析,请切换函数中的变量,以 x 为y 解析。

   

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  Chapter Application Problem
  ::应用章节问题

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  Earlier in this chapter, we explored a function used to model the height of a ball. This model is a helpful one to review the scope and depth of techniques discussed in this chapter. Recall that the height of a ball can be given by   $\begin{array}{r}h\left(t\right)=-16t^2+80t+6,\end{array}$ where $\begin{array}{r}t\end{array}$  is the time in seconds since the ball left the thrower's hand.
  ::在本章前面,我们探讨了一个用来模拟球高度的函数。 这个模型有助于审查本章讨论的技术的范围和深度。 回顾球的高度可以由 h( t)\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\可以给的球球球球离开投手的秒。

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  The graph of $\begin{array}{r}h\left(t\right)\end{array}$  can be seen below:
  ::h(t) 图表如下:

   

  

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  T his graph is related to the "squaring" family of functions. The function can be rewritten into vertex form by completing the square: 

  $$\begin{array}{rl}h\left(t\right)& =-16t^2+80t+6\\ h\left(t\right)& =-16\left(t^2-5t+\frac{25}{4}\right)+6+100\\ h\left(t\right)& =-16{\left(t-\frac{5}{2}\right)}^2+106\end{array}$$

  
  ::此图与函数的“ 配方” 组合相关。 函数可以通过完成正方块( h( t) }}}}}}}}}}}}}}}}}}}}}}}§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§}}}}}§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§§

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  T he height function is a transformation of the squaring function, shifted to the right units, reflected in the horizontal axis, stretched vertically by a factor of 16, and shifted vertically up by 106 units. It is also clear that the path of the ball follows this graph during its flight.  
  ::高度函数是交错函数的转换, 转换为正确的单位, 反映于水平轴, 垂直拉伸16倍, 垂直向上移动106个单位。 另外, 球的路径在飞行时也明显跟随此图 。

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  Connecting this function to the family of squaring functions provides a wealth of information from the study of that function:
  ::将这一功能与配方功能家庭联系起来,从研究这一功能中可获得大量信息:

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  • The ball will reach the maximum height when $\begin{array}{r}t=\frac{5}{2}\end{array}$ .
    ::球在t=52时会达到最大高度。
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  • The ball's height increases until $\begin{array}{r}t=\frac{5}{2},\end{array}$  and then its height decreases until its height is 0 feet.
    ::球的高度升高到t=52, 然后它的高度下降到它的高度是0英尺。
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  • The domain of the function is about $\begin{array}{r}\left[0,5\right],\end{array}$  which is the time the ball is in the air.
    ::函数的域约为 [0,5] , 也就是球在空气中的时间 。
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  • Like the squaring function, the path of the ball is symmetrical about the vertical line  $\begin{array}{r}t=\frac{5}{2},\end{array}$ which goes  through its maximum. It is clear that the rate at which the ball increases in height until that point is the same as the rate at which it descends after that point. This is understood by any ball player preparing to catch a ball.
    ::和对齐函数一样,球的路径对称于垂直线 t=52, 垂直线 t=52 穿过最大线。 显然, 球在高度上上升的速率, 直到该点, 与该点之后的速率相同。 任何球员准备抓住球, 都可以理解这一点 。
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  • The function has the following intercepts, which further explain the path of the ball: The point (0,6) reflects the fact that the ball was thrown from a height of 6 feet. The point  (5.073,0) says that the ball hit the ground at about $\begin{array}{r}t\end{array}$  = 5.073 seconds. 
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    •   $\begin{array}{r}x\end{array}$ -axis intercept (5.073, 0)     
      ::X轴拦截(5.073,0)
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    •   $\begin{array}{r}y\end{array}$ -axis intercept (0,6)    
      ::Y 轴拦截 (0,6)
    
    ::函数有以下拦截功能,可以进一步解释球的路径:点(0,6)反映球从6英尺高处投出的事实。点(5.073,0)表示球在约t=5.073秒处击中地面。x轴拦截(5.073,0)y-轴拦截(0,6)
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  • It is also clear that since this function does not pass the horizontal line test, it does not have an inverse.  
    ::同样清楚的是,由于该函数没有通过水平线测试,因此没有反向测试。

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  Because this type of function has so many applications, we will be study it in detail in the next chapter.
  ::由于这种职能有许多应用,我们将在下一章详细研究。

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  Review
  ::回顾

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  Try the following cumulative review problems to practice the concepts in this chapter:
  ::尝试下列累积审查问题来实践本章中的概念: