Polynomial & Rational Functions
AP Precalculus Topic 1 17:14 English narration · English + 中文 subtitles burned in
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Transcript
Look at this curve.
看这条曲线。
It rises, then falls, then rises again — these are its turning points.
它先上升，再下降，又上升——这些是转折点。
It cuts across the horizontal axis — these are the roots, where the function is equal to zero.
它穿过横轴——这些是零点， 也就是函数值为零的地方。
And far to the right, it races upward without stopping — that is its end behavior.
再往右，它一直向上冲去——这就是末端行为。
Turning points, roots, end behavior: three ways to read a shape.
转折点、零点、 末端行为：这是读懂一个图形的三种方式。
This unit teaches you to read every polynomial and rational function this way.
本单元教你用同样的方式读懂每一个多项式和有理函数。
Welcome to Unit One: polynomial and rational functions.
欢迎来到第一单元：多项式与有理函数。
We start with how one quantity changes with another, build up to polynomials and their zeros, and finish with rational functions and their asymptotes.
我们从一个量如何随另一个量变化讲起， 逐步建立多项式及其零点，最后讲到有理函数及其渐近线。
Let's begin.
让我们开始吧。
Start with the idea of a function.
先从函数的概念开始。
A function is a rule that maps each input to exactly one output.
函数是一条规则，它把每一个输入对应到恰好一个输出。
The set of allowed inputs is the domain; the set of outputs is the range.
允许的输入构成定义域，输出构成值域。
We call the input the independent variable and the output the dependent variable.
我们把输入叫自变量，输出叫因变量。
As the input changes, the output changes with it — in tandem.
输入一变，输出跟着变——这就是联动变化。
Over an interval, the function is increasing when a bigger input always gives a bigger output, and decreasing when a bigger input always gives a smaller output.
在一个区间上，若更大的输入总给出更大的输出， 函数就是递增的；若更大的输入总给出更小的输出，就是递减的。
You can read that straight off a graph of all the input–output pairs.
你可以直接从图上读出这一切。
Everything here starts with change.
这里的一切都从变化开始。
The average rate of change over an interval is the change in output divided by the change in input.
函数在一个区间上的平均变化率，等于输出的改变量除以输入的改变量。
On a graph, that is the slope of the secant line — the straight line joining two points on the curve.
在图上，它就是割线的斜率——连接曲线上两点的那条直线。
Take the curve y equals x squared.
看曲线 y 等于 x 的平方。
From one to four, the average rate is fifteen over three, which is five.
从一到四，平均变化率是十五除以三，等于五。
Now over a smaller interval, from one to two, it is three over one, which is three.
再看更小的区间，从一到二，是三除以一，等于三。
The rate itself changed — and that is exactly why this curve is not a straight line.
变化率本身变了——这正是这条曲线不是直线的原因。
The rate of change at a single point measures how fast the output is changing right there.
某一点处的变化率，衡量输出在那一点附近变化有多快。
You approximate it with average rates over small intervals around that point.
你用该点周围小区间上的平均变化率来近似它。
Comparing two points, the one with the larger average rate — over small enough intervals — is changing faster.
比较两点时，平均变化率更大的那一点——在足够小的区间上——变化更快。
A positive rate means both quantities move the same way; a negative rate means they move in opposite ways.
正的变化率表示两个量同向变化；负的变化率表示它们反向变化。
So the sign of the rate tells direction, and its size tells speed.
所以变化率的正负说明方向，大小说明快慢。
Here is a quick test for the type of a function.
这里有一个判断函数类型的快捷方法。
Take equally spaced inputs and list the outputs.
取等间距的输入，列出对应的输出。
Then take the gaps between neighbours — the first differences.
再取相邻输出之间的差—— 这叫一阶差分。
For a straight line, these first differences are constant.
对直线来说，一阶差分是常数。
Take the gaps again — the second differences.
再取一次差——这叫二阶差分。
For a quadratic, it is the second differences that are constant.
对二次函数来说， 是二阶差分为常数。
The degree of a polynomial is the smallest level at which these successive differences finally settle to a constant.
多项式的次数，就是这些逐次差分最终变为常数所需的最小层级。
For a linear function the average rate of change over any interval is constant — so the rate at which the rate changes is zero.
对线性函数，任意区间上的平均变化率都是常数——所以变化率本身的变化为零。
For a quadratic, average rates over equal-length intervals themselves form a linear pattern, so those rates change at a constant rate.
对二次函数，等长区间上的平均变化率本身呈线性规律，所以那些变化率以恒定速率在变。
That "rate of the rate" idea is how you tell the types apart.
这种「变化率的变化率」就是区分函数类型的方法。
You can see a quadratic in steel: a hanging bridge cable takes a parabola-like shape, the graph of a quadratic you can walk under.
你能在钢铁里看见二次函数： 悬索桥的缆索近似抛物线——那就是你可以走在下面的二次曲线。
Completing the square rewrites a quadratic so its vertex is obvious.
配方法把二次式改写成顶点一目了然的形式。
You rearrange the expression into a square of a linear piece, plus a constant.
你把式子整理成一个一次式的平方，再加上一个常数。
The square is never negative, so the constant is the lowest or highest output, and the input that zeros the linear piece is the vertex's x-coordinate.
平方永不为负，所以这个常数就是最低或最高输出，而令一次式为零的输入就是顶点的横坐标。
That vertex is where the parabola turns — its global maximum or minimum — and the average rate of change switches sign as you pass through it.
顶点正是抛物线转向的地方——它的全局极大或极小——平均变化率经过它时会改号。
A polynomial is a sum of power terms.
多项式是若干幂次项的和。
Its highest power is the degree, and the leading term — the highest-power term — controls the big picture.
它的最高次幂是次数，而首项——最高次的那一项——决定整体的大局。
Along the way the graph turns.
一路上图形会转向。
Where it stops rising and starts falling, we get a local — relative — extremum: a local maximum; where it stops falling and starts rising, a local minimum.
它由升转降的地方，是局部（也叫相对）极大值； 由降转升的地方，是局部极小值。
And where the curve changes the way it bends, from concave up to concave down, we have a point of inflection.
而曲线改变弯曲方向的地方——从向上开口变为向下开口——就是拐点。
Not every local maximum is a global maximum.
并非每一个局部极大都是全局极大。
The greatest local peak is the absolute maximum only when the polynomial actually has one.
局部高峰中最高的那个，只有当多项式真的有全局最大值时才是绝对最大。
Most polynomials are unbounded: every odd-degree polynomial runs off to plus or minus infinity, so it has no global max or min at all.
大多数多项式是无界的：每个奇数次多项式都会冲向正无穷或负无穷，因此根本没有全局最大或最小。
An even-degree polynomial does have either a global maximum or a global minimum, depending on the sign of the leading coefficient.
偶数次多项式则一定有全局最大值或全局最小值，取决于首项系数的正负。
And between any two distinct real zeros there is at least one local maximum or minimum — the graph has to turn somewhere to hit both.
而任意两个不同的实零点之间，至少有一个局部极大或极小——图形要碰到两个零点，中间必须转向。
A function has symmetry when it is even or odd.
函数在偶或奇时具有对称性。
Even means f of negative x equals f of x — the graph is a mirror across the vertical axis, like x squared or x to the fourth.
偶函数是指负 x 处的函数值等于 x 处的函数值—— 图形关于纵轴对称，例如 x 的平方或 x 的四次方。
Odd means f of negative x equals the negative of f of x — the graph looks the same after a half-turn about the origin, like x cubed or x to the fifth.
奇函数是指负 x 处的函数值等于 正 x 处函数值的相反数——图形绕原点旋转半圈后不变，例如 x 的立方或 x 的五次方。
To test, substitute negative x and simplify.
检验时把负 x 代入再化简。
Later you will meet the same idea: cosine is even and sine is odd.
以后你会再遇到同样的想法：余弦是偶函数，正弦是奇函数。
A zero, or root, is an input where the function is equal to zero.
零点，或者说根，是使函数值为零的输入。
For a real number, that value is a zero exactly when the matching linear factor divides the polynomial.
对一个实数来说，它是零点，当且仅当对应的一次因式 能整除这个多项式。
Look at this example.
看这个例子。
The part x plus two is a factor once — an odd number of times — so at that zero the graph passes straight through: it crosses the axis.
x 加二这一部分是一次因式——出现奇数次——所以在那个零点， 图形径直穿过：它穿过横轴。
But the part x minus one appears squared — an even number of times — so at that zero the graph does not cross; it is tangent to the axis and turns back.
但 x 减一这一部分是平方——出现偶数次——所以在那个零点， 图形不穿过；它只是与横轴相切，然后折返。
Counting multiplicities, a degree-n polynomial has exactly n complex zeros — some may be real, some not.
计及重数，一个 n 次多项式恰好有 n 个复数零点——其中一些可能是实数，一些不是。
Non-real zeros always come in conjugate pairs: if a plus b i is a zero, then a minus b i is also a zero.
非实零点总是成共轭对出现：若 a 加 b i 是零点，则 a 减 b i 也是零点。
On an Argand diagram a complex number is a point; its modulus is the distance from the origin, and its argument is the angle.
在复平面图上，复数是一个点；模是到原点的距离，辐角是夹角。
Real zeros give x-intercepts at the axis, and they are the endpoints of the intervals where you test whether the polynomial is positive or negative.
实零点在横轴上给出截距，它们也是你检验多项式正负时那些区间的端点。
End behavior asks where the graph goes as the input runs far to the left and far to the right.
末端行为问的是：当输入向左、向右无限远时，图形走向哪里。
Only the leading term matters — for very large inputs it dominates everything else.
只有首项重要——对很大的输入， 它压过其余所有项。
Two things decide it.
有两件事决定它。
First, is the degree even or odd?
第一，次数是偶还是奇？
For an even degree, both ends point the same way — both up if the leading coefficient is positive, both down if it is negative.
偶数次时，两端指向相同—— 首项系数为正则两端都朝上，为负则都朝下。
For an odd degree, the two ends point opposite ways.
奇数次时，两端指向相反。
So the sign of the leading coefficient, together with even or odd, fixes both ends completely.
所以首项系数的正负， 再加上次数的奇偶，就完全确定了两端。
Here is the whole picture in one chart.
这一张表把全貌收在一起。
As x goes to plus infinity or to minus infinity, the output of a nonconstant polynomial heads to plus infinity or to minus infinity.
当 x 趋向正无穷或负无穷时，非常数多项式的输出会趋向正无穷或负无穷。
Four combinations of even or odd degree and positive or negative leading coefficient fix all four arrow patterns.
次数的奇偶与首项系数的正负，四种组合固定了四种箭头图案。
For large absolute values of x the lower-degree terms are negligible — they never change which way the ends go.
对 x 的很大绝对值，低次项可以忽略——它们从不改变两端的走向。
Memorize the chart, and end behavior becomes a thirty-second decision on every exam polynomial.
记住这张表，末端行为在每道考试多项式题上就变成三十秒能定下来的判断。
Let's put it together.
我们把它组合起来。
Here is a polynomial to sketch.
这里有一个多项式要画。
Start with the end behavior.
先从末端行为开始。
The degree is three, and the leading coefficient is negative, so the graph comes down from the upper left and leaves to the lower right.
次数是三，首项系数是负的， 所以图形从左上方下来，向右下方离去。
Next, mark the zeros.
接着，标出零点。
At x equals minus one the factor appears once, so the curve crosses; at x equals two the factor is squared, so the curve only touches.
在 x 等于负一处因式只出现一次， 所以曲线穿过；在 x 等于二处因式是平方，所以曲线只相切。
Finally, join it into one smooth curve that obeys both the ends and the zeros.
最后，把它连成一条光滑的曲线， 同时符合两端和零点。
Now rational functions — one polynomial divided by another.
现在讲有理函数——一个多项式除以另一个多项式。
Their end behavior comes from a simple contest between the leading terms, top against bottom.
它们的末端行为，来自首项之间一场简单的较量， 上对下。
When the top wins — the numerator has the higher degree — the graph follows a polynomial; if that is a straight line, we call it a slant asymptote.
当上面赢——分子次数更高——图形跟随一个多项式；如果那是一条直线，我们称之为斜渐近线。
When the top and bottom have equal degrees, the ends level off at a horizontal line whose height is the ratio of the leading coefficients.
当上下次数相等时，两端稳定在一条水平线上，它的高度是首项系数之比。
And when the bottom wins — the denominator has the higher degree — the graph flattens toward zero.
当下面赢——分母次数更高—— 图形趋向于零。
Zoom in on the case where the numerator degree is one higher than the denominator.
聚焦分子次数恰好比分母高一次的情形。
Polynomial long division then produces a linear quotient plus a remainder that shrinks toward zero.
多项式长除法会得到一次商式，加上一个趋向于零的余项。
The graph therefore approaches that straight-line quotient — a slant, or oblique, asymptote.
因此图形逼近那条直线商式——斜渐近线，也叫斜向渐近线。
At a horizontal asymptote y equals b, the outputs get and stay arbitrarily close to b as x runs to either infinity.
对水平渐近线 y 等于 b，当 x 趋向任一无穷时，输出会任意接近 b 并保持在附近。
Same idea for a slant: the vertical gap between the curve and the line shrinks without bound as you go far out.
斜渐近线同理：曲线与直线之间的竖直距离，在远处会无限缩小。
Rational functions can break in two different ways.
有理函数会以两种不同的方式断开。
Where the denominator is zero and nothing cancels it, the function shoots off to infinity — that is a vertical asymptote, a line the graph rushes toward but never reaches.
当分母为零而没有任何因式约去时，函数冲向无穷—— 这就是竖直渐近线，一条图形不断逼近却永远到不了的直线。
But where a factor cancels from top and bottom, the graph is only missing a single point — that is a hole, a removable gap.
但当一个因式在分子分母中约去时， 图形只缺了一个点——这就是空洞，一个可去的缺口。
Same-looking algebra, two very different pictures — so always factor first.
看起来一样的代数式，画出来却大不相同—— 所以永远先分解因式。
The real zeros of a rational function are the real zeros of its numerator that are still in the domain — inputs that make the top zero without also making the bottom zero.
有理函数的实零点，是分子的实零点中仍在定义域内的那些——让分子为零却不让分母为零的输入。
Those zeros, together with the zeros of the denominator, split the number line into intervals.
这些零点连同分母的零点，把数轴分成若干区间。
On each open interval the function keeps one sign, so you test a sample point when you solve inequalities such as r of x greater than or equal to zero.
在每个开区间上函数保持同号， 所以解诸如 r 的 x 大于等于零这类不等式时，你取一个试验点即可。
Vertical asymptotes sit at denominator zeros that do not cancel; more precisely, when the multiplicity in the bottom exceeds the multiplicity in the top.
竖直渐近线出现在未约去的分母零点处；更精确地说，当分母中的重数超过分子中的重数时。
Near a vertical asymptote the denominator is nearly zero, so the function shoots to plus or minus infinity.
在竖直渐近线附近，分母几乎为零，所以函数冲向正无穷或负无穷。
Always check each side separately: the left and right can go opposite ways.
一定要左右两侧分开检查： 两侧可以走向相反。
This picture shows a curve racing up one side of a vertical line and down the other, while far out it levels toward a horizontal asymptote.
这幅图显示曲线在一条竖直线的一侧向上冲、另一侧向下冲， 而在远处则平缓趋向一条水平渐近线。
Reading both the vertical break and the far-end height is how you describe a rational graph completely.
读出竖直断裂和远端高度，才算完整描述有理函数图像。
Try this one.
试试这道。
Describe the rational function two x squared plus three, over x squared minus one.
描述有理函数：二 x 的平方加三，除以 x 的平方减一。
Compare the degrees: the top and bottom are both degree two — equal degrees — so the horizontal asymptote is the ratio two over one, that is, y equals two.
比较次数：上下都是二次—— 次数相等——所以水平渐近线是二比一，也就是 y 等于二。
Now the vertical asymptotes.
再看竖直渐近线。
The denominator, x squared minus one, is zero at x equals one and at x equals minus one, and neither cancels — so there are vertical asymptotes at both.
分母 x 的平方减一， 在 x 等于一和 x 等于负一处为零，而且都不约去——所以在这两处都有竖直渐近线。
A hole is a removable point.
空洞是可去的点。
It occurs at x equals c when a factor cancels — the multiplicity of the zero c in the numerator is at least its multiplicity in the denominator.
它出现在 x 等于 c，当某个因式被约去时—— 零点 c 在分子中的重数至少等于它在分母中的重数。
The graph is missing a single point.
图形只缺一个点。
Its height is the limit of the simplified function: if inputs near c give outputs near L, the hole sits at the point c comma L.
它的高度是简化后函数的极限：若 c 附近的输入给出接近 L 的输出，空洞就在点 c 逗号 L。
You find L by cancelling, then evaluating the reduced expression at c.
先约去再把 c 代入简化式，就能求出 L。
Never call a cancelled factor an asymptote.
永远不要把约去的因式叫作渐近线。
The same expression, written differently, reveals different features.
同一个表达式，写法不同，显露的特征也不同。
Factored form shows the real zeros, and therefore the x-intercepts, the holes, the vertical asymptotes, and the domain.
因式分解形式显示实零点， 从而显示横截距、空洞、竖直渐近线和定义域。
Standard form — the expanded sum of powers — shows the degree and the leading term, and therefore the end behavior.
标准形式——展开的幂次之和—— 显示次数和首项，从而显示末端行为。
When you need a slant asymptote, use polynomial long division: rewrite f of x as g of x times q of x plus r of x, where the remainder has smaller degree than the divisor.
需要斜渐近线时，用多项式长除法： 把 f 的 x 写成 g 的 x 乘以 q 的 x 加上 r 的 x，其中余式次数低于除式。
The quotient q is then the equation of the slant line when the numerator degree is one more than the denominator's.
当分子次数恰好比分母高一次时，商式 q 就是斜渐近线的方程。
Once you know a basic shape, transformations move it around without changing its rule.
一旦你认识了一个基本图形，变换就能在不改变它规则的情况下移动它。
Start with the parent parabola, y equals x squared.
从母抛物线 y 等于 x 的平方 开始。
Add a number inside or outside, and you get a translation — a shift — the same curve slid left, right, up, or down.
在里面或外面加一个数，就得到平移——同一条曲线向左、右、上、下滑动。
Multiply the output, and you get a dilation — a stretch — the curve pulled taller or pressed flatter.
把输出乘以一个数， 就得到伸缩——曲线被拉高或压扁。
Put a minus in front, and you get a reflection — the curve flipped over an axis.
在前面加一个负号，就得到反射——曲线沿某条轴翻转。
Translations, dilations, reflections: combine them to build any version of a shape you already know.
平移、伸缩、反射：把它们组合起来，就能构造出你已熟悉图形的任意版本。
Be precise with the formulas.
公式要说得精确。
Adding k outside — f of x plus k — shifts the graph up or down.
在外面加 k——f 的 x 加 k——使图形上下平移。
Replacing x by x minus h — f of x minus h — shifts right or left.
把 x 换成 x 减 h—— f 的 x 减 h——使图形左右平移。
Multiplying the output by a — a times f of x — stretches vertically.
把输出乘以 a——a 乘 f 的 x——是竖直伸缩。
Putting a number b inside — f of b x — stretches horizontally.
在里面放一个数 b——f 的 b x——是水平伸缩。
A minus in front, negative f of x, flips over the horizontal axis; f of negative x flips over the vertical axis.
前面加负号，负的 f 的 x，关于横轴翻转； f 的负 x 关于纵轴翻转。
Describe each move with the matching formula so the examiner can see you know which is which.
用对应的公式描述每一步，阅卷人才能看出你分得清。
Combine additive shifts and multiplicative stretches to model shifted, scaled versions of a known shape.
把加法平移与乘法伸缩组合起来，就能为已知图形的平移、缩放版本建模。
Start from the parent parabola, shift it, stretch it, and if needed reflect it.
从母抛物线出发，平移它、伸缩它，必要时再反射它。
Order matters in writing: usually you apply the horizontal moves inside the function, then the vertical moves outside.
书写时顺序很重要： 通常先在函数内部做水平变换，再在外部做竖直变换。
The picture shows several copies of the same family — same curve DNA, different position and scale.
图中是同一族的几个副本—— 曲线基因相同，位置与尺度不同。
Once you can name each move, you can reverse-engineer any transformed graph back to its parent.
一旦你能叫出每一步变换，就能把任意变换后的图还原到母函数。
Finally, how do you choose which function to model a situation?
最后，怎么选择用哪种函数来建模？
Match the model to how the quantity changes.
让模型匹配这个量如何变化。
A constant rate of change means a linear model; a constant second difference means a quadratic; several turns call for a higher-degree polynomial.
变化率恒定，就用线性模型； 二阶差分恒定，就是二次函数；出现多个转折，就需要更高次的多项式。
Geometry gives a strong hint too.
几何也给出很强的提示。
A quantity built from area, which is two-dimensional, is usually quadratic; a quantity built from volume, which is three-dimensional, is usually cubic.
由面积得到的量——面积是二维的——通常是二次的；由体积得到的量——体积是三维的—— 通常是三次的。
And whatever you choose, state your assumptions — a model is only trustworthy where those assumptions actually hold.
而无论你选哪种，都要说明你的假设——模型只在这些假设成立的地方才可信。
To construct a model, use the given features — points, intercepts, zeros with multiplicities, end behavior — to write the function, then use it to answer questions in context.
构造模型时，用给定特征——点、截距、带重数的零点、末端行为——写出函数， 再用它回答情境中的问题。
Always check the answer against the domain that makes sense for the situation, and interpret outputs with their real-world units.
始终对照情境中有意义的定义域检查答案， 并用现实单位解释输出。
A box's volume as a function of a cut length is a classic cubic: three linear dimensions multiply, so degree three.
盒子体积作为切口长度的函数是经典的三次模型： 三个一次尺寸相乘，所以次数是三。
State which pattern you assume continues, because outside that range the model may stop applying.
说明你假设哪种规律会延续， 因为超出那个范围，模型可能就不再适用。
A piecewise-defined function uses different rules over non-overlapping intervals of the domain.
分段函数在定义域互不重叠的区间上使用不同的规则。
To evaluate it, pick the branch whose interval contains the input.
求值时，选包含该输入的那一支。
For example, take f of x equal to x squared when x is less than zero, and two x when x is greater than or equal to zero.
例如，当 x 小于零时 f 的 x 等于 x 的平方，当 x 大于等于零时等于二 x。
Then f of negative three is nine from the first branch, but f of three is six from the second.
那么负三处的 f 从第一支得到九，而三处的 f 从第二支得到六。
Piecewise rules are useful when behaviour changes at a threshold — a tiered price, a speed limit that changes, a tax bracket.
当行为在某个阈值处改变时——阶梯价格、变化的限速、税率档——分段规则很有用。
Before you go, three marks students often lose.
结束之前，三个学生常失的分。
First, multiplicity decides the picture at a zero: an even power means the graph touches the axis, an odd power means it crosses.
第一，重数决定零点处的图形：偶次幂表示图形与横轴相切， 奇次幂表示穿过。
Second, for end behavior, look at the leading term only — the lower terms never change where the ends go.
第二，对末端行为，只看首项——低次项从不改变两端的走向。
Third, a cancelled factor makes a hole, not an asymptote — so always factor before you decide.
第三， 约去的因式产生空洞，而不是渐近线——所以在下结论之前一定先分解因式。
Get these right, and Unit One is yours.
把这些做对， 第一单元就是你的了。