剑桥 IGCSE IGCSE, A-Level & AP 练习单预览

多项式函数与有理函数

AP 微积分预备 · 第 1 主题

训练
讲义 词汇表
1.1

协同变化
大纲
Learning ObjectiveEssential Knowledge

1.1.A
Describe how the input and output values of a function vary together by comparing function values.

  • 1.1.A.1 A function is a mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value. The set of input values is called the domain of the function, and the set of output values is called the range of the function. The variable representing input values is called the independent variable, and the variable representing output values is called the dependent variable.
  • 1.1.A.2 The input and output values of a function vary in tandem according to the function rule, which can be expressed graphically, numerically, analytically, or verbally.
  • 1.1.A.3 A function is increasing over an interval of its domain if, as the input values increase, the output values always increase. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) < f(b)$.
  • 1.1.A.4 A function is decreasing over an interval of its domain if, as the input values increase, the output values always decrease. That is, for all $a$ and $b$ in the interval, if $a < b$, then $f(a) > f(b)$.

1.1.B
Construct a graph representing two quantities that vary with respect to each other in a contextual scenario.

  • 1.1.B.1 The graph of a function displays a set of input-output pairs and shows how the values of the function's input and output values vary.
  • 1.1.B.2 A verbal description of the way aspects of phenomena change together can be the basis for constructing a graph.
  • 1.1.B.3 The graph of a function is concave up on intervals in which the rate of change is increasing.
  • 1.1.B.4 The graph of a function is concave down on intervals in which the rate of change is decreasing.
  • 1.1.B.5 The graph intersects the $x$-axis when the output value is zero. The corresponding input values are said to be zeros of the function.

来源:美国大学理事会 AP 课程与考试说明

一个函数(function)是一个把每个输入映射到恰好一个输出的规则。允许的输入的集合是定义域(domain);输出的集合是值域(range)。输入变量是自变量(independent variable)而输出变量是因变量(dependent variable)。一个函数规则能用图象、数值、解析式(一个公式),或文字展示。

随着输入变化,输出"齐头并进"地——一起——变化。在一个区间上,一个函数是:

  • 递增(increasing),若每当 $a 时,那么 $f(a)(更大的输入、更大的输出);
  • 递减(decreasing),若每当 $a 时,那么 $f(a)>f(b)$(更大的输入、更小的输出)。

一个函数的图象显示它所有的输入-输出对,所以你能从图片直接读这个行为。

探索

Watch two quantities change together

y = ax² + bx + c

Precalculus studies how an output changes in tandem with its input. Read the curve left to right: where it is steep, $y$ changes a lot for a small change in $x$; where it is flat, $y$ barely moves.

词汇表 训练
英文 中文 拼音
function 函数 hán shù
domain 定义域 dìng yì yù
range 值域 zhí yù
independent variable 自变量 zì biàn liàng
dependent variable 因变量 yīn biàn liàng
increasing 递增 dì zēng
decreasing 递减 dì jiǎn
练习卷
1.2

变化率
大纲
Learning ObjectiveEssential Knowledge

1.2.A
Compare the rates of change at two points using average rates of change near the points.

  • 1.2.A.1 The average rate of change of a function over an interval of the function's domain is the constant rate of change that yields the same change in the output values as the function yielded on that interval of the function's domain. It is the ratio of the change in the output values to the change in input values over that interval.
  • 1.2.A.2 The rate of change of a function at a point quantifies the rate at which output values would change were the input values to change at that point. The rate of change at a point can be approximated by the average rates of change of the function over small intervals containing the point, if such values exist.
  • 1.2.A.3 The rates of change at two points can be compared using average rate of change approximations over sufficiently small intervals containing each point, if such values exist.

1.2.B
Describe how two quantities vary together at different points and over different intervals of a function.

  • 1.2.B.1 Rates of change quantify how two quantities vary together.
  • 1.2.B.2 A positive rate of change indicates that as one quantity increases or decreases, the other quantity does the same.
  • 1.2.B.3 A negative rate of change indicates that as one quantity increases, the other decreases.

来源:美国大学理事会 AP 课程与考试说明

一个函数在一个区间上的平均变化率(average rate of change)是输出的变化除以输入的变化——会给出相同总变化的单个恒定的率。在 $[a,b]$ 上:

$$\text{avg rate} = \frac{f(b)-f(a)}{b-a}.$$

一个点处的变化率(rate of change at a point)测量输出恰好在那个输入处变化多快。你用点周围小区间上的平均率近似它。比较两个点,有更大平均率(在足够小的区间上)的那个变化更快。一个正的率意味着两个量朝相同方向移动;一个负的率意味着它们朝相反方向移动。

Worked example. 对于 $f(x)=x^2$,在 $[1,4]$ 上的平均变化率是 $\dfrac{f(4)-f(1)}{4-1}=\dfrac{16-1}{3}=5$。在 $[1,2]$ 上它是 $\dfrac{4-1}{1}=3$ ——率本身变化,这正是为什么 $f$ 不是线性的。

词汇表 训练
英文 中文 拼音
average rate of change 平均变化率 píng jūn biàn huà lǜ
练习卷
1.3

线性函数与二次函数的变化率
大纲
Learning ObjectiveEssential Knowledge

1.3.A
Determine the average rates of change for sequences and functions, including linear, quadratic, and other function types.

  • 1.3.A.1 For a linear function, the average rate of change over any length input-value interval is constant.
  • 1.3.A.2 For a quadratic function, the average rates of change over consecutive equal-length input-value intervals can be given by a linear function.
  • 1.3.A.3 The average rate of change over the closed interval $[a, b]$ is the slope of the secant line from the point $(a, f(a))$ to $(b, f(b))$.

1.3.B
Determine the change in the average rates of change for linear, quadratic, and other function types.

  • 1.3.B.1 For a linear function, since the average rates of change over consecutive equal-length input-value intervals can be given by a constant function, these average rates of change for a linear function are changing at a rate of zero.
  • 1.3.B.2 For a quadratic function, since the average rates of change over consecutive equal-length input-value intervals can be given by a linear function, these average rates of change for a quadratic function are changing at a constant rate.
  • 1.3.B.3 When the average rate of change over equal-length input-value intervals is increasing for all small-length intervals, the graph of the function is concave up. When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals, the graph of the function is concave down.

来源:美国大学理事会 AP 课程与考试说明

在 $[a,b]$ 上的平均变化率是从 $(a,f(a))$ 到 $(b,f(b))$ 的割线(secant line)的斜率。

配方法给出一条抛物线的顶点
配方法给出一条抛物线的顶点
  • 对于一个线性(linear)函数,在任何区间上的平均变化率是恒定的——所以率变化的率是零。
  • 对于一个二次(quadratic)函数,在等长区间上的平均变化率本身形成一个线性模式——所以那些率以一个恒定的率变化。

这个"率的率"思想区别函数类型:一个恒定的二阶差标志一个二次。

探索

Explore a quadratic's changing rate

y = ax² + bx + c

Move the sliders and watch the parabola 抛物线 tilt. Its average rate of change is not constant — the slope is positive on one side of the vertex 顶点 and negative on the other. Notice how $a$ opens it up or down.

词汇表 训练
英文 中文 拼音
secant line 割线 gē xiàn
linear 线性 xiàn xìng
quadratic 二次 èr cì
练习卷
1.4

多项式函数与变化率
大纲
Learning ObjectiveEssential Knowledge

1.4.A
Identify key characteristics of polynomial functions related to rates of change.

  • 1.4.A.1 A nonconstant polynomial function of $x$ is any function representation that is equivalent to the analytical form $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \ldots + a_2 x^2 + a_1 x + a_0$, where $n$ is a positive integer, $a_i$ is a real number for each $i$ from $1$ to $n$, and $a_n$ is nonzero. The polynomial has degree $n$, the leading term is $a_n x^n$, and the leading coefficient is $a_n$. A constant is also a polynomial function of degree zero.
  • 1.4.A.2 Where a polynomial function switches between increasing and decreasing, or at the included endpoint of a polynomial with a restricted domain, the polynomial function will have a local, or relative, maximum or minimum output value. Of all local maxima, the greatest is called the global, or absolute, maximum. Likewise, the least of all local minima is called the global, or absolute, minimum.
  • 1.4.A.3 Between every two distinct real zeros of a nonconstant polynomial function, there must be at least one input value corresponding to a local maximum or local minimum.
  • 1.4.A.4 Polynomial functions of an even degree will have either a global maximum or a global minimum.
  • 1.4.A.5 Points of inflection of a polynomial function occur at input values where the rate of change of the function changes from increasing to decreasing or from decreasing to increasing. This occurs where the graph of a polynomial function changes from concave up to concave down or from concave down to concave up.

来源:美国大学理事会 AP 课程与考试说明

一个非常数多项式(polynomial)有形式

$$p(x)=a_n x^n + a_{n-1}x^{n-1}+\cdots+a_1 x + a_0,\quad a_n\neq 0.$$
它的次数(degree)是 $n$、它的首项(leading term)是 $a_n x^n$,而它的首项系数(leading coefficient)是 $a_n$。一个常数是一个 $0$ 次的多项式。

关键特征:

  • 一个多项式在递增和递减之间切换的地方,它有一个局部(相对)极值(local (relative) extremum)。只有当多项式确实有全局极值时,最大的局部极大值才是全局(绝对)最大值(global (absolute) maximum)——但大多数多项式是无界的(每个奇次多项式都会跑向 $\pm\infty$),所以它们根本没有全局最大值或最小值。
  • 在任何两个不同的实零点之间有至少一个局部极大值或极小值。
  • 一个偶次(even-degree)多项式有一个全局最大值或一个全局最小值。
  • 一个拐点(point of inflection)是图象改变凹凸性的地方——从上凹(concave up)到下凹(concave down)或反过来——即变化率在递增和递减之间切换的地方。
  • 一个函数有对称性,当它是偶函数(even)时——$f(-x)=f(x)$,图象关于 $y$ 轴对称(像 $x^2$、$x^4$)——或是奇函数(odd)时——$f(-x)=-f(x)$,图象关于原点旋转对称(像 $x^3$、$x^5$)。用代入 $-x$ 并化简来检验;后面,$\cos\theta$ 是偶的而 $\sin\theta$ 是奇的。
词汇表 训练
英文 中文 拼音
polynomial 多项式 duō xiàng shì
degree 次数 cì shù
leading term 首项 shǒu xiàng
leading coefficient 首项系数 shǒu xiàng xì shù
local (relative) extremum 局部极值 jú bù jí zhí
global (absolute) maximum 全局最大值 quán jú zuì dà zhí
point of inflection 拐点 guǎi diǎn
concave up 上凹 shàng āo
concave down 下凹 xià āo
even 偶函数 ǒu hán shù
odd 奇函数 jī hán shù
练习卷
1.5

多项式函数与复数零点
大纲
Learning ObjectiveEssential Knowledge

1.5.A
Identify key characteristics of a polynomial function related to its zeros when suitable factorizations are available or with technology.

  • 1.5.A.1 If $a$ is a complex number and $p(a) = 0$, then $a$ is called a zero of the polynomial function $p$, or a root of $p(x) = 0$. If $a$ is a real number, then $(x - a)$ is a linear factor of $p$ if and only if $a$ is a zero of $p$.
  • 1.5.A.2 If a linear factor $(x - a)$ is repeated $n$ times, the corresponding zero of the polynomial function has a multiplicity $n$. A polynomial function of degree $n$ has exactly $n$ complex zeros when counting multiplicities.
  • 1.5.A.3 If $a$ is a real zero of a polynomial function $p$, then the graph of $y = p(x)$ has an $x$-intercept at the point $(a, 0)$. Consequently, real zeros of a polynomial can be endpoints for intervals satisfying polynomial inequalities.
  • 1.5.A.4 If $a + bi$ is a non-real zero of a polynomial function $p$, then its conjugate $a - bi$ is also a zero of $p$.
  • 1.5.A.5 If the real zero, $a$, of a polynomial function has even multiplicity, then the signs of the output values are the same for input values near $x = a$. For these polynomial functions, the graph will be tangent to the $x$-axis at $x = a$.
  • 1.5.A.6 The degree of a polynomial function can be found by examining the successive differences of the output values over equal-interval input values. The degree of the polynomial function is equal to the least value $n$ for which the successive $n$th differences are constant.

1.5.B
Determine if a polynomial function is even or odd.

  • 1.5.B.1 An even function is graphically symmetric over the line $x = 0$ and analytically has the property $f(-x) = f(x)$. If $n$ is even, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an even function.
  • 1.5.B.2 An odd function is graphically symmetric about the point $(0, 0)$ and analytically has the property $f(-x) = -f(x)$. If $n$ is odd, then a polynomial of the form $p(x) = a_n x^n$, where $n \geq 1$ and $a_n \neq 0$, is an odd function.

来源:美国大学理事会 AP 课程与考试说明

$p$ 的一个零点(zero)(或根(root))是一个 $p(a)=0$ 的数 $a$。对于一个实 $a$,$(x-a)$ 恰好当 $a$ 是一个零点时是 $p$ 的一个因式。若因式 $(x-a)$ 被重复 $n$ 次,那个零点有重数(multiplicity)$n$。计入重数,一个 $n$ 次多项式恰好有 $n$ 个复数(complex)零点。

一个阿甘图上的一个复数:模是距离、辐角是角
一个阿甘图上的一个复数:模是距离、辐角是角
  • 一个实零点 $a$ 在 $(a,0)$ 给出一个 $x$ 截距;实零点是 $p(x)\ge 0$ 或 $\le 0$ 的区间的端点。
  • 非实零点成共轭(conjugate)对:若 $a+bi$ 是一个零点,$a-bi$ 也是。
  • 在一个偶重数的实零点处图象与 $x$ 轴相切(tangent)(它触及但不穿越);在奇重数它穿越。
  • 次数等于等距输出的第 $n$ 个逐次差(successive differences)变得恒定的最小 $n$。
词汇表 训练
英文 中文 拼音
zero 零点 líng diǎn
root 根 gēn
multiplicity 重数 chóng shù
complex 复数 fù shù
conjugate 共轭 gòng è
练习卷
1.6

多项式函数与末端行为
大纲
Learning ObjectiveEssential Knowledge

1.6.A
Describe end behaviors of polynomial functions.

  • 1.6.A.1 As input values of a nonconstant polynomial function increase without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} p(x) = \infty$ or $\lim_{x \to \infty} p(x) = -\infty$.
  • 1.6.A.2 As input values of a nonconstant polynomial function decrease without bound, the output values will either increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to -\infty} p(x) = \infty$ or $\lim_{x \to -\infty} p(x) = -\infty$.
  • 1.6.A.3 The degree and sign of the leading term of a polynomial determines the end behavior of the polynomial function, because as the input values increase or decrease without bound, the values of the leading term dominate the values of all lower-degree terms.

来源:美国大学理事会 AP 课程与考试说明

末端行为(end behavior)描述随着输入无界地增长一个函数走向哪里。对于一个非常数多项式,随着 $x\to\pm\infty$ 输出走向 $+\infty$ 或 $-\infty$,例如写 $\lim_{x\to\infty}p(x)=\infty$。哪个方向完全取决于首项 $a_n x^n$,因为对于大的 $|x|$ 它主导所有低次的项:$a_n$ 的符号以及 $n$ 是偶还是奇固定两端。

每个多项式的两端由它次数的奇偶性和它首项系数的符号决定
每个多项式的两端由它次数的奇偶性和它首项系数的符号决定
探索

See how the leading term sets the end behavior

y = ax³ + bx² + cx + d

The leading term decides the ends. For a cubic with $a>0$ the graph falls on the left and rises on the right; make $a<0$ and the two ends swap. As $x\to\pm\infty$ the highest-power term dominates every other term.

词汇表 训练
英文 中文 拼音
End behavior 末端行为 mò duān xíng wéi
练习卷
1.7

有理函数与末端行为
大纲
Learning ObjectiveEssential Knowledge

1.7.A
Describe end behaviors of rational functions.

  • 1.7.A.1 A rational function is analytically represented as a quotient of two polynomial functions and gives a measure of the relative size of the polynomial function in the numerator compared to the polynomial function in the denominator for each value in the rational function's domain.
  • 1.7.A.2 The end behavior of a rational function will be affected most by the polynomial with the greater degree, as its values will dominate the values of the rational function for input values of large magnitude. For input values of large magnitude, a polynomial is dominated by its leading term. Therefore, the end behavior of a rational function can be understood by examining the corresponding quotient of the leading terms.
  • 1.7.A.3 If the polynomial in the numerator dominates the polynomial in the denominator for input values of large magnitude, then the quotient of the leading terms is a nonconstant polynomial, and the original rational function has the end behavior of that polynomial. If that polynomial is linear, then the graph of the rational function has a slant asymptote parallel to the graph of the line.
  • 1.7.A.4 If neither polynomial in a rational function dominates the other for input values of large magnitude, then the quotient of the leading terms is a constant, and that constant indicates the location of a horizontal asymptote of the graph of the original rational function.
  • 1.7.A.5 If the polynomial in the denominator dominates the polynomial in the numerator for input values of large magnitude, then the quotient of the leading terms is a rational function with a constant in the numerator and nonconstant polynomial in the denominator, and the graph of the original rational function has a horizontal asymptote at $y = 0$.
  • 1.7.A.6 When the graph of a rational function $r$ has a horizontal asymptote at $y = b$, where $b$ is a constant, the output values of the rational function get arbitrarily close to $b$ and stay arbitrarily close to $b$ as input values increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to \infty} r(x) = b$ or $\lim_{x \to -\infty} r(x) = b$.

来源:美国大学理事会 AP 课程与考试说明

一个有理函数(rational function)是两个多项式的一个商,$r(x)=\dfrac{\text{numerator}}{\text{denominator}}$。它的末端行为由首项的商支配:

当分子有更高的次数时,曲线趋近一条倾斜的渐近线
当分子有更高的次数时,曲线趋近一条倾斜的渐近线
  • 分子次数 > 分母次数: 商是一个非常数多项式,而 $r$ 遵循那个多项式的末端行为。若那个商是线性的,图象有一条斜渐近线(slant asymptote)。
  • 相等的次数: 商是一个常数,给出一条水平渐近线(horizontal asymptote)$y=$ 首项系数之比。
  • 分子次数 < 分母次数: 商趋于 $0$,所以水平渐近线是 $y=0$。

在一条水平渐近线 $y=b$ 处,输出得到并保持任意地接近 $b$:$\lim_{x\to\pm\infty}r(x)=b$。

词汇表 训练
英文 中文 拼音
rational function 有理函数 yǒu lǐ hán shù
slant asymptote 斜渐近线 xié jiàn jìn xiàn
horizontal asymptote 水平渐近线 shuǐ píng jiàn jìn xiàn
练习卷
1.8

有理函数与零点
大纲
Learning ObjectiveEssential Knowledge

1.8.A
Determine the zeros of rational functions.

  • 1.8.A.1 The real zeros of a rational function correspond to the real zeros of the numerator for such values in its domain.
  • 1.8.A.2 The real zeros of both polynomial functions of a rational function $r$ are endpoints or asymptotes for intervals satisfying the rational function inequalities $r(x) \geq 0$ or $r(x) \leq 0$.

来源:美国大学理事会 AP 课程与考试说明

一个有理函数的实零点是它分子仍然在定义域里的实零点。这些零点,连同分母的零点,把数轴分成你在解不等式 $r(x)\ge 0$ 或 $r(x)\le 0$ 时测试的区间。

练习卷
1.9

有理函数与竖直渐近线
大纲
Learning ObjectiveEssential Knowledge

1.9.A
Determine vertical asymptotes of graphs of rational functions.

  • 1.9.A.1 If the value $a$ is a real zero of the polynomial function in the denominator of a rational function and is not also a real zero of the polynomial function in the numerator, then the graph of the rational function has a vertical asymptote at $x = a$. Furthermore, a vertical asymptote also occurs at $x = a$ if the multiplicity of $a$ as a real zero in the denominator is greater than its multiplicity as a real zero in the numerator.
  • 1.9.A.2 Near a vertical asymptote, $x = a$, of a rational function, the values of the polynomial function in the denominator are arbitrarily close to zero, so the values of the rational function $r$ increase or decrease without bound. The corresponding mathematical notation is $\lim_{x \to a^+} r(x) = \infty$ or $\lim_{x \to a^+} r(x) = -\infty$ for input values near $a$ and greater than $a$, and $\lim_{x \to a^-} r(x) = \infty$ or $\lim_{x \to a^-} r(x) = -\infty$ for input values near $a$ and less than $a$.

来源:美国大学理事会 AP 课程与考试说明

一条垂直渐近线(vertical asymptote)在 $x=a$ 出现,当 $a$ 是分母的一个零点但不被分子约去时——更精确地,当它在分母里的重数超过它在分子里的重数时。它附近,分母近乎零,所以 $r$ 射向 $\pm\infty$:$\lim_{x\to a^{\pm}}r(x)=\pm\infty$。检查每一侧,因为两侧能走相反的方向。

Worked example. 描述 $r(x)=\dfrac{2x^2+3}{x^2-1}$。分子和分母有相等的次数,所以水平渐近线是 $y=\dfrac{2}{1}=2$。分母 $x^2-1$ 在 $x=\pm1$ 为零而两者都不约去,所以在 $x=1$ 和 $x=-1$ 有垂直渐近线。

一个有理函数趋近一条垂直渐近线和一条水平渐近线
一个有理函数趋近一条垂直渐近线和一条水平渐近线
探索

Explore a vertical asymptote

y = a/(x − b) + c

This is $y = \dfrac{a}{x-b} + c$. The graph shoots off toward $\pm\infty$ at the vertical asymptote 竖直渐近线 $x = b$ (where the bottom is zero) and levels off toward the horizontal asymptote 水平渐近线 $y = c$. Slide $b$ and $c$ to move each line.

词汇表 训练
英文 中文 拼音
vertical asymptote 垂直渐近线 chuí zhí jiàn jìn xiàn
练习卷
1.10

有理函数与空洞
大纲
Learning ObjectiveEssential Knowledge

1.10.A
Determine holes in graphs of rational functions.

  • 1.10.A.1 If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator, then the graph of the rational function has a hole at the corresponding input value.
  • 1.10.A.2 If the graph of a rational function $r$ has a hole at $x = c$, then the location of the hole can be determined by examining the output values corresponding to input values sufficiently close to $c$. If input values sufficiently close to $c$ correspond to output values arbitrarily close to $L$, then the hole is located at the point with coordinates $(c, L)$. The corresponding mathematical notation is $\lim_{x \to c} r(x) = L$. It should be noted that $\lim_{x \to c^-} r(x) = \lim_{x \to c^+} r(x) = \lim_{x \to c} r(x) = L$.

来源:美国大学理事会 AP 课程与考试说明

一个空洞(hole)(可去点)在 $x=c$ 出现,当一个因式约去时——分子里零点 $c$ 的重数至少是它在分母里的重数。图象缺失一个单一的点。它的高度是化简后函数的极限:若 $c$ 附近的输入给出 $L$ 附近的输出,空洞在 $(c,L)$,而 $\lim_{x\to c}r(x)=L$。

词汇表 训练
英文 中文 拼音
hole 空洞 kōng dòng
练习卷
1.11

多项式与有理表达式的等价表示
大纲
Learning ObjectiveEssential Knowledge

1.11.A
Rewrite polynomial and rational expressions in equivalent forms.

  • 1.11.A.1 Because the factored form of a polynomial or rational function readily provides information about real zeros, it can reveal information about $x$-intercepts, asymptotes, holes, domain, and range.
  • 1.11.A.2 The standard form of a polynomial or rational function can reveal information about end behaviors of the function.
  • 1.11.A.3 The information extracted from different analytic representations of the same polynomial or rational function can be used to answer questions in context.

1.11.B
Determine the quotient of two polynomial functions using long division.

  • 1.11.B.1 Polynomial long division is an algebraic process similar to numerical long division involving a quotient and remainder. If the polynomial $f$ is divided by the polynomial $g$, then $f$ can be rewritten as $f(x) = g(x)q(x) + r(x)$, where $q$ is the quotient, $r$ is the remainder, and the degree of $r$ is less than the degree of $g$.
  • 1.11.B.2 The result of polynomial long division is helpful in finding equations of slant asymptotes for graphs of rational functions.

1.11.C
Rewrite the repeated product of binomials using the binomial theorem.

  • 1.11.C.1 The binomial theorem utilizes the entries in a single row of Pascal's Triangle to more easily expand expressions of the form $(a + b)^n$, including polynomial functions of the form $p(x) = (x + c)^n$, where $c$ is a constant.

来源:美国大学理事会 AP 课程与考试说明

同一个表达式,以不同的方式写,揭示不同的特征:

  • 因式形式显示实零点,因而 $x$ 截距、空洞、垂直渐近线和定义域。
  • 标准形式(展开)显示次数和首项,因而末端行为。

多项式长除法(polynomial long division)重写 $f(x)=g(x)\,q(x)+r(x)$,其中 $q$ 是商(quotient)而 $r$ 是余数(remainder)(次数小于 $g$)。当分子的次数比分母的大一时,商给出一条斜渐近线的方程。

词汇表 训练
英文 中文 拼音
Polynomial long division 多项式长除法 duō xiàng shì zhǎng chú fǎ
quotient 商 shāng
remainder 余数 yú shù
练习卷
1.12

函数的变换
大纲
Learning ObjectiveEssential Knowledge

1.12.A
Construct a function that is an additive and/or multiplicative transformation of another function.

  • 1.12.A.1 The function $g(x) = f(x) + k$ is an additive transformation of the function $f$ that results in a vertical translation of the graph of $f$ by $k$ units.
  • 1.12.A.2 The function $g(x) = f(x + h)$ is an additive transformation of the function $f$ that results in a horizontal translation of the graph of $f$ by $-h$ units.
  • 1.12.A.3 The function $g(x) = a f(x)$, where $a \neq 0$, is a multiplicative transformation of the function $f$ that results in a vertical dilation of the graph of $f$ by a factor of $|a|$. If $a < 0$, the transformation involves a reflection over the $x$-axis.
  • 1.12.A.4 The function $g(x) = f(bx)$, where $b \neq 0$, is a multiplicative transformation of the function $f$ that results in a horizontal dilation of the graph of $f$ by a factor of $\left| \dfrac{1}{b} \right|$. If $b < 0$, the transformation involves a reflection over the $y$-axis.
  • 1.12.A.5 Additive and multiplicative transformations can be combined, resulting in combinations of horizontal and vertical translations and dilations.
  • 1.12.A.6 The domain and range of a function that is a transformation of a parent function may be different from those of the parent function.

来源:美国大学理事会 AP 课程与考试说明

函数图象的变换

一个变换(transformation)从一个旧函数 $f$ 构建一个新函数:

向输出或输入相加移动曲线;一个乘数拉伸它
向输出或输入相加移动曲线;一个乘数拉伸它
  • 平移(translations)(移动):$f(x)+k$ 上/下移;$f(x-h)$ 右/左移。
  • 伸缩(dilations)(拉伸):$a\,f(x)$ 竖直拉伸;$f(bx)$ 水平拉伸。
  • 反射(reflections):$-f(x)$ 翻过 $x$ 轴;$f(-x)$ 翻过 $y$ 轴。

组合加性移动和乘性拉伸以为一个已知形状的移动、缩放版本建模。

变换母抛物线:移动、拉伸和反射
变换母抛物线:移动、拉伸和反射
探索

Explore shifts, stretches, and flips

Choose translate, reflect, rotate, or enlarge and change the amount. Watch which features stay the same — a transformation 变换 moves the whole graph without changing its underlying rule.

词汇表 训练
英文 中文 拼音
transformation 变换 biàn huàn
Translations 平移 píng yí
Dilations 伸缩 shēn suō
Reflections 反射 fǎn shè
练习卷
1.13

函数模型选择与假设阐述
大纲
Learning ObjectiveEssential Knowledge

1.13.A
Identify an appropriate function type to construct a function model for a given scenario.

  • 1.13.A.1 Linear functions model data sets or aspects of contextual scenarios that demonstrate roughly constant rates of change.
  • 1.13.A.2 Quadratic functions model data sets or aspects of contextual scenarios that demonstrate roughly linear rates of change, or data sets that are roughly symmetric with a unique maximum or minimum value.
  • 1.13.A.3 Geometric contexts involving area or two dimensions can often be modeled by quadratic functions. Geometric contexts involving volume or three dimensions can often be modeled by cubic functions.
  • 1.13.A.4 Polynomial functions model data sets or contextual scenarios with multiple real zeros or multiple maxima or minima.
  • 1.13.A.5 A polynomial function of degree $n$ models data sets or contextual scenarios that demonstrate roughly constant nonzero $n$th differences.
  • 1.13.A.6 A polynomial function of degree $n$ or less can be used to model a graph of $n + 1$ points with distinct input values.
  • 1.13.A.7 A piecewise-defined function consists of a set of functions defined over nonoverlapping domain intervals and is useful for modeling a data set or contextual scenario that demonstrates different characteristics over different intervals.

1.13.B
Describe assumptions and restrictions related to building a function model.

  • 1.13.B.1 A model may have underlying assumptions about what is consistent in the model.
  • 1.13.B.2 A model may have underlying assumptions about how quantities change together.
  • 1.13.B.3 A model may require domain restrictions based on mathematical clues, contextual clues, or extreme values in the data set.
  • 1.13.B.4 A model may require range restrictions, such as rounding values, based on mathematical clues, contextual clues, or extreme values in the data set.

来源:美国大学理事会 AP 课程与考试说明

选择一个模型(model)意味着把一个函数类型匹配到一个量如何变化。一个线性模型拟合一个恒定的变化率;一个二次拟合一个恒定的二阶差;一个多项式拟合有几个转折的数据。在几何情形里,维数是一个强提示:关于面积(二维)的量通常是二次的,而关于体积(三维)的量通常是三次的——例如一个盒子的体积作为一个切割长度的函数。陈述你的模型依赖的假设(assumptions)(例如,模式继续),因为一个模型只在那些假设成立的地方有效。

词汇表 训练
英文 中文 拼音
model 模型 mó xíng
assumptions 假设 jiǎ shè
练习卷
1.14

函数模型的构建与应用
大纲
Learning ObjectiveEssential Knowledge

1.14.A
Construct a linear, quadratic, cubic, quartic, polynomial of degree $n$, or related piecewise-defined function model.

  • 1.14.A.1 A model can be constructed based on restrictions identified in a mathematical or contextual scenario.
  • 1.14.A.2 A model of a data set or a contextual scenario can be constructed using transformations of the parent function.
  • 1.14.A.3 A model of a data set can be constructed using technology and regressions, including linear, quadratic, cubic, and quartic regressions.
  • 1.14.A.4 A piecewise-defined function model can be constructed through a combination of modeling techniques.

1.14.B
Construct a rational function model based on a context.

  • 1.14.B.1 Data sets and aspects of contextual scenarios involving quantities that are inversely proportional can often be modeled by rational functions. For example, the magnitudes of both gravitational force and electromagnetic force between objects are inversely proportional to the objects' squared distance.

1.14.C
Apply a function model to answer questions about a data set or contextual scenario.

  • 1.14.C.1 A model can be used to draw conclusions about the modeled data set or contextual scenario, including answering key questions and predicting values, rates of change, average rates of change, and changing rates of change. Appropriate units of measure should be extracted or inferred from the given context.

来源:美国大学理事会 AP 课程与考试说明

要构建一个模型,用给定的特征——点、截距、带重数的零点、末端行为——写函数,然后用它在上下文里回答问题。总是对照对情况有意义的定义域检查答案,并用它们的现实世界单位解释输出。

一个分段函数(piecewise-defined function)在定义域的不重叠区间上用不同的规则。要求它的值,选区间包含输入的那个分支。例如,

$$f(x)=\begin{cases} x^2, & x<0\\ 2x, & x\ge 0\end{cases}$$
从第一个分支给出 $f(-3)=9$,但从第二个给出 $f(3)=6$——在行为在一个阈值处改变时有用(一个分级价格、一个改变的速度限制)。

词汇表 训练
英文 中文 拼音
piecewise-defined function 分段函数 fēn duàn hán shù
练习卷
1.14

考试技巧
  • 通过它的变化率读一个函数:平均率 = 割线斜率;一个恒定的率意味着线性,一个线性变化的率意味着二次。
  • 因式分解一个多项式以求零点(x 截距)及其重数;偶重数触及、奇穿越坐标轴。
  • 末端行为由首项设定;一个有理函数的渐近线来自顶部和底部的次数。
  • 从关键特征(点、零点、末端行为)构建模型并陈述你的假设。
  • 精确地描述变换:移动 $f(x-h)+k$、拉伸 $af(bx)$、反射 $-f(x)$ / $f(-x)$。

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