Polynomial and Rational Functions

A function 函数 is a rule that maps each input to exactly one output. The set of allowed inputs is the domain 定义域; the set of outputs is the range 值域. The input variable is the independent variable 自变量 and the output variable is the dependent variable 因变量. A function rule can be shown graphically, numerically, analytically (a formula), or verbally.

As the input changes, the output changes "in tandem" – together. Over an interval, a function is:

• increasing 递增 if, whenever $a, then $f(a) (bigger input, bigger output);
• decreasing 递减 if, whenever $a, then $f(a)>f(b)$ (bigger input, smaller output).

A graph of a function shows all its input–output pairs, so you can read this behavior straight off the picture.

The average rate of change 平均变化率 of a function over an interval is the change in output divided by the change in input – the single constant rate that would give the same total change. Over $[a,b]$:

The rate of change at a point measures how fast the output changes right at that input. You approximate it with average rates over small intervals around the point. Comparing two points, the one with the larger average rate (over small enough intervals) is changing faster. A positive rate means both quantities move the same way; a negative rate means they move in opposite ways.

Worked example. For $f(x)=x^2$, the average rate of change over $[1,4]$ is $\dfrac{f(4)-f(1)}{4-1}=\dfrac{16-1}{3}=5$. Over $[1,2]$ it is $\dfrac{4-1}{1}=3$ – the rate itself changes, which is exactly why $f$ is not linear.

The average rate of change over $[a,b]$ is the slope of the secant line 割线 from $(a,f(a))$ to $(b,f(b))$.

Completing the square gives the vertex of a parabola

• For a linear 线性 function, the average rate of change over any interval is constant – so the rate at which the rate changes is zero.
• For a quadratic 二次 function, the average rates of change over equal-length intervals themselves form a linear pattern – so those rates change at a constant rate.

This "rate of the rate" idea distinguishes function types: a constant second difference signals a quadratic.

A hanging bridge cable takes the shape of a parabola-like curve — a quadratic you can see in steel

A nonconstant polynomial 多项式 has the form

Its degree 次数 is $n$, its leading term 首项 is $a_n x^n$, and its leading coefficient 首项系数 is $a_n$. A constant is a polynomial of degree $0$.

Key features:

• Where a polynomial switches between increasing and decreasing, it has a local (relative) extremum 局部极值. The greatest local maximum is the global (absolute) maximum 全局最大值; the least local minimum is the global minimum.
• Between any two distinct real zeros there is at least one local maximum or minimum.
• An even-degree polynomial has either a global maximum or a global minimum.
• A point of inflection 拐点 is where the graph changes concavity – from concave up 上凹 to concave down 下凹 or the reverse – i.e. where the rate of change switches between increasing and decreasing.
• A function has symmetry when it is even 偶函数 – $f(-x)=f(x)$, graph symmetric across the $y$-axis (like $x^2$, $x^4$) – or odd 奇函数 – $f(-x)=-f(x)$, graph symmetric by rotation about the origin (like $x^3$, $x^5$). Test by substituting $-x$ and simplifying; later, $\cos\theta$ is even and $\sin\theta$ is odd.

A zero 零点 (or root 根) of $p$ is a number $a$ with $p(a)=0$. For a real $a$, $(x-a)$ is a factor of $p$ exactly when $a$ is a zero. If the factor $(x-a)$ is repeated $n$ times, that zero has multiplicity 重数 $n$. Counting multiplicities, a degree-$n$ polynomial has exactly $n$ complex 复数 zeros.

A complex number on an Argand diagram: modulus is the distance, argument the angle

• A real zero $a$ gives an $x$-intercept at $(a,0)$; real zeros are the endpoints of the intervals where $p(x)\ge 0$ or $\le 0$.
• Non-real zeros come in conjugate 共轭 pairs: if $a+bi$ is a zero, so is $a-bi$.
• At a real zero of even multiplicity the graph is tangent to the $x$-axis (it touches but does not cross); at odd multiplicity it crosses.
• The degree equals the least $n$ for which the $n$th successive differences of equally-spaced outputs become constant.

End behavior 末端行为 describes where a function heads as the input grows without bound. For a nonconstant polynomial, as $x\to\pm\infty$ the output goes to $+\infty$ or $-\infty$, written e.g. $\lim_{x\to\infty}p(x)=\infty$. Which way depends entirely on the leading term $a_n x^n$, because for large $|x|$ it dominates all lower-degree terms: the sign of $a_n$ and whether $n$ is even or odd fix both ends.

Every polynomial's ends are decided by its degree's parity and the sign of its leading coefficient

A rational function 有理函数 is a quotient of two polynomials, $r(x)=\dfrac{\text{numerator}}{\text{denominator}}$. Its end behavior is governed by the quotient of the leading terms:

When the numerator has higher degree, the curve approaches a slanting asymptote

• Numerator degree > denominator degree: the quotient is a nonconstant polynomial, and $r$ follows that polynomial's end behavior. If that quotient is linear, the graph has a slant asymptote 斜渐近线.
• Equal degrees: the quotient is a constant, giving a horizontal asymptote 水平渐近线 $y=$ the ratio of leading coefficients.
• Numerator degree < denominator degree: the quotient tends to $0$, so the horizontal asymptote is $y=0$.

At a horizontal asymptote $y=b$, the outputs get and stay arbitrarily close to $b$: $\lim_{x\to\pm\infty}r(x)=b$.

The real zeros of a rational function are the real zeros of its numerator that are still in the domain. These zeros, together with the zeros of the denominator, split the number line into the intervals you test when solving inequalities $r(x)\ge 0$ or $r(x)\le 0$.

A vertical asymptote 垂直渐近线 occurs at $x=a$ when $a$ is a zero of the denominator but not cancelled by the numerator – more precisely, when its multiplicity in the denominator exceeds its multiplicity in the numerator. Near it, the denominator is nearly zero, so $r$ shoots to $\pm\infty$: $\lim_{x\to a^{\pm}}r(x)=\pm\infty$. Check each side, since the two sides can go opposite ways.

Worked example. Describe $r(x)=\dfrac{2x^2+3}{x^2-1}$. The numerator and denominator have equal degree, so the horizontal asymptote is $y=\dfrac{2}{1}=2$. The denominator $x^2-1$ is zero at $x=\pm1$ and neither cancels, so there are vertical asymptotes at $x=1$ and $x=-1$.

A rational function approaching a vertical asymptote and a horizontal asymptote

A hole 空洞 (removable point) occurs at $x=c$ when a factor cancels – the multiplicity of the zero $c$ in the numerator is at least its multiplicity in the denominator. The graph is missing a single point. Its height is the limit of the simplified function: if inputs near $c$ give outputs near $L$, the hole is at $(c,L)$, and $\lim_{x\to c}r(x)=L$.

The same expression, written differently, reveals different features:

• Factored form shows real zeros, and hence $x$-intercepts, holes, vertical asymptotes, domain, and range.
• Standard form (expanded) shows the degree and leading term, and hence end behavior.

Polynomial long division 多项式长除法 rewrites $f(x)=g(x)\,q(x)+r(x)$, where $q$ is the quotient 商 and $r$ the remainder 余数 (of smaller degree than $g$). The quotient gives the equation of a slant asymptote when the numerator's degree is one more than the denominator's.

A transformation 变换 builds a new function from an old one $f$:

Adding to the output or input shifts the curve; a multiplier stretches it

• Translations 平移 (shifts): $f(x)+k$ moves up/down; $f(x-h)$ moves right/left.
• Dilations 伸缩 (stretches): $a\,f(x)$ stretches vertically; $f(bx)$ stretches horizontally.
• Reflections 反射: $-f(x)$ flips over the $x$-axis; $f(-x)$ flips over the $y$-axis.

Combine additive shifts and multiplicative stretches to model shifted, scaled versions of a known shape.

Transforming the parent parabola: shift, stretch, and reflect

Choosing a model 模型 means matching a function type to how a quantity changes. A linear model fits a constant rate of change; a quadratic fits a constant second difference; a polynomial fits data with several turns. In geometric situations the number of dimensions is a strong hint: quantities about area (two dimensions) are typically quadratic, and quantities about volume (three dimensions) are typically cubic – for example a box's volume as a function of a cut length. State the assumptions 假设 your model relies on (for example, that the pattern continues), because a model is only valid where those assumptions hold.

To construct a model, use given features – points, intercepts, zeros with multiplicities, end behavior – to write the function, then use it to answer questions in context. Always check the answer against the domain that makes sense for the situation, and interpret outputs with their real-world units.

A piecewise-defined function 分段函数 uses different rules over non-overlapping intervals of the domain. To evaluate it, pick the branch whose interval contains the input. For example,

gives $f(-3)=9$ from the first branch but $f(3)=6$ from the second – useful when behaviour changes at a threshold (a tiered price, a speed limit that changes).

• Read a function through its rate of change: average rate = secant slope; a constant rate means linear, a linear-changing rate means quadratic.
• Factor a polynomial to find zeros (x-intercepts) and their multiplicity; even multiplicity touches, odd crosses the axis.
• End behaviour is set by the leading term; a rational function's asymptotes come from the degrees of top and bottom.
• Build models from key features (points, zeros, end behaviour) and state your assumptions.
• Describe transformations precisely: shift $f(x-h)+k$, stretch $af(bx)$, reflect $-f(x)$ / $f(-x)$.