Polynomial Functions and Rates of Change
Introduced| English |
|---|
| degree/dɪˈɡriː/ |
| leading term/ˈliːdɪŋ tɜːm/ |
| polynomial function/ˌpɒlɪˈnəʊmɪəl ˈfʌŋkʃn/ |
| local maximum/ˈləʊkl ˈmæksɪməm/ |
| local minimum/ˈləʊkl ˈmɪnɪməm/ |
| point of inflection/pɔɪnt ɒv ɪnˈflekʃn/ |
| concavity/kənˈkævɪti/ |
How many wiggles can a curve have?
- A line is straight; a parabola has one bend; but some curves rise and fall several times.
- What decides how many hills and valleys a graph is allowed to have?
- The answer is hidden in one number in the function's rule.
- Meet the polynomials — the functions built from powers of $x$.
Degree and leading term
- A polynomial function 多项式函数 is a sum of terms like $3x^4$, $-2x$, and $7$.
- Its degree 次数 is the highest power of $x$ that appears.
- The leading term 首项 is the term with that highest power — it dominates the graph for large $x$.
- Degree is the single most important number: it caps how wiggly the curve can be.
The degree of a polynomial is…
The degree is the highest exponent on the variable — e.g. $2x^4 - x + 7$ has degree $4$. It controls the graph's overall shape and end behavior.
Local maxima and minima
- Follow a polynomial left to right and mark where it changes direction.
- A local maximum 极大值 is a peak: increasing turns to decreasing.
- A local minimum 极小值 is a valley: decreasing turns to increasing.
- Between these turning points the function is steadily increasing or decreasing.
Reshape a cubic and count its turning points
y = ax³ + bx² + cx + d
Adjust the sliders. A degree-3 curve can wiggle at most twice — try to make it turn three times (you cannot).
A point where the graph switches from decreasing to increasing is a local ____.
A local minimum is a valley — the curve stops falling and starts rising.
Points of inflection
- Even between turning points, the curve can change the way it bends.
- A point of inflection 拐点 is where the concavity 凹凸性 flips: concave up becomes concave down, or the reverse.
- At an inflection point the rate of change stops speeding up and starts slowing down (or vice versa).
- So it is the steepest or gentlest spot on a rising or falling stretch.
A point of inflection is a point where…
At a point of inflection the curve stops bending one way and starts bending the other — concave up becomes concave down, or vice versa.
Degree caps the wiggles
- A polynomial of degree $n$ has at most $n - 1$ turning points.
- It also has at most $n - 2$ points of inflection.
- A cubic (degree $3$) can turn at most twice; a quartic (degree $4$) at most three times — like the figure above.
- "At most" matters: a curve may have fewer wiggles, but never more than its degree allows.
At most how many turning points can a polynomial of degree 5 have?
A degree-$n$ polynomial has at most $n - 1$ turning points, so degree $5$ allows at most $4$.
A degree-3 polynomial can have 3 turning points.
Degree $3$ allows at most $3 - 1 = 2$ turning points. It can have $2$ or none, but never $3$.
Select all true statements about polynomial graphs.
A cubic can have $2$ turning points or none (e.g. $y = x^3$), so "must have 2" is false. The other three are correct.
"At most" is not "exactly". A cubic can have two turning points, but $y = x^3$ has none — it rises the whole way, with a single point of inflection at the origin. Degree sets the ceiling, not the count.
Consider $f(x) = x^3 - 3x$.
- Degree $3$, so at most $2$ turning points — and it has exactly $2$: a local maximum near $x = -1$ and a local minimum near $x = 1$.
- Between them, at $x = 0$, the concavity flips: that is its point of inflection.
- Its leading term $x^3$ makes the graph fall on the far left and rise on the far right.
A polynomial function's degree (the highest power) caps its shape: at most $n - 1$ turning points — the local maxima and minima — and at most $n - 2$ points of inflection, where the concavity flips. The leading term controls what happens far out on each side.