From a Polynomial Formula to Its Shape
A polynomial is defined for every real number, so its graph has no domain breaks or vertical asymptotes. Its behavior is governed instead by its degree and leading coefficient at the ends, its roots and intercepts, and the changes in direction and concavity detected by its derivatives.
A complete sketch is not a plot of many sampled points. It is a qualitative description of the graph that accounts for all important features and puts them in the correct order. The examples below combine those features, including repeated roots and polynomials with several critical points.
A Reliable Sketching Sequence
For a polynomial \(f\), organize the analysis before drawing. First determine the end behavior from the leading term. Then find the intercepts and note any repeated roots. Compute \(f'\) to locate critical points and determine where the function increases or decreases. Compute \(f''\) to identify concavity and possible inflection points. Finally, place the information on one horizontal axis and connect the features in a way that respects all the signs.
These determine whether the two ends point in the same or opposite directions, and whether the right end rises or falls.
Record where the graph meets the axes, and use multiplicity to determine whether it crosses or touches the horizontal axis.
Find critical numbers, test the sign of \(f'\) between them, and classify any local extrema from sign changes.
Find where \(f''\) is zero, test concavity on the resulting intervals, and keep only points where concavity changes as inflection points.
Place all features in order and verify that the proposed curve is consistent with the end behavior and both derivative sign charts.
Why Multiplicity and Derivative Signs Matter
If \(a\) is a root of multiplicity \(m\), a polynomial can be written \(f(x)=(x-a)^m q(x)\), where \(q(a)\ne 0\). Since \(q\) is continuous, it keeps the same sign sufficiently close to \(a\). The factor \((x-a)^m\) changes sign across \(a\) when \(m\) is odd, but not when \(m\) is even. Thus, near an odd-multiplicity root the graph crosses the axis, while near an even-multiplicity root it touches the axis and turns back to the same side.
Derivative sign charts provide the corresponding information about motion. If \(f'\) is positive on an interval, the function increases there; if \(f'\) is negative, it decreases. One justification follows from the Mean Value Theorem: for \(u<v\) in an interval where \(f'>0\), there is a \(c\) between \(u\) and \(v\) such that \(f(v)-f(u)=f'(c)(v-u)>0\). Hence \(f(v)>f(u)\). The argument with \(f'<0\) gives decrease. Therefore, a change in \(f'\) from positive to negative gives a local maximum, and a change from negative to positive gives a local minimum.
Worked Examples
Worked Example: A Repeated Root That Is a Local Maximum
Sketch the polynomial \(f(x)=(x+2)^2(x-1)\), identifying its intercepts, critical points, concavity, and end behavior.
The polynomial has degree \(3\) and leading coefficient \(1\), because its leading term is \(x^2\cdot x=x^3\). An odd degree with positive leading coefficient means the left end falls and the right end rises:
The factored form gives roots \(x=-2\), with multiplicity \(2\), and \(x=1\), with multiplicity \(1\). Thus the graph touches the axis at \((-2,0)\) and crosses at \((1,0)\). At \(x=0\), \(f(0)=(2)^2(-1)=-4\), so the \(y\)-intercept is \((0,-4)\).
To find the critical points, first expand and differentiate:
The critical numbers are \(x=-2\) and \(x=0\). Testing the sign of \(f'\) gives positive for \(x<-2\), negative for \(-2<x<0\), and positive for \(x>0\). The function therefore increases, then decreases, then increases. At \(x=-2\), \(f(-2)=0\), so there is a local maximum at \((-2,0)\). At \(x=0\), \(f(0)=-4\), so there is a local minimum at \((0,-4)\).
For concavity, differentiate again:
The second derivative is negative for \(x<-1\) and positive for \(x>-1\). Concavity changes at \(x=-1\), and \(f(-1)=(1)^2(-2)=-2\), giving an inflection point at \((-1,-2)\).
Sketch description and check: Start low on the left, rise to the repeated root \((-2,0)\), where the curve turns downward, and pass through the inflection point \((-1,-2)\). Continue down to the minimum \((0,-4)\), then rise, cross the axis at \((1,0)\), and continue upward to the right. The values \(f(-2)=0\), \(f(0)=-4\), \(f(1)=0\) verify the two roots and the minimum value. The derivative signs agree with the described turns, and the end behavior agrees with a positive-leading cubic.
Worked Example: A Quartic with Three Critical Points
Analyze and sketch \(g(x)=(x^2-1)^2\).
The degree is \(4\) and the leading coefficient is positive, so both ends rise:
The roots are \(x=-1\) and \(x=1\), each with multiplicity \(2\); the graph touches the horizontal axis at both. The \(y\)-intercept is \(g(0)=(-1)^2=1\), giving \((0,1)\).
Differentiation gives
The critical numbers are \(-1,0,1\). The sign of \(g'\) is negative for \(x<-1\), positive for \(-1<x<0\), negative for \(0<x<1\), and positive for \(x>1\). Thus the function decreases to \(x=-1\), increases to \(x=0\), decreases to \(x=1\), and then increases. Since \(g(-1)=0\), \(g(0)=1\), and \(g(1)=0\), the graph has local minima at \((-1,0)\) and \((1,0)\), and a local maximum at \((0,1)\).
For concavity,
The second derivative is zero when \(x^2=1/3\), so \(x=\pm 1/\sqrt{3}\). It is positive for \(x<-1/\sqrt{3}\) and \(x>1/\sqrt{3}\), and negative between those values. The graph changes concavity at both points. Since \(x^2=1/3\) there, \(g(x)=(1/3-1)^2=4/9\); the inflection points are \((-1/\sqrt{3},4/9)\) and \((1/\sqrt{3},4/9)\).
Sketch description and check: Both ends rise. From the left, the curve descends to \((-1,0)\), rises through the first inflection point to \((0,1)\), then descends through the second inflection point to \((1,0)\), and rises again. Substitution confirms the three critical-point values: \(g(-1)=0\), \(g(0)=1\), and \(g(1)=0\). The first-derivative signs show the stated turns, and the second-derivative signs show the stated concavity intervals.
Worked Example: A Cubic with Two Turning Points
Sketch \(h(x)=x^3-3x\), including its roots, critical points, and inflection point.
This is a degree-\(3\) polynomial with positive leading coefficient, so its left end falls and its right end rises. To find the roots, factor:
The graph crosses the axis at \(x=-\sqrt{3},0,\sqrt{3}\), since each root has multiplicity \(1\). The \(y\)-intercept is \(h(0)=0\), the same as the root at the origin.
Differentiate to locate the critical points:
The critical numbers are \(-1\) and \(1\). The derivative is positive for \(x<-1\), negative for \(-1<x<1\), and positive for \(x>1\). Therefore \(x=-1\) is a local maximum and \(x=1\) is a local minimum. Their values are \(h(-1)=(-1)^3-3(-1)=-1+3=2\), and \(h(1)=1-3=-2\).
The second derivative is \(h''(x)=6x\), which is negative for \(x<0\) and positive for \(x>0\). The concavity changes at \(x=0\), where \(h(0)=0\), so \((0,0)\) is an inflection point.
Sketch description and check: The curve rises from the lower left, crosses at \((-\sqrt{3},0)\), and reaches its local maximum \((-1,2)\). It then falls through the inflection point \((0,0)\), crosses there, and reaches its local minimum \((1,-2)\). It rises afterward and crosses again at \((\sqrt{3},0)\), continuing upward. Substitution verifies \(h(-1)=2\), \(h(1)=-2\), and \(h(0)=0\); the derivative and concavity signs support the stated shape.
A Common Sketching Mistake
A repeated root does not automatically mean the graph has a local minimum. Consider the first example: \(x=-2\) is a root of multiplicity \(2\), so the graph touches the axis there. One might incorrectly reason, “The graph touches the axis, so it must turn upward into a minimum.” But \(f'(x)=3x(x+2)\) is positive just to the left of \(-2\) and negative just to its right. The graph is increasing before the point and decreasing afterward, so \((-2,0)\) is a local maximum.
Consistency Checks for a Finished Sketch
Before accepting a sketch, compare every feature against the calculations. Roots must occur at the stated \(x\)-intercepts; the \(y\)-intercept must agree with \(f(0)\). The graph must turn at local extrema in the order indicated by the first-derivative signs. Its concavity must match the second-derivative signs, and any proposed inflection point must actually have a change in concavity. Finally, the ends must agree with the polynomial’s degree and leading coefficient.
These checks are useful because a sketch is a combined account of several facts, not a single calculation. A curve that has the correct roots but the wrong end behavior, or the right extrema but the wrong concavity, is not a consistent sketch of the given polynomial.
Check Your Understanding
Use sign charts and exact substitution where appropriate to answer these questions.
- For \(p(x)=(x-3)^2(x+1)\), which root has even multiplicity, and does the graph cross or touch the axis there?
- How do the degree and leading coefficient determine the end behavior of a polynomial?
- If \(f'\) changes from negative to positive at a critical number, what type of local extremum occurs?
- Why is solving \(f''(x)=0\) not, by itself, sufficient to establish an inflection point?
- For \(q(x)=x^4-2x^2\), find the critical numbers by differentiating and factoring \(q'(x)\).
