1
The leading term decides both ends
For large |x| the highest power dwarfs everything else. Even degree sends both ends the same way, odd degree sends them opposite ways, and the sign of the leading coefficient says which way.
Lesson · Polynomials and Factoring · Polynomial graphs
Graphs of Polynomial Functions
The factored form answers the questions a graph asks. Degree and leading coefficient fix what happens at the far ends; each factor fixes where the curve meets the axis; and the multiplicity of a factor decides whether it crosses there or turns around.
f(x) = (x + 2)(x - 1)²(x - 3) degree 4, leading coefficient 1
even degree, a > 0 both ends rise
odd multiplicity the curve CROSSES the axis
even multiplicity the curve TOUCHES and turns back
degree n: root multiplicities add to at most n
What the factored form tells you
Core ideas
What this lesson establishes
A sketch here is not a plot. It is four deductions from the formula, and each one has a reason you can state.
2
Multiplicity decides cross or touch
A factor (x - c) appearing an odd number of times gives a sign change at c, so the curve crosses. An even number of times gives no sign change, so the curve touches the axis and turns back.
3
Degree bounds the root count
A degree-n polynomial has at most n roots counted with multiplicity. The multiplicities written beside the axis must therefore add to at most n; exceeding the degree proves that a factor or multiplicity was misread.
The material
Sketching from the factored form
Four deductions, in the order that makes each one easy.
Polynomial graphs are continuous and smooth — no jumps, no corners, no asymptotes. Degree, leading coefficient, real roots and multiplicities determine the end directions and what happens at the axis. They do not locate every turning point, so the result here is a qualitative sketch rather than an exact plot.
End behaviour
Theorem · The leading term governs the ends
For f(x) = aₙxⁿ + … the behaviour as x → ±∞ matches that of aₙxⁿ. If n is even the two ends go the same way — both up when aₙ > 0, both down when aₙ < 0. If n is odd they go opposite ways: down-then-up when aₙ > 0.
The reason is division rather than intuition. Writing f(x) = xⁿ(aₙ + aₙ₋₁/x + … + a₀/xⁿ), every term after aₙ tends to 0 as |x| grows, so the bracket tends to aₙ and the sign of f follows aₙxⁿ.
Roots and multiplicity
Definition · Multiplicity
If (x - c)^k divides f but (x - c)^(k+1) does not, then c is a root of multiplicity k.
Theorem · Cross or touch
At a root of odd multiplicity the graph crosses the x-axis; at a root of even multiplicity it touches the axis and returns to the side it came from.
The reason is the sign of the factor. Passing c from left to right, (x - c) changes sign, so (x - c)^k changes sign exactly when k is odd. The other factors are non-zero near c and keep their signs, so the sign of f changes precisely when k is odd.
Example · f(x) = (x + 2)(x - 1)²(x - 3)
Degree 4 with leading coefficient 1, so both ends rise. The roots are -2 with multiplicity 1 (crosses), 1 with multiplicity 2 (touches), and 3 with multiplicity 1 (crosses). The y-intercept is f(0) = (2)(1)(-3) = -6.
That is enough for a sketch. Coming down from the upper left, the curve crosses at -2 into negative values, rises to touch the axis at 1 without crossing, falls back below, and crosses at 3 on its way up. Sample values confirm it: f(-3) = 96, f(0) = -6, f(2) = -4, f(4) = 54.
Intercepts and the limits of the sketch
The y-intercept is always f(0), which is the constant term in standard form and the product of the constants in factored form. It anchors the sign intervals to one exact point away from the roots.
A touch forces a local turn at its even-multiplicity root, but these data do not locate any other turning points. Exact turning-point locations require calculus; Quadratics and Complex Numbers handles the one case algebra can do on its own, the vertex of a parabola. Draw a qualitative sketch here and let the evaluated lab supply the metric picture.
Interactive
Graphing a polynomial
Zeros with their multiplicity, and where the ends go
Method
Sketching a polynomial
Ends first, then the axis, then the one point you should always plot.
1
Factor it, if it is not factored already
Everything below is read off the factored form. The common-factor, grouping, trinomial and special-form techniques and the search in “The Factor Theorem and Rational Roots” exist to get you here.
2
Fix the two ends
Look at the degree and the leading coefficient only. Even degree: both ends together. Odd degree: opposite ends. Positive leading coefficient: the right-hand end rises.
3
Mark each root, cross or touch
Odd multiplicity crosses; even multiplicity touches and turns. Write the multiplicity next to each root before drawing anything.
4
Plot the y-intercept and join up
Compute f(0). Then draw a smooth qualitative curve consistent with the ends, crossings, touches and that point. Do not invent an exact location for an unforced turn; the lab's evaluated plot is the metric picture.
Check yourself
Score 0 / 0Choose the answer you can justify.
Sketching f(x) = (x + 2)(x - 1)²(x - 3)
Every line below is read from the factored form. The expanded form is computed once, only to confirm the degree and the leading coefficient.
f(x) = (x + 2)(x - 1)²(x - 3)
expanded: x⁴ - 3x³ - 3x² + 11x - 6
degree 4 (even), leading coefficient +1
-> both ends rise: f -> +∞ as x -> -∞ and as x -> +∞
roots and multiplicities
x = -2 multiplicity 1 odd CROSSES
x = 1 multiplicity 2 even TOUCHES, turns back
x = 3 multiplicity 1 odd CROSSES
y-intercept f(0) = (2)(1)(-3) = -6
sign of f x < -2 + f(-3) = 96
-2 < x < 1 - f( 0) = -6
1 < x < 3 - f( 2) = -4
x > 3 + f( 4) = 54
multiplicities: 1 + 2 + 1 = 4, exactly the degree
The sign row is the check that the multiplicities were read correctly. f is negative on both sides of x = 1, which is exactly what an even multiplicity predicts, and it changes sign at -2 and at 3, as odd multiplicities require.
Note that the multiplicities sum to 4, the degree. Counted with multiplicity there are four roots here; counted as distinct points on the axis there are three. Both statements are true and they are not the same statement.
For a faded sketch, use g(x) = -(x + 1)²(x - 2). The supplied first reading is degree 3 with negative leading coefficient, so the left end rises and the right end falls. Mark the roots and multiplicities, find g(0), and sketch before opening the lab. The curve touches at -1, crosses at 2, and has y-intercept 2; crossing at -1 means the even multiplicity was treated as odd.
Common mistakes
1
Deciding the ends from the constant term or the number of rootsOnly the leading term matters at the ends. x³ - 100 and x³ + 100 behave identically as x → ±∞, however different they look near the origin.
2
Drawing every root as a crossingA squared factor touches and turns. Treating (x - 1)² as a crossing gives a curve on the wrong side of the axis for the whole interval that follows, and the sign check will catch it.
3
Pretending the qualitative data locate every turnThe ends, roots, multiplicities and y-intercept constrain a sketch; they do not give exact x-coordinates for unforced turning points. A neat-looking curve is not a calculation, which is why the evaluated lab remains a check.
Check your understanding
✓
Finish when you can sketch from the factored form without plotting points.
Given f(x) = (x + 2)(x - 1)²(x - 3) you should produce the shape from the ends, the three roots with their multiplicities and f(0) alone — using sample values only to confirm the signs, never to discover them.
Note: This is where the course arrives. The graph is drawn from the factored form, the factored form came from the search in “The Factor Theorem and Rational Roots”, and that search was made possible by the common-factor, grouping, trinomial and special-form techniques. Anything the lab plots is computed by evaluating the polynomial you entered, so a curve that looks wrong is telling you something about the polynomial rather than about the drawing.