8.3 Graphing Polynomials, Roots, and End Behavior

8.3 Graphing Polynomials, Roots, and End Behavior

Table 11’s third polynomial skill is graphing polynomial functions. On AAF you rarely plot a dozen points. You read four features from the algebra, or you rebuild the algebra from a described graph:

1. End behavior — what y does as x → ±∞, from the degree (even or odd) and the leading coefficient (positive or negative).
2. Roots / x-intercepts — from factors (x − r).
3. Multiplicity — odd multiplicity crosses the x-axis; even multiplicity bounces.
4. Y-intercept — P(0), the constant term in standard form.

/practice/accuplacer-advanced-algebraPractice questions with detailed explanations

End behavior from degree and leading coefficient

Let P(x) = a_n x^n + ⋯ with a_n ≠ 0. For large |x|, the leading term dominates. The four cases:

Memory aid: even degree, same ends; odd degree, opposite ends. The sign of a_n tells the right-hand end (x → +∞). Then the even/odd rule fills in the left-hand end. You do not need a table of values at x = 100 to decide this; the leading term is the decision.

Examples you can read at a glance:

• P(x) = 2x^4 − x + 1: even, a > 0 → both ends up.
• P(x) = −x^4 + 3x^2: even, a < 0 → both ends down. A global maximum exists; no global minimum.
• P(x) = x^3 − 4x: odd, a > 0 → left down, right up. No global max or min.
• P(x) = −2x^3 + 5: odd, a < 0 → left up, right down. No global max or min.

Do not use the constant term for end behavior. +5 in −2x^3 + 5 is the y-intercept P(0) = 5, not the right-hand end. Do not use the number of turning points as a substitute for degree: a cubic can have two turning points or none, but its ends are always opposite. A quartic can look locally like a “W” or an “M” and still have matching ends.

If the polynomial is factored, multiply the degrees of the factors to get n, and multiply the leading pieces of each factor to get a_n. P(x) = −(x − 2)^2(x + 3) has degree 2 + 1 = 3 and leading term −(x^2)(x) = −x^3, so odd with a < 0: left up, right down.

Multiplicity: bounce versus cross

If (x − r)^m is a factor and m is the highest such power, then x = r is a root of multiplicity m.

• Odd multiplicity (1, 3, …): the graph crosses the x-axis at x = r. Multiplicity 1 crosses almost like a non-horizontal line. Multiplicity 3 flattens as it crosses (a horizontal inflection on the axis), but it still changes sign.
• Even multiplicity (2, 4, …): the graph touches the x-axis at x = r and turns back — a bounce. The graph does not change sign at that root.

AAF almost always uses multiplicity 1 (cross) or 2 (bounce). Treat multiplicity 3 as “crosses, but flatter” if a graph is drawn that way. You do not need calculus; you need the factor and whether its exponent is even or odd.

P(x) = (x + 1)(x − 4)^2 crosses at x = −1 and bounces at x = 4. Leading term is x · x^2 = x^3, so odd degree, a = 1 > 0: left down, right up. Just to the left of x = 4 and just to the right, y has the same sign (positive, because the right-hand end is up and the bounce does not flip it). At x = −1 the sign does flip.

P(x) = −(x − 2)^2(x + 3) bounces at x = 2, crosses at x = −3. Degree 3, a = −1 < 0: left up, right down.

A simple-root factor (x − r) is an x-intercept you can plot as soon as you see it. A squared factor is still an x-intercept, but the graph looks locally like a parabola sitting on the axis, opening the same direction as the sign of P on both sides of r.

Y-intercept is P(0)

Plug in x = 0. In standard form, P(0) is the constant term. In factored form, P(0) is a times the product of the constant pieces of each factor.

For P(x) = −(x − 2)^2(x + 3):

P(0) = −(−2)^2(3) = −(4)(3) = −12.

The graph crosses the y-axis at (0, −12). That single number is how you solve for an unknown stretch factor when the roots are already known.

Worked: rebuild P(x) = a(x − 2)^2(x + 3) from graph features

Suppose a graph:

• touches the x-axis at x = 2 and turns back (bounce),
• crosses the x-axis at x = −3,
• has y-intercept (0, −12),
• falls to the right (as x → +∞, y → −∞).

Translate each feature:

1. Bounce at x = 2 ⇒ even multiplicity; AAF default is 2, so a factor (x − 2)^2.
2. Cross at x = −3 ⇒ odd multiplicity, default 1, so a factor (x + 3).
3. No other x-intercepts shown ⇒ those are the only real roots, so P(x) = a(x − 2)^2(x + 3).
4. Y-intercept −12: a(0 − 2)^2(0 + 3) = 12a = −12 ⇒ a = −1.
5. Check end behavior: degree 2 + 1 = 3 (odd), a = −1 < 0 ⇒ right-hand end down, matching “falls to the right.”

Equation: P(x) = −(x − 2)^2(x + 3).

If the y-intercept had been (0, 12), then 12a = 12 and a = 1, which would make the right-hand end up — that would contradict a graph that falls to the right. End behavior and the sign of a must agree; use that as a check, not as a second unknown. If they disagree, you assigned the bounce to the wrong root or you used the wrong power.

A stem may hide the stretch factor as an unknown exponent on one factor. Same principle, original factors: g(x) = c(x + 1)(x − 5)^m.

• The graph always crosses at x = −1 because that factor has multiplicity 1.
• If the graph bounces at x = 5, then m is even (usually m = 2). If it crosses at x = 5, then m is odd (usually m = 1 or 3).
• The y-intercept is g(0) = c(1)(−5)^m = c(−5)^m. One extra point, or the y-intercept, pins down c once m is known.

Do not assign the bounce to the wrong intercept. The exponent sits on the factor whose root is the bounce point. Swapping to c(x + 1)^m(x − 5) would bounce at x = −1 and cross at x = 5, which is a different graph. Creating this two-variable polynomial is the reverse of the factor-plus-point writing in Polynomial Operations and Structure.

Local and global max/min

Read extrema from the graph, then match them to the algebra.

• A local (relative) maximum is a turning point higher than the points immediately beside it. A cubic with a > 0 typically has a local max on the left hump and a local min on the right valley.
• A local minimum is a turning point lower than its neighbors.
• A global (absolute) maximum on the whole real line exists only if the graph never goes higher — which for a polynomial means even degree and a < 0 (both ends down). Then the highest turning point is the global max.
• A global minimum on the whole real line exists only if even degree and a > 0 (both ends up).
• Odd-degree polynomials have neither a global max nor a global min on all real numbers, because one end runs to +∞ and the other to −∞. They can still have local max and local min.

The quantity the exam wants is usually a y-value (the max/min value), not the x-coordinate of the turning point. “The local maximum is 2” means the high y-value is 2. “A local maximum at x = −1” names the location. Read which one the stem asked for.

Worked: cubic with two turning points

Let h(x) = x^3 − 3x. Degree 3, a > 0: left down, right up. From a graph you would see a local max at x = −1 and a local min at x = 1:

h(−1) = −1 + 3 = 2, h(1) = 1 − 3 = −2.

So the local maximum value is 2 and the local minimum value is −2. There is no global maximum (the right end goes to +∞) and no global minimum (the left end goes to −∞). An item that asks “the maximum value of h” without an interval is a trap: there isn’t one. An item that asks for the local maximum value wants 2. An item that shows a graph on a closed window from x = −2 to x = 2 wants you to compare endpoint values with the turning points:

On [−2, 2], the highest value is 2 (attained at x = −1 and at x = 2); the lowest is −2 (attained at x = −2 and at x = 1). The examinable comparison is: name the max or min value (a y-number) and do not confuse it with the x-coordinate of the turning point.

For even degree: q(x) = −x^4 + 4x^2 has a < 0, both ends down, so a global maximum exists. Factored, q(x) = −x^2(x^2 − 4) = −x^2(x − 2)(x + 2). Roots at x = 0 (multiplicity 2, bounce) and at x = ±2 (cross). The graph bounces at the origin and crosses at ±2. The global max is the higher of the two symmetric humps; you can read it from a graph as the largest y-value. The range is y ≤ M for that global max M. There is no global minimum, because both ends run to −∞.

From graph to equation, AAF order

1. List x-intercepts. Bounce ⇒ even exponent; cross ⇒ odd exponent.
2. Write P(x) = a(x − r1)^{m1}(x − r2)^{m2} ⋯.
3. Use the y-intercept or one extra point to solve for a.
4. Check: sign of a and the degree must match the drawn end behavior.
5. If the question asks for a max/min, read y-values at turning points and, if a window is given, at endpoints. Odd degree ⇒ no global max/min on all reals.

Creating, solving, and graphing are one loop. The expanded form from Polynomial Operations and Structure gives degree and leading coefficient. The factored form from Solving Polynomial Equations gives roots and multiplicity. The graph in this section is those two writings in picture form. Official skill names for the whole AAF test, including this polynomial block, are on College Board’s What’s on the Tests page.