Graphing Polynomial Functions

📈 Graphing Polynomial Functions

Part 1 of 5 — Degree, Leading Coefficient & End Behavior

Topics in This Part

🔑 Key Concept: Before plotting a single point, the two "ends" of a polynomial's graph are already decided — by just two numbers: the degree and the leading coefficient. Master those and you control the whole shape's frame.

What Is a Polynomial Function?

A polynomial function is a sum of terms of the form $a_n x^n$, where the exponents are whole numbers ($0, 1, 2, 3, \dots$):

$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$

• The degree is the largest exponent, $n$.
• The leading coefficient is $a_n$ — the number on the highest-power term.
• The constant term is $a_0$ — it equals $f(0)$, the $y$-intercept.

What counts as a polynomial?

💡 Every polynomial graph is a single smooth, unbroken curve — no sharp corners, no holes, no jumps, no vertical asymptotes.

Concept Check 🎯

End Behavior: The Four Cases

End behavior describes what happens to $f(x)$ as $x$ runs far left ($x \to -\infty$) and far right ($x \to +\infty$). It is controlled entirely by the leading term $a_n x^n$.

Two questions decide it:

1. Is the degree even or odd?
2. Is the leading coefficient positive or negative?

🔑 Memory hook: Even acts like a parabola (both ends agree). Odd acts like a line (ends disagree). A negative leading coefficient flips the picture upside-down.

Predict the Ends 🔽

Use only the degree and the leading coefficient.

Why It Works

For huge $|x|$, the highest-power term dwarfs every other term. In $f(x) = 2x^4 - x + 1$ at $x = 100$, the $2x^4$ term is $200{,}000{,}000$ while $-x+1$ is only $-99$ — utterly negligible. So the leading term is the end behavior.

💡 Sign reminder for the leading term:

• Even power keeps a negative $x$ positive ($(-)^{\text{even}} = +$), so the left end matches the right end's sign.
• Odd power keeps a negative $x$ negative ($(-)^{\text{odd}} = -$), so the ends point opposite ways.

One last drill on degree before Part 2, where we find where the graph crosses the $x$-axis.

Name the Degree 🧮

Find the degree of each polynomial.

1) $f(x) = 9x^2 - x^6 + 4x$ — degree $= \,?$ 2) $g(x) = (x - 1)(x + 5)(x - 2)$ — degree $= \,?$ (think about the highest power if multiplied out)

📈 Graphing Polynomial Functions

Part 2 of 5 — Zeros & $x$-Intercepts

🔑 The Idea: A polynomial crosses or touches the $x$-axis exactly where $f(x) = 0$. Those inputs are the zeros (also called roots), and in factored form they're sitting in plain sight.

Zeros from Factored Form

If a polynomial is factored, the Zero Product Property does all the work: a product is $0$ only when one of its factors is $0$.

Example: $f(x) = (x - 3)(x + 2)(x - 5)$

Set each factor to $0$: $x - 3 = 0 \Rightarrow x = 3$ $x + 2 = 0 \Rightarrow x = -2$ $x - 5 = 0 \Rightarrow x = 5$

So the $x$-intercepts are $(3, 0)$, $(-2, 0)$, and $(5, 0)$.

⚠️ Watch the sign! The factor $(x + 2)$ gives the zero $x = -2$, not $x = 2$. Always flip the sign of the constant inside the factor.

Find the Zeros 🧮

Read the zeros straight from each factored polynomial.

1) $f(x) = (x - 4)(x + 6)$ — the two zeros are $x = \,?$ and $x = \,?$ (enter the smaller, then the larger) 2) $g(x) = x(2x - 3)$ — the nonzero zero is $x = \,?$ (fraction or decimal)

The Factor Theorem

The connection runs both directions:

$\boxed{\;(x - c) \text{ is a factor} \;\Longleftrightarrow\; f(c) = 0 \;\Longleftrightarrow\; c \text{ is a zero}\;}$

So "zero of the function," "root of the equation," "$x$-intercept of the graph," and "factor of the polynomial" are four names for the same thing.

Verifying a zero

Is $x = 2$ a zero of $f(x) = x^3 - 2x^2 - x + 2$? Plug it in: $f(2) = 8 - 8 - 2 + 2 = 0 \;\checkmark$

Yes — so $(x - 2)$ is a factor.

💡 The Fundamental Theorem of Algebra guarantees a degree-$n$ polynomial has at most $n$ real zeros (counting how many times a graph can cross the $x$-axis). A cubic crosses at most $3$ times; a quartic, at most $4$.

Concept Check 🎯

Don't Forget the $y$-Intercept

The $y$-intercept is always $f(0)$ — just substitute $x = 0$.

Example: $f(x) = (x - 3)(x + 2)(x - 5)$

$f(0) = (0 - 3)(0 + 2)(0 - 5) = (-3)(2)(-5) = 30$

So the graph passes through $(0, 30)$.

🔑 A polynomial graph has exactly one $y$-intercept (a function has only one output at $x=0$) but can have several $x$-intercepts. In Part 3 we find out whether the curve crosses or just touches at each one.

Match Factor to Zero 🔽

Pick the zero produced by each factor.

📈 Graphing Polynomial Functions

Part 3 of 5 — Multiplicity: Cross vs. Touch

🔑 The Idea: A repeated factor changes how the graph meets the $x$-axis. The exponent on a factor — its multiplicity — tells you whether the curve slices through the axis or just bounces off it.

What Is Multiplicity?

The multiplicity of a zero is the exponent on its factor.

$f(x) = (x - 1)^{1}(x + 2)^{2}(x - 4)^{3}$

The Cross-vs-Touch Rule

🔑 Odd = cross, Even = touch. Higher odd multiplicity (like $3$) makes the crossing flatten as it passes through; higher even multiplicity flattens the bounce.

Cross or Touch? 🔽

For $f(x) = (x + 1)^2(x - 3)(x - 5)^4$, decide what the graph does at each zero.

Multiplicities Add Up to the Degree

The multiplicities of all real zeros (plus any complex ones) sum to the degree. This is a powerful self-check.

Example: $f(x) = (x + 1)^2(x - 3)(x - 5)^4$

$\underbrace{2}_{x=-1} + \underbrace{1}_{x=3} + \underbrace{4}_{x=5} = 7$

So this is a degree-7 polynomial. Its leading term is $1 \cdot x^7$ (odd degree, positive lead) → down on the left, up on the right.

💡 If the multiplicities of the real zeros add up to less than the degree, the leftover degree is taken up by complex zeros, which never show on the real graph. A degree-$4$ with only two real zeros (each multiplicity $1$) has $2$ complex zeros hiding off-screen.

Degree from Multiplicities 🧮

1) $f(x) = (x - 2)^3(x + 4)^2$ — what is the degree of $f$? 2) A degree-$5$ polynomial has zeros $x = 0$ (mult. $1$) and $x = 6$ (mult. $2$). If those are its only real zeros, how many complex zeros does it have?

Turning Points

A turning point is where the graph changes direction (a local max or min). For a polynomial of degree $n$:

$\text{number of turning points} \;\le\; n - 1$

💡 A degree-$4$ graph wiggles at most $3$ times. If you've sketched $4$ turning points, you've made a mistake. Combined with end behavior and the cross/touch rule, the turning-point cap keeps your sketch honest.

Concept Check 🎯

📈 Graphing Polynomial Functions

Part 4 of 5 — Sketching a Full Graph

🔑 The Payoff: With end behavior, zeros, multiplicities, and the $y$-intercept, you can sketch any factored polynomial without plotting dozens of points. This part is the full assembly line.

The 5-Step Sketch Recipe

To graph a polynomial in factored form:

1. End behavior — read degree (sum of multiplicities) and the sign of the leading coefficient.
2. $x$-intercepts — set each factor to $0$.
3. Multiplicity — at each zero decide cross (odd) or touch (even).
4. $y$-intercept — compute $f(0)$.
5. Connect — draw a smooth curve obeying the end behavior, hitting each intercept correctly, never exceeding $n - 1$ turning points.

⚠️ A polynomial can only change sign at an $x$-intercept where it crosses. Between consecutive crossings the graph stays entirely above or entirely below the axis — use that to place your turns.

Build the Sketch 🔽

Analyze $g(x) = (x + 3)(x - 2)^2$ one feature at a time.

Worked Example

Sketch $f(x) = -(x + 2)(x - 1)^2$.

Step 1 — End behavior. Degree $= 1 + 2 = 3$ (odd). Leading coefficient: multiply the leads of each factor and the front sign: $-(1)(1)^2 = -1 \;(\text{negative}).$ Odd degree, negative lead → up on the left, down on the right.

Step 2 — Zeros. $x = -2$ and $x = 1$.

Step 3 — Multiplicity. $x = -2$ has multiplicity $1$ → crosses. $x = 1$ has multiplicity $2$ → touches.

Step 4 — $y$-intercept. $f(0) = -(0 + 2)(0 - 1)^2 = -(2)(1) = -2 \;\Rightarrow\; (0, -2).$

Step 5 — Connect. Come down from the upper left, cross at $x = -2$, rise to a peak, pass through $(0, -2)$ on the way down... actually let's confirm the shape with the sign chart below.

Key Features 🧮

For $h(x) = 2(x - 1)(x + 3)(x - 4)$:

1) What is the $y$-intercept? (Enter $h(0)$.) 2) How many times does the graph cross the $x$-axis?

The Sign Chart (Confirming the Shape)

Test one point in each interval split by the zeros $x = -2$ and $x = 1$ for $f(x) = -(x + 2)(x - 1)^2$:

Notice the graph is above the axis left of $-2$, crosses to below at $x = -2$, and stays below through the touch at $x = 1$ — the sign does not change at the even-multiplicity zero. ✓ This matches "up-left, down-right" end behavior perfectly.

✅ Self-check: the graph changes sign only at $x=-2$ (a crossing) and never at $x=1$ (a touch). That is exactly what odd vs. even multiplicity predicts.

Concept Check 🎯

📈 Graphing Polynomial Functions

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) read end behavior from the leading term, (2) find zeros and the $y$-intercept, (3) use multiplicity to decide cross vs. touch, and (4) assemble a complete sketch. Let's tie it together.

Quick Reference

⚠️ Top mistakes to avoid: flipping the sign of a zero (factor $(x+5)$ → zero $-5$, not $+5$); calling an even-multiplicity touch a crossing; forgetting that a negative leading coefficient flips both ends.

Full Analysis 🔽

Analyze $f(x) = -2(x + 1)^2(x - 3)$.

Reading a Graph Backward

So far you've gone equation → graph. The same rules run in reverse:

• Both ends down ⇒ even degree, negative leading coefficient.
• A bounce at an intercept ⇒ that factor has even multiplicity.
• A flattened crossing ⇒ odd multiplicity of $3$ or more.

The next two questions ask you to spot exactly these signatures.

Mixed Practice 🎯

One Last Checklist Before the Quiz

When you analyze any factored polynomial, hit these four in order:

1. Degree = sum of multiplicities → tells you the end-behavior family.
2. Leading-coefficient sign → confirms which way the ends point.
3. Each zero's multiplicity → cross (odd) or touch (even).
4. $f(0)$ → the single $y$-intercept.

🔑 Get those four right and the sketch is forced — there's only one smooth curve that fits. Now finish with the Exit Quiz.

Exit Quiz ✅

Answer all three to finish the lesson.