🎯⭐ INTERACTIVE LESSON

Polynomial Functions and End Behavior

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Polynomial Functions and End Behavior - Complete Interactive Lesson

Part 1: Degree & Leading Coefficient

📐 Degree & Leading Coefficient

Part 1 of 7 — Degree & Leading Coefficient

Degree: highest power of $x$
Leading coefficient: coefficient of the highest-degree term
These two values determine end behavior

Example: $f(x) = -3x^4 + 2x^2 - 1$ → degree 4, leading coefficient $-3$

Worked Example

$g(x) = 5x^3 - x + 7$

Degree = 3, Leading coefficient = 5 ✅

Concept Check 🎯

Identify Degree 🧮

$2x^5 - x$ . Degree?
$-x^2 + 3x$ . Degree?
. Leading coefficient?

Concept Check 🔍

Practice

#	Polynomial	Degree	Leading Coeff
1	$2x^5 - x$	5	2
2	$-x^2 + 3x + 1$

Challenge Question 📋

Part 2: End Behavior Rules

📊 End Behavior Rules

Part 2 of 7 — End Behavior Rules

Degree	Leading Coeff	Left	Right
Even	+	↑	↑
Even	−	↓	↓
Odd	+	↓	↑
Odd	−	↑	↓

As $x \to \pm\infty$ , only the leading term matters.

Worked Example

. End behavior?

Part 3: Zeros & Multiplicity

🔢 Zeros & Multiplicity

Part 3 of 7 — Zeros & Multiplicity

A zero (root) is where $f(x) = 0$
Multiplicity: how many times a factor repeats
- Odd multiplicity → graph crosses x-axis
- Even multiplicity → graph bounces off x-axis

$f(x) = (x-2)^3(x+1)^2$ → zero at (mult. 3, cross), zero at (mult. 2, bounce)

Part 4: Turning Points

📈 Turning Points

Part 4 of 7 — Turning Points

A degree- $n$ polynomial has at most $n-1$ turning points.

Local max: graph goes from increasing to decreasing
Local min: graph goes from decreasing to increasing

The number of turning points is always $\leq n - 1$ .

Part 5: Sketching Polynomials

🧮 Sketching Polynomials

Part 5 of 7 — Sketching Polynomials

Steps to sketch:

Find the degree and leading coefficient → end behavior
Find zeros and their multiplicities
Find the y-intercept ( $f(0)$ )
Plot key points and connect smoothly

Worked Example

$f(x) = x^2 - 4 = (x-2)(x+2)$

Part 6: Problem-Solving Workshop

🛠️ Problem-Solving Workshop

Part 6 of 7 — Problem-Solving Workshop

Combine all concepts:

Identify degree, LC, end behavior
Find zeros and y-intercept
Determine turning points
Sketch the graph

Worked Example

$f(x) = -x^3 + 4x$

Degree 3, LC = −1 → left ↑, right ↓
→ zeros: 0, 2, −2

Part 7: Review & Applications

🏆 Review & Applications

Part 7 of 7 — Review & Applications

Key Concepts

Degree & LC → end behavior
Even mult → bounce; Odd mult → cross
Max turning points = degree − 1
y-intercept = f(0)

Worked Example

$f(x) = 2x^3 - 6x$ . Degree? LC? y-int?

Degree 3, LC = 2 (odd+pos → left↓ right↑), ✅

4 x^{3} + x^{2} - 1

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

Odd degree, negative leading coeff → Left ↑, Right ↓ ✅

Concept Check 🎯

End Behavior 🧮

$3x^4$ : Degree?
$-x^2 + 1$ : Degree?
$5x^3 - 2x$ : Degree?

Concept Check 🔍

Practice

#	Polynomial	Left	Right
1	$x^2$	↑	↑
2	$-x^3$	↑	↓
3	$x^4$	↑	↑

Challenge Question 📋

Worked Example

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

$x = 1$ (multiplicity 2, bounces), $x = -3$ (multiplicity 1, crosses) ✅

Concept Check 🎯

Find the Zeros 🧮

$(x-4)^2 = 0$ . $x =$ ?
$(x+2)^3 = 0$ . $x =$ ?
$(x-1)(x+1)^2 = 0$ . One zero is $x =$ ?

Concept Check 🔍

Practice

#	Function	Zeros	Behavior
1	$(x-4)^2$	4	Bounce
2	$(x+2)^3$	−2	Cross
3	$(x-1)(x+1)^2$	1, −1	Cross, Bounce

Challenge Question 📋

Worked Example

Degree 4 polynomial. Max turning points?

$4 - 1 = 3$ turning points max ✅

Concept Check 🎯

Max Turning Points 🧮

Degree 3 → max turning points?
Degree 5 → max turning points?
Degree 2 → max turning points?

Concept Check 🔍

Practice

#	Degree	Max Turning Points
1	3	2
2	5	4
3	2	1

Challenge Question 📋

Degree 2, LC = 1 → both ends up
Zeros: $x = 2, -2$ (mult 1 each → cross)
y-int: $f(0) = -4$
Parabola opening up ✅

Concept Check 🎯

y-Intercepts 🧮

Find $f(0)$ :

$f(x) = x^2 - 9$
$f(x) = x^3 - x$
$f(x) = 2x^2 + 1$

Concept Check 🔍

Practice

#	Function	y-intercept
1	$x^2 - 9$	−9
2	$x^3 - x$	0
3	$2x^2 + 1$	1

Challenge Question 📋

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

Max turning points: 2

Concept Check 🎯

Combined Practice 🧮

One zero of $x^2 - 1$ ?
The other zero of $x^2 - 1$ ?
Max turning points for degree 4?

Concept Check 🔍

Practice

#	Concept	Problem
1	End behavior	$-2x^{4}$
2	Zeros	$x^{2}-1$
3	Turning points	degree 4

Challenge Question 📋

Concept Check 🎯

Review 🧮

Degree of $4x^5 + x$ ?
Max turning points for degree 5?
y-intercept of $3x^2 - 7$ ?

Concept Check 🔍

Practice

#	Concept	Answer
1	Degree of $4x^{5}$	5
2	Max turning points for degree 5	4
3	y-int of $3x^{2}-7$	−7

Challenge Question 📋