Graphing Polynomial Functions

Understanding polynomial behavior and graphs

Graphing Polynomial Functions

End Behavior

Determined by the leading term $a_nx^n$ :

Odd degree:

$a_n > 0$ : falls left, rises right ↙↗
$a_n < 0$ : rises left, falls right ↖↘

Even degree:

$a_n > 0$ : rises both sides ↗↗
$a_n < 0$ : falls both sides ↘↘

Zeros and Multiplicity

Zero: value where $f(x) = 0$ (x-intercept)

Multiplicity: how many times the factor appears

Even multiplicity: graph touches x-axis and bounces Odd multiplicity: graph crosses x-axis

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

Zero at $x = 2$ (multiplicity 2, bounces)
Zero at $x = -1$ (multiplicity 1, crosses)

Turning Points

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

Y-Intercept

Evaluate $f(0)$ to find where graph crosses y-axis.

Key Features

Degree determines end behavior
Leading coefficient affects direction
Zeros show x-intercepts
Multiplicity affects crossing behavior

📚 Practice Problems

1Problem 1easy

❓ Question:

Identify the degree and leading coefficient of f(x) = -2x⁴ + 3x² - 5, and describe the end behavior.

💡 Show Solution

Step 1: Identify the degree: The highest power is 4, so degree = 4 (even)

Step 2: Identify the leading coefficient: The coefficient of x⁴ is -2 (negative)

Step 3: Determine end behavior: For even degree with negative leading coefficient:

As x → -∞, f(x) → -∞
As x → +∞, f(x) → -∞ (Both ends go down)

Step 4: Visualize: The graph looks like an upside-down "W" shape

Answer: Degree 4, leading coefficient -2 End behavior: both ends go to -∞

2Problem 2easy

❓ Question:

Identify the degree and leading coefficient of f(x) = -2x⁴ + 3x² - 5, and describe the end behavior.

💡 Show Solution

Step 1: Identify the degree: The highest power is 4, so degree = 4 (even)

Step 2: Identify the leading coefficient: The coefficient of x⁴ is -2 (negative)

Step 3: Determine end behavior: For even degree with negative leading coefficient:

As x → -∞, f(x) → -∞
As x → +∞, f(x) → -∞ (Both ends go down)

Step 4: Visualize: The graph looks like an upside-down "W" shape

Answer: Degree 4, leading coefficient -2 End behavior: both ends go to -∞

3Problem 3easy

❓ Question:

Find the zeros and their multiplicities for f(x) = (x + 2)²(x - 3).

💡 Show Solution

Step 1: Set each factor equal to zero: (x + 2)² = 0 → x = -2 (x - 3) = 0 → x = 3

Step 2: Determine multiplicities: (x + 2)² has exponent 2 → multiplicity 2 (x - 3) has exponent 1 → multiplicity 1

Step 3: Describe behavior at each zero: At x = -2 (even multiplicity): graph touches x-axis and turns around At x = 3 (odd multiplicity): graph crosses x-axis

Answer: Zero at x = -2 with multiplicity 2 (touches) Zero at x = 3 with multiplicity 1 (crosses)

4Problem 4easy

❓ Question:

Find the zeros and their multiplicities for f(x) = (x + 2)²(x - 3).

💡 Show Solution

Step 1: Set each factor equal to zero: (x + 2)² = 0 → x = -2 (x - 3) = 0 → x = 3

Step 2: Determine multiplicities: (x + 2)² has exponent 2 → multiplicity 2 (x - 3) has exponent 1 → multiplicity 1

Step 3: Describe behavior at each zero: At x = -2 (even multiplicity): graph touches x-axis and turns around At x = 3 (odd multiplicity): graph crosses x-axis

Answer: Zero at x = -2 with multiplicity 2 (touches) Zero at x = 3 with multiplicity 1 (crosses)

5Problem 5easy

❓ Question:

Describe the end behavior of $f(x) = -2x^4 + 3x^2 - 1$

💡 Show Solution

Leading term: $-2x^4$

Degree: 4 (even) Leading coefficient: -2 (negative)

For even degree with negative leading coefficient:

Left end: falls (goes to $-\infty$ )
Right end: falls (goes to $-\infty$ )

Answer: Falls on both ends ↘↘

6Problem 6medium

❓ Question:

Sketch the general shape of f(x) = x³ - 4x. Find the zeros and describe the end behavior.

💡 Show Solution

Step 1: Factor to find zeros: f(x) = x³ - 4x = x(x² - 4) = x(x + 2)(x - 2)

Step 2: Identify zeros: x = 0, x = -2, x = 2 All have multiplicity 1 (all cross the x-axis)

Step 3: Determine end behavior: Degree 3 (odd), leading coefficient 1 (positive)

As x → -∞, f(x) → -∞
As x → +∞, f(x) → +∞

Step 4: Find y-intercept: f(0) = 0

Step 5: Describe the graph:

Crosses x-axis at -2, 0, and 2
Starts from bottom left
Ends at top right
Has 2 turning points (degree 3 has at most 2)

Answer: Zeros: x = -2, 0, 2 (all cross) End behavior: -∞ to +∞

7Problem 7medium

❓ Question:

Sketch the general shape of f(x) = x³ - 4x. Find the zeros and describe the end behavior.

💡 Show Solution

Step 1: Factor to find zeros: f(x) = x³ - 4x = x(x² - 4) = x(x + 2)(x - 2)

Step 2: Identify zeros: x = 0, x = -2, x = 2 All have multiplicity 1 (all cross the x-axis)

Step 3: Determine end behavior: Degree 3 (odd), leading coefficient 1 (positive)

As x → -∞, f(x) → -∞
As x → +∞, f(x) → +∞

Step 4: Find y-intercept: f(0) = 0

Step 5: Describe the graph:

Crosses x-axis at -2, 0, and 2
Starts from bottom left
Ends at top right
Has 2 turning points (degree 3 has at most 2)

Answer: Zeros: x = -2, 0, 2 (all cross) End behavior: -∞ to +∞

8Problem 8medium

❓ Question:

Find all zeros and their multiplicities: $f(x) = x^3(x - 2)^2(x + 1)$

💡 Show Solution

Set each factor equal to zero:

From $x^3$ :

Zero at $x = 0$ , multiplicity 3 (odd, crosses)

From $(x - 2)^2$ :

Zero at $x = 2$ , multiplicity 2 (even, bounces)

From $(x + 1)$ :

Zero at $x = -1$ , multiplicity 1 (odd, crosses)

Answer:

$x = 0$ (mult. 3, crosses)
$x = 2$ (mult. 2, bounces)
$x = -1$ (mult. 1, crosses)

9Problem 9medium

❓ Question:

Determine the maximum number of turning points for f(x) = 2x⁵ - 3x⁴ + x² - 7.

💡 Show Solution

Step 1: Recall the turning points rule: A polynomial of degree n has at most (n - 1) turning points

Step 2: Identify the degree: The highest power is 5

Step 3: Calculate maximum turning points: Maximum turning points = 5 - 1 = 4

Step 4: Additional information:

The actual number could be less than 4
Turning points are local maxima or minima
These occur where f'(x) = 0

Answer: Maximum of 4 turning points

10Problem 10medium

❓ Question:

What is the maximum number of turning points for $f(x) = 5x^6 - 3x^4 + x^2 - 7$ ?

💡 Show Solution

The degree of the polynomial is 6.

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

$\text{Max turning points} = 6 - 1 = 5$

Answer: Maximum of 5 turning points

11Problem 11medium

❓ Question:

Determine the maximum number of turning points for f(x) = 2x⁵ - 3x⁴ + x² - 7.

💡 Show Solution

Step 1: Recall the turning points rule: A polynomial of degree n has at most (n - 1) turning points

Step 2: Identify the degree: The highest power is 5

Step 3: Calculate maximum turning points: Maximum turning points = 5 - 1 = 4

Step 4: Additional information:

The actual number could be less than 4
Turning points are local maxima or minima
These occur where f'(x) = 0

Answer: Maximum of 4 turning points

12Problem 12hard

❓ Question:

Write a polynomial function in factored form with the following characteristics: degree 4, zeros at x = -1 (multiplicity 2), x = 2 (multiplicity 1), x = 3 (multiplicity 1), and passes through the point (0, -6).

💡 Show Solution

Step 1: Write the general form using zeros and multiplicities: f(x) = a(x + 1)²(x - 2)(x - 3) where a is a constant to be determined

Step 2: Use the point (0, -6) to find a: f(0) = -6 a(0 + 1)²(0 - 2)(0 - 3) = -6 a(1)(-2)(-3) = -6 a(6) = -6 a = -1

Step 3: Write the final function: f(x) = -(x + 1)²(x - 2)(x - 3)

Step 4: Verify the point (0, -6): f(0) = -(1)²(-2)(-3) = -(1)(-2)(-3) = -6 ✓

Step 5: Verify end behavior: Expand to find leading term: Leading term = -x⁴ Degree 4 (even), negative coefficient Both ends → -∞

Answer: f(x) = -(x + 1)²(x - 2)(x - 3)

13Problem 13hard

❓ Question:

💡 Show Solution

Step 1: Write the general form using zeros and multiplicities: f(x) = a(x + 1)²(x - 2)(x - 3) where a is a constant to be determined

Step 2: Use the point (0, -6) to find a: f(0) = -6 a(0 + 1)²(0 - 2)(0 - 3) = -6 a(1)(-2)(-3) = -6 a(6) = -6 a = -1

Step 3: Write the final function: f(x) = -(x + 1)²(x - 2)(x - 3)

Step 4: Verify the point (0, -6): f(0) = -(1)²(-2)(-3) = -(1)(-2)(-3) = -6 ✓

Step 5: Verify end behavior: Expand to find leading term: Leading term = -x⁴ Degree 4 (even), negative coefficient Both ends → -∞

Answer: f(x) = -(x + 1)²(x - 2)(x - 3)

🎴

Practice with Flashcards

Review key concepts with our flashcard system

📖

Browse All Topics

Explore other calculus topics