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Feb 6, 2026

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4 pages

Understanding and Graphing Polynomial Functions Easily

~Taffi~

@taffi

Polynomial functions are math expressions that combine variables and exponents... Show more

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$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

Polynomial Functions Basics

A polynomial function follows the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the coefficients are real numbers and n is a positive integer. These functions create smooth curves with no breaks or gaps and are continuous for all real numbers.

The degree of a polynomial is its highest exponent, which determines its classification. For example, degree 0 is constant $f(x) = 2$ , degree 1 is linear $f(x) = -3x+8$ , degree 2 is quadratic $f(x) = 5x²+4x-1$ , and degree 3 is cubic $f(x) = -4x³+x²-x+7$ .

The end behavior of a polynomial depends on the power and coefficient of its leading term. For even powers with positive coefficients, both ends point upward (↗↗); with negative coefficients, both ends point downward (↘↘). For odd powers with positive coefficients, left end points down and right end points up (↘↗); with negative coefficients, left end points up and right end points down (↗↘).

🔍 Try This: Identify the degree and end behavior of f(x) = -2x³+5x-1. Since the highest power is 3 (odd) with a negative coefficient, the function will rise on the left and fall on the right.

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

Turning Points

Turning points are where polynomial functions change direction from increasing to decreasing (or vice versa). A polynomial of degree n can have at most $n-1$ turning points, which makes this a key feature for identifying polynomial degree.

When a turning point is higher than nearby points, it's called a relative maximum. When it's lower than nearby points, it's called a relative minimum. These maximum and minimum values together are called extrema.

You can find turning points using a graphing calculator by following these steps: First, graph the function; then use the CALC menu to find minimum and maximum values; finally, identify the left and right bounds of the turning point and get its coordinates.

💡 Quick Tip: For the function f(x) = x³+5x²-8x-1, the relative maximum is at (-4,47) and the relative minimum is at (0.67,-3.48). The function increases from (-∞,-4) and (0.67,∞), while decreasing from (-4,0.67).

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

Zeros of Polynomial Functions

Zeros are where a polynomial function equals zero $f(x) = 0$ and are also called roots or x-intercepts. A polynomial of degree n can have at most n real zeros. Even-degree polynomials have an even number of real zeros (or none), while odd-degree polynomials always have at least one real zero.

The solutions to the equation f(x) = 0 are the zeros of the function. If x = c is a solution, then $x-c$ is a linear factor of the polynomial. You can find linear factors by factoring the polynomial.

For example, given f(x) = x³ + 5x² - 24x, we can factor it as f(x) = x $x² + 5x - 24$ = x $x+8$ $x-3$ . The linear factors are x, $x+8$ , and $x-3$ , which means the zeros are x = 0, x = -8, and x = 3.

🎯 Remember: Every zero of a polynomial corresponds to a factor, and every factor reveals a zero. Knowing how to factor polynomials is essential for finding all zeros quickly.

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

Multiplicity and Graphing

The multiplicity of a zero is the degree of its corresponding linear factor in the factored form of the polynomial. Multiplicity tells us how a graph behaves at each zero.

When a zero has an even multiplicity $like x = -2 in f(x) = x(x+2)²$ , the graph is tangent to the x-axis at that point—it touches but doesn't cross the axis. When a zero has an odd multiplicity $like x = 0 in the same function$ , the graph crosses through the x-axis at that point.

For example, in f(x) = x²+10x² + 25x = x $x+5$ ², the zeros are x = 0 with multiplicity 1 (odd) and x = -5 with multiplicity 2 (even). At x = 0, the graph crosses the x-axis, while at x = -5, the graph touches the x-axis without crossing.

🔍 Visual Clue: When graphing polynomials, look at how the curve behaves at each zero. If it bounces off the x-axis, the multiplicity is even. If it passes through the x-axis, the multiplicity is odd.

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199

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Feb 6, 2026

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4 pages

Understanding and Graphing Polynomial Functions Easily

~Taffi~

@taffi

Polynomial functions are math expressions that combine variables and exponents in specific ways. They create smooth curves on graphs with predictable behaviors that depend on their degree and coefficients. Understanding how these functions behave will help you analyze their graphs... Show more

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

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Polynomial Functions Basics

🔍 Try This: Identify the degree and end behavior of f(x) = -2x³+5x-1. Since the highest power is 3 (odd) with a negative coefficient, the function will rise on the left and fall on the right.

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

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Turning Points

💡 Quick Tip: For the function f(x) = x³+5x²-8x-1, the relative maximum is at (-4,47) and the relative minimum is at (0.67,-3.48). The function increases from (-∞,-4) and (0.67,∞), while decreasing from (-4,0.67).

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

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Zeros of Polynomial Functions

🎯 Remember: Every zero of a polynomial corresponds to a factor, and every factor reveals a zero. Knowing how to factor polynomials is essential for finding all zeros quickly.

$Main Ideas/Questions Notes/Examples A polynomial function is a function of the form: POLYNOMIAL FUNCTION $f(x) = a_nx^n + a_{n-1}x^{n-1} +$

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Multiplicity and Graphing

The multiplicity of a zero is the degree of its corresponding linear factor in the factored form of the polynomial. Multiplicity tells us how a graph behaves at each zero.

🔍 Visual Clue: When graphing polynomials, look at how the curve behaves at each zero. If it bounces off the x-axis, the multiplicity is even. If it passes through the x-axis, the multiplicity is odd.