Unit 1a Review Polynomial And Rational Functions

Polynomial and rational functions form the bedrock of advanced algebra and calculus. Understanding their properties, behaviors, and manipulation techniques is crucial for success in higher mathematics and many applied fields. This review aims to provide a comprehensive recap of key concepts related to polynomial and rational functions, equipping you with the knowledge and skills necessary to tackle complex problems involving these functions.

Polynomial Functions: A Deep Dive

Polynomial functions are fundamental building blocks in mathematics. They are defined by their simple structure, yet exhibit a wide range of behaviors and applications.

Definition and General Form

A polynomial function is a function that can be expressed in the form:

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Where:

• x is the variable.
• n is a non-negative integer representing the degree of the polynomial.
• aₙ, aₙ₋₁, ..., a₁, a₀ are constants called coefficients, where aₙ ≠ 0.

Key characteristics of polynomial functions:

• Non-negative integer exponents: The exponent of each term must be a whole number (0, 1, 2, 3, ...). This distinguishes them from other types of functions like rational or radical functions.
• Real number coefficients: The coefficients (aₙ, aₙ₋₁, etc.) are real numbers.
• Continuous and smooth graphs: Polynomial functions have graphs that are continuous (no breaks or jumps) and smooth (no sharp corners or cusps).

Examples of polynomial functions:

• f(x) = 3x² - 2x + 1 (Quadratic function, degree 2)
• f(x) = x³ + 5x - 7 (Cubic function, degree 3)
• f(x) = 2x⁵ - x² + 4 (Quintic function, degree 5)
• f(x) = 5 (Constant function, degree 0)
• f(x) = -2x (Linear function, degree 1)

Degree and Leading Coefficient

The degree of a polynomial is the highest power of the variable x in the polynomial. The leading coefficient is the coefficient of the term with the highest power of x.

Example:

Consider the polynomial function: f(x) = -4x⁴ + 2x³ - x + 6

• Degree: 4 (because the highest power of x is 4)
• Leading Coefficient: -4 (the coefficient of the x⁴ term)

The degree and leading coefficient significantly influence the end behavior of the polynomial function's graph.

End Behavior

The end behavior of a polynomial function describes what happens to the function's values (f(x)) as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). It's determined by the degree and the leading coefficient.

Here's a summary of the end behavior based on the degree and leading coefficient:

• Even Degree:

  • Positive Leading Coefficient: As x → ∞, f(x) → ∞ and as x → -∞, f(x) → ∞ (The graph rises to the left and right)
  • Negative Leading Coefficient: As x → ∞, f(x) → -∞ and as x → -∞, f(x) → -∞ (The graph falls to the left and right)

• Odd Degree:

  • Positive Leading Coefficient: As x → ∞, f(x) → ∞ and as x → -∞, f(x) → -∞ (The graph falls to the left and rises to the right)
  • Negative Leading Coefficient: As x → ∞, f(x) → -∞ and as x → -∞, f(x) → ∞ (The graph rises to the left and falls to the right)

Examples:

• f(x) = 2x³ + x - 1 (Odd degree, positive leading coefficient): Falls to the left, rises to the right.
• f(x) = -x⁴ + 3x² + 2 (Even degree, negative leading coefficient): Falls to the left, falls to the right.
• f(x) = x² - 5x + 6 (Even degree, positive leading coefficient): Rises to the left, rises to the right.
• f(x) = -x⁵ + 4x³ - x (Odd degree, negative leading coefficient): Rises to the left, falls to the right.

Zeros (Roots) of Polynomial Functions

The zeros (also called roots or x-intercepts) of a polynomial function are the values of x for which f(x) = 0. Finding the zeros is a crucial step in analyzing and graphing polynomial functions.

Methods for finding zeros:

• Factoring: If the polynomial can be factored, set each factor equal to zero and solve for x. This is the most straightforward method when applicable.

  • Example: f(x) = x² - 4 = (x - 2)(x + 2). Zeros are x = 2 and x = -2.

• Quadratic Formula: For quadratic functions (degree 2) of the form ax² + bx + c = 0, the quadratic formula provides the zeros:

  • x = (-b ± √(b² - 4ac)) / 2a

• Rational Root Theorem: This theorem helps identify potential rational zeros (zeros that can be expressed as a fraction p/q). The possible rational roots are all the factors of the constant term (a₀) divided by all the factors of the leading coefficient (aₙ). You then test these potential roots using synthetic division or direct substitution.

• Synthetic Division: A shorthand method for dividing a polynomial by a linear factor (x - c). If the remainder is 0, then c is a zero of the polynomial. It's a useful tool for testing potential rational roots found using the Rational Root Theorem.

• Numerical Methods (for more complex polynomials): When factoring and other algebraic methods are not feasible, numerical methods like the Newton-Raphson method or using a graphing calculator to approximate the zeros are employed.

Multiplicity of Zeros:

A zero can have a multiplicity, which refers to the number of times a particular factor (x - c) appears in the factored form of the polynomial.

• Odd Multiplicity: The graph crosses the x-axis at the zero.
• Even Multiplicity: The graph touches the x-axis at the zero but does not cross (it "bounces" off the x-axis).

Example:

f(x) = (x - 1)²(x + 2)

• x = 1 is a zero with multiplicity 2 (even multiplicity)
• x = -2 is a zero with multiplicity 1 (odd multiplicity)

Graphing Polynomial Functions

To graph a polynomial function effectively, consider the following steps:

1. Determine the end behavior: Based on the degree and leading coefficient.
2. Find the zeros (roots): Factor, use the quadratic formula, Rational Root Theorem, or numerical methods.
3. Determine the multiplicity of each zero: This tells you whether the graph crosses or touches the x-axis at each zero.
4. Find the y-intercept: Set x = 0 and solve for f(0).
5. Find additional points: Choose x-values between and beyond the zeros and calculate the corresponding f(x) values. This helps to refine the shape of the graph.
6. Sketch the graph: Plot the points, keeping in mind the end behavior and the behavior at the zeros. Remember that polynomial functions have continuous and smooth graphs.

Transformations of Polynomial Functions

Understanding transformations allows you to quickly sketch graphs of related polynomial functions based on a known parent function.

Common transformations include:

• Vertical Shifts: f(x) + c (shifts the graph up by c units if c > 0, down by c units if c < 0)
• Horizontal Shifts: f(x - c) (shifts the graph right by c units if c > 0, left by c units if c < 0)
• Vertical Stretches/Compressions: af(x) (stretches the graph vertically by a factor of a if a > 1, compresses it vertically if 0 < a < 1. If a < 0, it also reflects the graph across the x-axis)
• Horizontal Stretches/Compressions: f(ax) (compresses the graph horizontally by a factor of a if a > 1, stretches it horizontally if 0 < a < 1. If a < 0, it also reflects the graph across the y-axis)
• Reflections: -f(x) (reflects the graph across the x-axis), f(-x) (reflects the graph across the y-axis)

Rational Functions: Unveiling Asymptotes and Discontinuities

Rational functions are functions that can be expressed as the ratio of two polynomials. They introduce new complexities compared to polynomial functions, particularly with the concepts of asymptotes and discontinuities.

Definition and General Form

A rational function is a function of the form:

f(x) = P(x) / Q(x)

Where:

• P(x) and Q(x) are polynomial functions.
• Q(x) ≠ 0 (The denominator cannot be zero)

Key characteristics of rational functions:

• Ratio of polynomials: The function is defined as a fraction where both the numerator and denominator are polynomials.
• Potential for discontinuities: Rational functions can have points where they are undefined (discontinuous) due to the denominator being zero.
• Asymptotes: Rational functions often have asymptotes (lines that the graph approaches but never touches or crosses).

Examples of rational functions:

• f(x) = 1/x (The reciprocal function)
• f(x) = (x + 2) / (x - 1)
• f(x) = (x² + 1) / (x² - 4)
• f(x) = (2x - 3) / (x² + 5x + 6)

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator Q(x) is equal to zero, and the numerator P(x) is not equal to zero at the same value. In other words, if c is a real number such that Q(c) = 0 and P(c) ≠ 0, then there is a vertical asymptote at x = c.

Finding Vertical Asymptotes:

1. Set the denominator Q(x) equal to zero and solve for x.
2. For each solution c, check if P(c) ≠ 0. If it is, then x = c is a vertical asymptote.

Example:

f(x) = (x + 1) / (x - 2)

1. Set the denominator equal to zero: x - 2 = 0 => x = 2
2. Check the numerator at x = 2: (2 + 1) = 3 ≠ 0
3. Therefore, there is a vertical asymptote at x = 2.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of the function as x approaches positive infinity (x → ∞) and negative infinity (x → -∞). They represent the horizontal line that the graph approaches as x gets very large or very small.

Rules for Finding Horizontal Asymptotes:

Let f(x) = P(x) / Q(x), where the degree of P(x) is m and the degree of Q(x) is n.

1. If m < n: The horizontal asymptote is y = 0 (the x-axis).
2. If m = n: The horizontal asymptote is y = (leading coefficient of P(x)) / (leading coefficient of Q(x)).
3. If m > n: There is no horizontal asymptote. Instead, there may be a slant (oblique) asymptote (discussed below).

Examples:

• f(x) = (x + 1) / (x²) (m = 1, n = 2, m < n): Horizontal asymptote is y = 0.
• f(x) = (3x² + 2x) / (x² - 1) (m = 2, n = 2, m = n): Horizontal asymptote is y = 3/1 = 3.
• f(x) = (x³ + 1) / (x + 2) (m = 3, n = 1, m > n): No horizontal asymptote.

Slant (Oblique) Asymptotes

Slant asymptotes (also called oblique asymptotes) occur when the degree of the numerator P(x) is exactly one greater than the degree of the denominator Q(x). They are lines of the form y = mx + b that the graph approaches as x approaches positive or negative infinity.

Finding Slant Asymptotes:

1. Perform polynomial long division of P(x) by Q(x).
2. The quotient (excluding the remainder) is the equation of the slant asymptote.

Example:

f(x) = (x² + x - 2) / (x - 1)

1. Divide (x² + x - 2) by (x - 1) using long division. The quotient is x + 2.
2. Therefore, the slant asymptote is y = x + 2.

Holes (Removable Discontinuities)

Holes occur when a factor (x - c) is present in both the numerator P(x) and the denominator Q(x). This means that the function is undefined at x = c, but the discontinuity is removable because the factor can be cancelled out.

Finding Holes:

1. Factor both the numerator and the denominator.
2. Identify any common factors.
3. The x-value that makes the common factor equal to zero is the x-coordinate of the hole.
4. To find the y-coordinate of the hole, cancel the common factor and then substitute the x-value into the simplified function.

Example:

f(x) = (x² - 1) / (x - 1) = ((x - 1)(x + 1)) / (x - 1)

1. The common factor is (x - 1).
2. The x-coordinate of the hole is x = 1.
3. Cancel the common factor: (x + 1).
4. Substitute x = 1 into the simplified function: (1 + 1) = 2.
5. Therefore, there is a hole at the point (1, 2).

Graphing Rational Functions

To graph a rational function effectively, follow these steps:

1. Find the asymptotes: Vertical, horizontal, and slant (if applicable).
2. Find the holes: Identify any removable discontinuities.
3. Find the intercepts: Set x = 0 to find the y-intercept, and set f(x) = 0 to find the x-intercepts (zeros of the numerator).
4. Create a sign chart: Determine the intervals where the function is positive or negative. This helps to understand how the graph behaves between the asymptotes and intercepts. Choose test values in each interval and evaluate the function.
5. Plot the asymptotes, holes, and intercepts: Use dashed lines to represent the asymptotes.
6. Sketch the graph: Use the information gathered to sketch the graph, making sure it approaches the asymptotes correctly and passes through the intercepts and avoids the holes. Remember that the graph can cross a horizontal or slant asymptote, but it will never cross a vertical asymptote.

Transformations of Rational Functions

Similar to polynomial functions, rational functions can also be transformed. The same principles of vertical and horizontal shifts, stretches, compressions, and reflections apply. Understanding these transformations allows you to quickly sketch variations of basic rational functions like f(x) = 1/x.

Applications of Polynomial and Rational Functions

Polynomial and rational functions have wide-ranging applications in various fields:

• Physics: Modeling projectile motion (polynomial), describing electrical circuits (rational).
• Engineering: Designing structures (polynomial), analyzing fluid flow (rational).
• Economics: Modeling cost, revenue, and profit functions (polynomial and rational).
• Computer Graphics: Creating curves and surfaces (polynomial).
• Chemistry: Modeling reaction rates (rational).
• Biology: Modeling population growth (rational).

Conclusion

Polynomial and rational functions are essential concepts in mathematics with numerous real-world applications. This review has provided a comprehensive overview of their definitions, properties, graphing techniques, and transformations. By mastering these concepts, you will be well-equipped to tackle more advanced mathematical problems and apply these functions to solve practical problems in various fields. Practice is key to solidifying your understanding. Work through numerous examples, and don't hesitate to seek help when needed. Good luck!