Unit 3 Power Polynomials And Rational Functions

Polynomials and rational functions form the bedrock of advanced algebra and calculus, playing a crucial role in modeling real-world phenomena and solving complex equations. Unit 3 delves into the intricacies of these functions, exploring their properties, behaviors, and applications.

Understanding Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. The general form of a polynomial function is:

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

Where:

• aₙ, aₙ₋₁, ..., a₁, a₀ are coefficients (real numbers).
• x is the variable.
• n is a non-negative integer representing the degree of the polynomial.

Key Characteristics of Polynomial Functions

• Domain: All real numbers. Polynomial functions are defined for any value of x.
• Continuity: Polynomial functions are continuous, meaning their graphs have no breaks, holes, or jumps.
• Smoothness: Polynomial functions are smooth, meaning their graphs have no sharp corners or cusps.
• End Behavior: The end behavior of a polynomial function describes what happens to the function's values as x approaches positive or negative infinity. The end behavior is determined by the leading term (aₙxⁿ) of the polynomial.
• Zeros (Roots): The zeros of a polynomial function are the values of x for which f(x) = 0. These are the points where the graph intersects the x-axis.
• Turning Points: Turning points are the points where the graph changes direction (from increasing to decreasing or vice versa). The number of turning points is at most n - 1, where n is the degree of the polynomial.

Graphing Polynomial Functions

To graph a polynomial function, consider the following steps:

1. Determine the end behavior: Look at the leading term (aₙxⁿ).

   • If n is even and aₙ > 0, the graph rises to the left and right.
   • If n is even and aₙ < 0, the graph falls to the left and right.
   • If n is odd and aₙ > 0, the graph falls to the left and rises to the right.
   • If n is odd and aₙ < 0, the graph rises to the left and falls to the right.

2. Find the zeros: Set f(x) = 0 and solve for x. Each real zero corresponds to an x-intercept.

3. Determine the multiplicity of each zero: The multiplicity of a zero is the number of times the corresponding factor appears in the factored form of the polynomial.

   • If a zero has odd multiplicity, the graph crosses the x-axis at that point.
   • If a zero has even multiplicity, the graph touches the x-axis at that point and bounces back.

4. Find the y-intercept: Set x = 0 and find f(0). This is the point where the graph intersects the y-axis.

5. Find additional points: Choose some values of x between the zeros and evaluate f(x) to get additional points to plot.

6. Sketch the graph: Connect the points, keeping in mind the end behavior, zeros, multiplicity, and turning points.

Example: Graphing a Polynomial Function

Let's graph the polynomial function f(x) = x³ - 3x² + 2x.

1. End Behavior: The leading term is x³. Since the degree is odd and the coefficient is positive, the graph falls to the left and rises to the right.

2. Zeros: Set f(x) = 0:

   • x³ - 3x² + 2x = 0
   • x(x² - 3x + 2) = 0
   • x(x - 1)(x - 2) = 0
   • The zeros are x = 0, x = 1, x = 2.

3. Multiplicity: Each zero has a multiplicity of 1 (odd). The graph crosses the x-axis at each zero.

4. Y-intercept: Set x = 0:

   • f(0) = 0³ - 3(0)² + 2(0) = 0
   • The y-intercept is (0, 0).

5. Additional Points: Let's find f(0.5) and f(1.5):

   • f(0.5) = (0.5)³ - 3(0.5)² + 2(0.5) = 0.375
   • f(1.5) = (1.5)³ - 3(1.5)² + 2(1.5) = -0.375

6. Sketch the Graph: Plot the points and connect them, keeping in mind the end behavior and zeros. The graph will fall from the left, cross the x-axis at x = 0, rise to a turning point near x = 0.5, fall to a turning point near x = 1.5, cross the x-axis at x = 1 and x = 2, and then rise to the right.

Exploring Rational Functions

Rational functions are functions that can be expressed as the ratio of two polynomial functions:

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

Where P(x) and Q(x) are polynomial functions, and Q(x) ≠ 0.

Key Characteristics of Rational Functions

• Domain: The domain of a rational function is all real numbers except for the values of x that make the denominator Q(x) equal to zero. These values are called discontinuities.

• Asymptotes: Asymptotes are lines that the graph of a rational function approaches but never touches. There are three types of asymptotes:

  • Vertical Asymptotes: Occur at the zeros of the denominator Q(x), where the function is undefined. If Q(a) = 0 and P(a) ≠ 0, then there is a vertical asymptote at x = a.
  • Horizontal Asymptotes: Describe the end behavior of the function as x approaches positive or negative infinity. To find the horizontal asymptote, compare the degrees of P(x) and Q(x):
    • If the degree of P(x) is less than the degree of Q(x), the horizontal asymptote is y = 0.
    • If the degree of P(x) is equal to the degree of Q(x), the horizontal asymptote is y = aₙ / bₙ, where aₙ and bₙ are the leading coefficients of P(x) and Q(x), respectively.
    • If the degree of P(x) is greater than the degree of Q(x), there is no horizontal asymptote. Instead, there may be a slant (oblique) asymptote.
  • Slant (Oblique) Asymptotes: Occur when the degree of P(x) is exactly one more than the degree of Q(x). To find the slant asymptote, perform polynomial long division of P(x) by Q(x). The quotient (without the remainder) is the equation of the slant asymptote.

• Zeros (Roots): The zeros of a rational function are the values of x that make the numerator P(x) equal to zero, provided that these values are not also zeros of the denominator.

• Y-intercept: The y-intercept is the point where the graph intersects the y-axis. To find the y-intercept, set x = 0 and evaluate f(0).

• Holes: Holes occur when a factor is common to both the numerator and the denominator. If both P(a) = 0 and Q(a) = 0, then there is a hole at x = a. To find the y-coordinate of the hole, simplify the rational function by canceling the common factor, and then evaluate the simplified function at x = a.

Graphing Rational Functions

To graph a rational function, follow these steps:

1. Find the domain: Identify any values of x that make the denominator zero. These values will correspond to vertical asymptotes or holes.

2. Find the vertical asymptotes: If Q(a) = 0 and P(a) ≠ 0, then there is a vertical asymptote at x = a.

3. Find the horizontal or slant asymptote: Compare the degrees of the numerator and denominator to determine the existence and equation of any horizontal or slant asymptotes.

4. Find the zeros: Set the numerator P(x) = 0 and solve for x. Ensure these values are not also zeros of the denominator.

5. Find the y-intercept: Set x = 0 and evaluate f(0).

6. Find any holes: If there are common factors in the numerator and denominator, identify the x-coordinate of the hole. Simplify the function by canceling the common factor and evaluate the simplified function at that x-coordinate to find the y-coordinate of the hole.

7. Find additional points: Choose some values of x in each interval created by the vertical asymptotes and zeros, and evaluate f(x) to get additional points to plot.

8. Sketch the graph: Connect the points, keeping in mind the asymptotes, zeros, holes, and general behavior of the function. Remember that the graph will approach the asymptotes but never cross them (unless it's a horizontal asymptote that the graph crosses in the middle).

Example: Graphing a Rational Function

Let's graph the rational function f(x) = (x + 2) / (x - 1).

1. Domain: The denominator is x - 1. x - 1 = 0 when x = 1. So the domain is all real numbers except x = 1.

2. Vertical Asymptote: Since x = 1 makes the denominator zero and not the numerator, there is a vertical asymptote at x = 1.

3. Horizontal Asymptote: The degree of the numerator and denominator are both 1. The leading coefficients are both 1. Therefore, the horizontal asymptote is y = 1/1 = 1.

4. Zeros: Set the numerator x + 2 = 0. This gives x = -2.

5. Y-intercept: Set x = 0:

   • f(0) = (0 + 2) / (0 - 1) = -2
   • The y-intercept is (0, -2).

6. Holes: There are no common factors in the numerator and denominator, so there are no holes.

7. Additional Points: Let's find f(-3), f(0.5), and f(2):

   • f(-3) = (-3 + 2) / (-3 - 1) = (-1) / (-4) = 0.25
   • f(0.5) = (0.5 + 2) / (0.5 - 1) = 2.5 / (-0.5) = -5
   • f(2) = (2 + 2) / (2 - 1) = 4 / 1 = 4

8. Sketch the Graph: Plot the asymptotes, zeros, y-intercept, and additional points. The graph will approach the vertical asymptote at x = 1 from both sides. It will also approach the horizontal asymptote at y = 1 as x approaches positive or negative infinity. The graph will cross the x-axis at x = -2 and the y-axis at y = -2.

Polynomial and Rational Inequalities

Solving polynomial and rational inequalities involves finding the intervals where the function is either greater than, less than, greater than or equal to, or less than or equal to a given value (usually zero).

Solving Polynomial Inequalities

1. Rewrite the inequality: Move all terms to one side, so the inequality is in the form f(x) > 0, f(x) < 0, f(x) ≥ 0, or f(x) ≤ 0, where f(x) is a polynomial.

2. Find the zeros: Solve f(x) = 0 to find the zeros of the polynomial. These zeros are the critical values.

3. Create a sign chart: Draw a number line and mark the critical values (zeros) on it. These values divide the number line into intervals.

4. Test each interval: Choose a test value within each interval and evaluate f(x) at that value. Determine the sign of f(x) in each interval (positive or negative).

5. Determine the solution: Identify the intervals where f(x) satisfies the given inequality. Remember to include or exclude the endpoints of the intervals based on whether the inequality includes "equal to".

Solving Rational Inequalities

1. Rewrite the inequality: Move all terms to one side, so the inequality is in the form f(x) > 0, f(x) < 0, f(x) ≥ 0, or f(x) ≤ 0, where f(x) is a rational function.

2. Find the critical values:

   • Find the zeros of the numerator by solving P(x) = 0.
   • Find the zeros of the denominator by solving Q(x) = 0.

3. Create a sign chart: Draw a number line and mark all the critical values (zeros of numerator and denominator) on it. These values divide the number line into intervals.

4. Test each interval: Choose a test value within each interval and evaluate f(x) at that value. Determine the sign of f(x) in each interval (positive or negative).

5. Determine the solution: Identify the intervals where f(x) satisfies the given inequality. Remember:

   • Include the zeros of the numerator if the inequality includes "equal to".
   • Exclude the zeros of the denominator, as the function is undefined at those points.

Example: Solving a Rational Inequality

Solve the inequality (x + 3) / (x - 2) ≥ 0.

1. The inequality is already in the correct form.

2. Find the critical values:

   • Zeros of the numerator: x + 3 = 0 => x = -3
   • Zeros of the denominator: x - 2 = 0 => x = 2

3. Create a sign chart:

   -----(-3)+++++(2)-----
   <----|----|----|---->
       -3    2
   

4. Test each interval:

   • Interval x < -3: Choose x = -4. f(-4) = (-4 + 3) / (-4 - 2) = (-1) / (-6) = 1/6 > 0 (Positive)
   • Interval -3 < x < 2: Choose x = 0. f(0) = (0 + 3) / (0 - 2) = 3 / (-2) = -3/2 < 0 (Negative)
   • Interval x > 2: Choose x = 3. f(3) = (3 + 3) / (3 - 2) = 6 / 1 = 6 > 0 (Positive)

5. Determine the solution:

   • We want (x + 3) / (x - 2) ≥ 0.
   • The function is positive for x < -3 and x > 2.
   • The function is equal to zero at x = -3.
   • The function is undefined at x = 2, so we exclude it.
   • Therefore, the solution is (-∞, -3] ∪ (2, ∞).

Applications of Polynomial and Rational Functions

Polynomial and rational functions have a wide range of applications in various fields:

• Physics: Modeling projectile motion, oscillations, and electrical circuits. Polynomials can approximate complex curves and behaviors.

• Engineering: Designing structures, analyzing circuits, and controlling systems. Rational functions are used in control systems and signal processing.

• Economics: Modeling cost, revenue, and profit functions. Polynomials can represent the demand curve.

• Computer Graphics: Creating curves and surfaces for computer-aided design (CAD) and animation. Bézier curves, which are based on polynomials, are fundamental to computer graphics.

• Statistics: Fitting curves to data and making predictions. Polynomial regression is used to model relationships between variables.

• Chemistry: Modeling reaction rates and chemical equilibrium.

• Biology: Modeling population growth and disease spread.

Conclusion

Unit 3 provides a solid foundation in understanding and manipulating polynomial and rational functions. Mastering the concepts of domain, range, asymptotes, zeros, and graphing techniques is essential for success in higher-level mathematics and its applications. By understanding the behavior of these functions, you can model and solve a wide variety of real-world problems.