Chapter 3: Rational Equations and Applications

Solving Rational Equations

Rational equations are equations that involve rational expressions (fractions with polynomials in the numerator and denominator). Solving these equations requires finding a common denominator and ensuring that solutions do not make any denominator zero.

• Key Steps:

  1. Identify the least common denominator (LCD) of all rational expressions.

  2. Multiply both sides of the equation by the LCD to clear denominators.

  3. Solve the resulting polynomial equation.

  4. Check all solutions in the original equation to avoid extraneous solutions (values that make any denominator zero).

• Example: Solve .

  • LCD is .

  • Multiply both sides by : .

  • Solve: .

  • Check: does not make any denominator zero, so it is valid.

Chapter 4: Polynomial and Rational Functions

Lesson 4.1: Polynomial Functions and Their Graphs

• Leading Term, Leading Coefficient, and Degree:

  • The leading term is the term with the highest degree in a polynomial.

  • The leading coefficient is the coefficient of the leading term.

  • The degree is the highest power of the variable in the polynomial.

  • Example: For , the leading term is , the leading coefficient is 4, and the degree is 3.

• End Behavior:

  • Describes how the function behaves as or .

  • Determined by the leading term.

  • Example: For , as , .

• Behavior at Zeros:

  • The graph crosses or touches the x-axis at zeros depending on their multiplicity.

  • If the zero has odd multiplicity, the graph crosses the axis; if even multiplicity, it touches and turns around.

• Finding Zeros and Multiplicities:

  • Set the polynomial equal to zero and solve for .

  • Multiplicity is the exponent of the factor.

  • Example: has zeros at (multiplicity 2) and (multiplicity 1).

• Finding Real Zeros from Graphs:

  • Identify x-intercepts on the graph; these are the real zeros.

Lesson 4.2: Properties and Graphs of Polynomial Functions

• Maximum Number of Real Zeros and Turning Points:

  • A polynomial of degree has at most $n$ real zeros and turning points.

• Graphing Polynomial Functions:

  • Plot zeros, determine end behavior, and sketch the graph using turning points and multiplicities.

• Applications:

  • Polynomial functions can model real-world phenomena such as projectile motion or profit maximization.

Modeling and Regression

• Building Polynomial Models:

  • Use data points to construct a polynomial function that fits the data (regression).

Lesson 4.4: Constructing Polynomials from Zeros

• Finding a Polynomial with Specified Zeros:

  • If zeros are , , and , the polynomial is , where is a constant.

Lesson 4.5: Rational Functions

• Domain of Rational Functions:

  • All real numbers except those that make the denominator zero.

• Graphing Rational Functions:

  • Identify vertical and horizontal asymptotes, holes, and intercepts.

• Asymptotes:

  • Vertical asymptotes: Set denominator equal to zero and solve for .

  • Horizontal asymptotes: Compare degrees of numerator and denominator.

  • Example: has a vertical asymptote at and a horizontal asymptote at .

Lesson 4.6: Polynomial Inequalities

• Solving Polynomial Inequalities:

  • Set the inequality to zero, find zeros, and test intervals between zeros.

  • Express the solution in interval notation.

Quadratic Applications

• Solving Applications Involving Quadratic Equations:

  • Translate word problems into quadratic equations and solve using factoring, completing the square, or the quadratic formula.

  • Quadratic Formula:

Chapter 5: Exponential and Logarithmic Functions

Lesson 5.1: Inverse Functions

• Inverse of a Relation:

  • Switch the roles of and and solve for $y$.

• One-to-One Functions:

  • A function is one-to-one if each -value corresponds to only one -value.

  • Passes the horizontal line test.

• Graphing Inverse Relations:

  • The graph of an inverse is a reflection over the line .

Lesson 5.2: Exponential Functions and Applications

• Evaluating Exponential Functions:

  • Exponential functions have the form .

• Compound Interest Formula:

  • For compounding periods per year:

  • For continuous compounding:

• Applications:

  • Population growth, radioactive decay, and finance.

Lesson 5.3: Logarithmic Functions

• Exponential and Logarithmic Equations:

  • Logarithms are the inverses of exponentials:

• Change of Base Formula:

• Evaluating Logarithms:

  • Use properties and calculators as needed.

Lesson 5.4: Properties of Logarithms

• Logarithm Rules:

  • Product:

  • Quotient:

  • Power:

• Evaluating Expressions:

  • Combine or expand logarithms using the above properties.

Lesson 5.5: Solving Exponential and Logarithmic Equations

• Solving Exponential Equations:

  • Isolate the exponential part and take logarithms of both sides if necessary.

• Solving Logarithmic Equations:

  • Combine logarithms, rewrite in exponential form, and solve for the variable.

Lesson 5.6: Exponential Growth and Decay

• Exponential Growth and Decay:

  • General formula: , where for growth, for decay.

• Compound Interest Applications:

  • Use the compound interest formulas as above.

Exponential Modeling and Regression

• Matching Functions to Models:

  • Identify the best-fit exponential function for a set of data.

• Applications:

  • Population, finance, and science problems involving exponential change.

Chapter 6: Systems of Equations and Matrices

Lesson 6.1: Systems of Linear and Nonlinear Equations

• Solving Systems of Linear Equations:

  • Methods: Substitution, elimination, or matrices.

  • Any method may be used unless specified.

• Solving Nonlinear Systems:

  • Use substitution or elimination to solve systems involving at least one nonlinear equation.

Lesson 6.2 & 6.3: Matrices and Systems

• Order of a Matrix:

  • The order is the number of rows by the number of columns (e.g., 2x3).

• Augmented Matrix:

  • Represents a system of equations in matrix form, including the constants on the right side.

  • Example: The system , is written as:

    1	2	3
    3	-1	4

• Solving Systems Using Matrices:

  • Use row operations to reduce the augmented matrix to row-echelon form or reduced row-echelon form.

  • Back-substitute to find solutions.