🍬Honors Algebra II Unit 7 – Rational Expressions and Functions

Key Concepts and Definitions

• Rational expressions are fractions with polynomials in the numerator and denominator
• The domain of a rational expression is all real numbers except those that make the denominator equal to zero
• Rational functions are functions defined by rational expressions
• Asymptotes are lines that a graph approaches but never touches
  • Vertical asymptotes occur when the denominator of a rational function equals zero
  • Horizontal asymptotes describe the behavior of the function as x approaches positive or negative infinity
• Holes or points of discontinuity occur when a factor cancels out in the numerator and denominator, leaving an undefined point
• Extraneous solutions are solutions that satisfy the equation after cross-multiplication but do not satisfy the original equation
• Synthetic division is a shortcut method for dividing polynomials by linear factors

Properties of Rational Expressions

• Rational expressions can be simplified by factoring the numerator and denominator and canceling common factors
• Multiplication and division of rational expressions follow the same rules as fractions
  • Multiply the numerators and denominators separately
  • Divide by multiplying the first fraction by the reciprocal of the second fraction
• Addition and subtraction of rational expressions require a common denominator
  • Find the least common multiple (LCM) of the denominators
  • Multiply each rational expression by the appropriate factor to obtain the common denominator
  • Add or subtract the numerators and keep the common denominator
• The properties of exponents (product, quotient, power, and negative exponent rules) apply to rational expressions

Simplifying Rational Expressions

• Factor the numerator and denominator completely
• Cancel any common factors in the numerator and denominator
• Simplify any remaining terms
• Check for any restrictions on the variable (values that make the denominator zero)
• Example: $\frac{x^2-9}{x^2-5x+6}$ simplifies to $\frac{(x+3)(x-3)}{(x-2)(x-3)}$, which further simplifies to $\frac{x+3}{x-2}$, $x \neq 3$

Operations with Rational Expressions

• Addition and subtraction require a common denominator
  • Find the LCM of the denominators
  • Multiply each rational expression by the appropriate factor to obtain the common denominator
  • Add or subtract the numerators and keep the common denominator
• Multiplication involves multiplying the numerators and denominators separately and simplifying the result
• Division involves multiplying the first rational expression by the reciprocal of the second and simplifying the result
• Complex fractions can be simplified by multiplying the numerator and denominator by the LCM of the denominators within the complex fraction
• Example: $\frac{\frac{2}{x+1}+\frac{3}{x-1}}{\frac{1}{x+1}-\frac{1}{x-1}}$ simplifies to $\frac{5x}{2}$

Solving Rational Equations

• Multiply both sides of the equation by the LCM of the denominators to clear the fractions
• Solve the resulting polynomial equation
• Check the solutions in the original equation to identify any extraneous solutions
• Example: To solve $\frac{2}{x-1}+\frac{3}{x+1}=1$, multiply both sides by $(x-1)(x+1)$ to get $2(x+1)+3(x-1)=x^2-1$, which simplifies to $5x+1=x^2-1$. Solve the quadratic equation to find $x=2$ or $x=-\frac{1}{3}$. Both solutions check in the original equation.

Graphing Rational Functions

• Find the domain of the function (exclude values that make the denominator zero)
• Identify any vertical asymptotes (where the denominator equals zero) and horizontal asymptotes (using the degree of the numerator and denominator)
• Determine the x- and y-intercepts (if any)
• Find the holes or points of discontinuity (if any) by setting the numerator and denominator equal to zero and solving for x
• Use the above information to sketch the graph
• Example: For $f(x)=\frac{x+1}{x-2}$, the vertical asymptote is at $x=2$, the horizontal asymptote is at $y=1$, the x-intercept is at $(-1,0)$, and there are no holes. The graph has two separate branches, one approaching the vertical asymptote from the left and one from the right, with both branches approaching the horizontal asymptote as x approaches infinity or negative infinity.

Applications and Word Problems

• Rational expressions and functions can model various real-world situations, such as rates, work, and mixture problems
• Identify the given information and the unknown variable
• Set up a rational equation or function based on the problem context
• Solve the equation or analyze the function to answer the question
• Example: If a pipe can fill a tank in 4 hours and another pipe can fill the same tank in 6 hours, how long will it take to fill the tank if both pipes are used simultaneously? Let x be the time to fill the tank. The rates of the pipes are $\frac{1}{4}$ and $\frac{1}{6}$ of the tank per hour. Set up the equation $\frac{1}{4}x+\frac{1}{6}x=1$ and solve for x to find the time is 2.4 hours.

Common Mistakes and How to Avoid Them

• Not factoring the numerator and denominator completely before canceling common factors
  • Always factor the numerator and denominator as much as possible
• Canceling terms instead of factors
  • Only cancel factors that are common to both the numerator and denominator
• Forgetting to find a common denominator when adding or subtracting rational expressions
  • Always find the LCM of the denominators and multiply each expression by the appropriate factor
• Not checking for extraneous solutions after solving rational equations
  • Always substitute the solutions back into the original equation to verify they work
• Misidentifying or forgetting to find vertical or horizontal asymptotes when graphing rational functions
  • Always find the vertical asymptotes by setting the denominator equal to zero and the horizontal asymptotes by comparing the degrees of the numerator and denominator
• Misinterpreting the domain of a rational function
  • The domain is all real numbers except those that make the denominator equal to zero