7.2 Rational Functions and Their Graphs

Rational functions are the powerhouse of algebra, combining polynomials in ways that create wild and unpredictable behavior. They're like mathematical roller coasters, with twists, turns, and sudden drops that keep you on your toes.

Understanding rational functions is key to mastering this chapter. We'll explore their domains, ranges, and asymptotes, learning to graph them and analyze their behavior. Get ready for a thrilling ride through the world of rational expressions!

Rational functions: Domain, range, and asymptotes

Definition and characteristics of rational functions

• A rational function is the quotient of two polynomial functions, $P(x)$ and $Q(x)$, where $Q(x) \neq 0$
• The domain consists of all real numbers except those that make the denominator equal to zero, called points of discontinuity or non-permissible values
• Vertical asymptotes occur at non-permissible values, where the function approaches positive or negative infinity
• The range consists of all real numbers the function can output, excluding y-values corresponding to vertical asymptotes or holes

Determining horizontal and oblique asymptotes

• Horizontal asymptotes are determined by comparing the degrees of the numerator and denominator polynomials:
  • If degree of numerator < degree of denominator, horizontal asymptote is $y = 0$
  • If degree of numerator = degree of denominator, horizontal asymptote is $y = a/b$, where $a$ and $b$ are leading coefficients of numerator and denominator
  • If degree of numerator > degree of denominator by 1, the function has an oblique (slant) asymptote, found by performing long division on numerator and denominator
• Example: For $f(x) = \frac{2x^2 + 3x - 1}{x - 4}$, the horizontal asymptote is $y = 2x + 7$ (found by long division)

Graphing rational functions: Key features

Identifying intercepts, asymptotes, and holes

• Determine x-intercepts by setting numerator equal to zero and solving for $x$ (points where graph crosses x-axis)
• Determine y-intercept by evaluating function at $x = 0$ (point where graph crosses y-axis)
• Plot intercepts, asymptotes, and holes (points of discontinuity where function is undefined but limit exists) on coordinate plane
• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, x-intercepts are $(-2, 0)$ and $(2, 0)$, y-intercept is $(0, 2)$, vertical asymptote is $x = 2$, and hole is at $(2, 4)$

Sketching the graph and analyzing behavior

• Analyze function behavior near asymptotes and intercepts to sketch the graph
• Consider signs of numerator and denominator to determine function behavior in each interval between vertical asymptotes
• Identify symmetry in the graph (even, odd, or neither)
• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, the graph has a vertical asymptote at $x = 2$, a hole at $(2, 4)$, and is symmetric about the y-axis (even function)

Analyzing rational functions: Limits and asymptotes

Using limits to analyze behavior near discontinuities

• Find limit of rational function at a point of discontinuity by evaluating limit from both left and right sides
  • If one-sided limits are equal, the limit exists; otherwise, it does not exist
• Vertical asymptotes occur when limit of function as $x$ approaches a certain value from either side is positive or negative infinity
• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, $\lim_{x \to 2^-} f(x) = -\infty$ and $\lim_{x \to 2^+} f(x) = +\infty$, confirming the vertical asymptote at $x = 2$

Determining horizontal and oblique asymptotes using limits

• Horizontal asymptotes can be found by evaluating limit of function as $x$ approaches positive or negative infinity (limit represents y-value of horizontal asymptote)
• Oblique asymptotes can be found by evaluating limit of difference between function and quotient of leading terms of numerator and denominator as $x$ approaches infinity or negative infinity (limit represents y-intercept of oblique asymptote)
• Example: For $f(x) = \frac{2x^2 + 3x - 1}{x - 4}$, $\lim_{x \to \infty} (f(x) - (2x + 7)) = 0$, confirming the oblique asymptote $y = 2x + 7$

Transformations of rational functions

Types of transformations and their representations

• Horizontal shift: $f(x - h)$ for $h$ units right, $f(x + h)$ for $h$ units left
• Vertical shift: $f(x) + k$ for $k$ units up, $f(x) - k$ for $k$ units down
• Horizontal stretch/compression: $f(ax)$, where $|a| > 1$ is compression, $0 < |a| < 1$ is stretch
• Vertical stretch/compression: $af(x)$, where $|a| > 1$ is stretch, $0 < |a| < 1$ is compression
• Reflection across x-axis: $-f(x)$; reflection across y-axis: $f(-x)$

Effects of transformations on key features of the graph

• Transformations affect location of intercepts, asymptotes, and holes, but not overall shape or behavior of function
• Example: For $f(x) = \frac{x^2 - 4}{x - 2}$, the transformation $g(x) = -2f(x - 1)$ results in a vertical stretch by a factor of 2, a reflection across the x-axis, and a horizontal shift of 1 unit to the right, moving the vertical asymptote to $x = 3$ and the hole to $(3, -8)$