Absolute Value, Power, Polynomial, and Rational Functions: Key Concepts and Graphical Analysis

Absolute Value Functions

Definition and Properties

The absolute value function is a piecewise-defined function that measures the distance of a number from zero on the number line. It is defined as:

• Definition: $f(x) = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$

• Distance Interpretation: The absolute value $|a-b|$ gives the distance between $a$ and $b$ on the number line, always a non-negative value.

Graphical Features

• The graph of $f(x) = |x|$ is a "V" shape with its vertex at the origin (0,0).

• The corner point (vertex) is where the graph changes direction.

• Transformations (shifts, stretches, reflections) move or reshape the graph, but the corner point remains a key feature for identifying transformations.

Example: Graph of $f(x) = |x-5|$

This function is shifted right by 5 units. The vertex is at (5,0).

Solving Absolute Value Equations and Inequalities

• To solve $|A| = B$ (where $B \geq 0$), set $A = B$ or $A = -B$.

• To solve $|x-5| \leq 4$, rewrite as $-4 \leq x-5 \leq 4$ and solve for $x$.

• Absolute value inequalities can be solved graphically or by testing intervals.

Example: Graphical Solution of $|x-5| \leq 4$

The solution is the interval $1 \leq x \leq 9$.

Power Functions and Polynomial Functions

Power Functions

A power function has the form $f(x) = kx^p$, where $k$ is a nonzero constant and $p$ is a real number.

• Examples: $f(x) = x^2$ (quadratic), $f(x) = x^3$ (cubic), $f(x) = x^{-1}$ (reciprocal), $f(x) = x^{1/2}$ (square root).

Graphical Behavior of Power Functions

• Even powers ($x^2$, $x^4$, $x^6$, ...): Graphs are symmetric about the y-axis and approach infinity as $x \to \pm\infty$.

• Odd powers ($x^3$, $x^5$, $x^7$, ...): Graphs are symmetric about the origin; as $x \to \infty$, $f(x) \to \infty$; as $x \to -\infty$, $f(x) \to -\infty$.

Polynomials

• A polynomial is a sum of terms, each a transformed power function with a non-negative integer exponent: $f(x) = a_0 + a_1x + a_2x^2 + ... + a_nx^n$.

• Degree: The highest power of $x$ in the polynomial.

• Leading term: The term with the highest degree.

• Leading coefficient: The coefficient of the leading term.

Long Run and Short Run Behavior

• Long run behavior is determined by the leading term.

• Short run behavior includes intercepts and turning points.

• A polynomial of degree $n$ has at most $n$ real zeros and at most $n-1$ turning points.

Graphical Behavior at Intercepts

• Single zero (multiplicity 1): Graph crosses the axis.

• Double zero (multiplicity 2): Graph touches and bounces off the axis.

• Triple zero (multiplicity 3): Graph crosses the axis and flattens out.

Rational Functions

Definition and Key Features

A rational function is a ratio of two polynomials: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$.

• Vertical asymptotes: Occur where $Q(x) = 0$ and $P(x) \neq 0$.

• Holes: Occur where both $P(x)$ and $Q(x)$ are zero at the same $x$-value (after simplification).

• Horizontal asymptotes: Determined by the degrees of $P(x)$ and $Q(x)$:

  • If degree of $Q(x)$ > degree of $P(x)$: $y=0$ is the horizontal asymptote.

  • If degrees are equal: $y=\frac{\text{leading coefficient of }P(x)}{\text{leading coefficient of }Q(x)}$.

  • If degree of $P(x)$ > degree of $Q(x)$: No horizontal asymptote.

• Intercepts: Vertical intercept at $x=0$ (if defined); horizontal intercepts where $P(x)=0$ (if not a hole).

Example: Graph of a Rational Function with Two Vertical Asymptotes

Example: Graph of a Rational Function with a Double Zero and Asymptotes

Behavior Near Asymptotes and Intercepts

• At a vertical asymptote with an odd power, the function approaches $+\infty$ on one side and $-\infty$ on the other.

• At a vertical asymptote with an even power, the function approaches the same infinity on both sides.

• At a double zero (multiplicity 2), the graph bounces off the x-axis.

Summary Table: Polynomial and Rational Function Features

Feature	Polynomial	Rational Function
Definition	Sum of power functions with non-negative integer exponents	Ratio of two polynomials
Long Run Behavior	Determined by leading term	Determined by ratio of leading terms
Intercepts	At most degree many x-intercepts	Numerator zeros: x-intercepts; denominator zeros: vertical asymptotes
Asymptotes	None	Vertical and horizontal (or slant) asymptotes possible

Key Takeaways

• Absolute value functions model distance and are solved using piecewise definitions.

• Power and polynomial functions are classified by degree and leading coefficient, which determine their end behavior.

• Rational functions introduce asymptotes and holes, with behavior determined by the degrees and factors of numerator and denominator.

• Graphical analysis is essential for understanding intercepts, turning points, and asymptotic behavior.