Precalculus Study Guide: Functions, Graphs, and Polynomial/Rational Analysis

Functions and Their Graphs

Piecewise Functions

Piecewise functions are defined by different expressions depending on the input value. They are useful for modeling situations where a rule changes based on the domain.

• Definition: A piecewise function is a function composed of multiple sub-functions, each applying to a certain interval of the domain.

• Example: $f(x) = \begin{cases} -2x-5, & \text{if } -4 \leq x < -2 \\ -3(x+1)^2, & \text{if } -2.5 < x < 0 \\ |x-1|-5, & \text{if } x \geq 0 \end{cases}$

• Graphing:

  1. Identify the intervals and corresponding expressions.

  2. Plot each segment on its specified interval.

  3. Check for open/closed endpoints based on inequalities.

• Evaluating: To find $f(a)$, determine which interval $a$ falls into and use the corresponding expression.

Absolute Value Functions

Absolute value functions have a characteristic 'V' shape and are defined as the distance from zero.

• Definition: $f(x) = |x|$

• Graph: Vertex at (0,0), opens upward.

• Transformations: $f(x) = a|x-h|+k$ shifts and stretches the graph.

• Example: $f(x) = \frac{1}{2}|x-7|-4$ is shifted right by 7 and down by 4, and vertically compressed.

Square Root Functions

Square root functions are defined for non-negative arguments and have a domain restriction.

• Definition: $f(x) = \sqrt{x}$

• Domain: $x \geq 0$

• Transformations: $f(x) = \sqrt{x+5}$ shifts left by 5 units.

Quadratic Functions

Quadratic functions are polynomials of degree 2 and graph as parabolas.

• Standard Form: $f(x) = ax^2 + bx + c$

• Vertex: $x = -\frac{b}{2a}$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Domain: All real numbers

• Range: Depends on the direction the parabola opens

• Example: $f(x) = -2x^2 - 12x - 21$

Polynomial Functions

Polynomials are expressions involving powers of $x$ with real coefficients.

• Degree: The highest power of $x$

• Leading Coefficient (L.C.): The coefficient of the term with the highest degree

• x-intercepts: Values of $x$ where $f(x) = 0$; multiplicity indicates how the graph behaves at the intercept

• End Behavior: Determined by degree and leading coefficient

• Example: $f(x) = 3(2-x)(x+2)(x-1)^2$

Rational Functions and Asymptotes

Rational Functions

Rational functions are ratios of polynomials. Their graphs can have vertical, horizontal, or slant asymptotes, as well as holes.

• Definition: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$

• Vertical Asymptotes (V.A.): Values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$

• Horizontal Asymptotes (H.A.): Determined by the degrees of $P(x)$ and $Q(x)$

  • If degree of $P(x)$ < degree of $Q(x)$, H.A. is $y=0$

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

  • If degree of $P(x)$ > degree of $Q(x)$ by 1, there is a slant asymptote (S.A.)

• Holes: Occur when a factor cancels in both numerator and denominator

• Example: $f(x) = \frac{x^2+16}{x^2+5x+4}$

Intercepts of Rational Functions

• x-intercept: Set numerator equal to zero and solve for $x$

• y-intercept: Evaluate $f(0)$

Summary Table: Asymptotes and Intercepts

Feature	How to Find
Vertical Asymptote	Set denominator = 0, solve for $x$
Horizontal Asymptote	Compare degrees of numerator and denominator
Slant Asymptote	Divide numerator by denominator if degree numerator = degree denominator + 1
x-intercept	Set numerator = 0, solve for $x$
y-intercept	Evaluate $f(0)$
Hole	Common factor cancels in numerator and denominator

Graphing and Analyzing Functions

General Steps for Graphing

• Identify the type of function (piecewise, absolute value, square root, quadratic, polynomial, rational)

• Determine domain and range

• Find intercepts

• Locate asymptotes (if any)

• Plot key points and sketch the graph

Example: Quadratic in Standard Form

• Given a graph, identify vertex $(h, k)$

• Write equation: $f(x) = a(x-h)^2 + k$

• Convert to standard form: $f(x) = ax^2 + bx + c$

Example: Polynomial Analysis

• Given $f(x) = 3(2-x)(x+2)(x-1)^2$:

• Degree: 4

• Leading Coefficient: $3$

• x-intercepts: $x=2$, $x=-2$, $x=1$ (multiplicity 2)

• End Behavior: As $x \to \pm\infty$, $f(x) \to \infty$ (since degree is even and leading coefficient is positive)

Example: Rational Function Analysis

• Given $f(x) = \frac{x^2+16}{x^2+5x+4}$:

• Vertical Asymptotes: Set $x^2+5x+4=0$

• Horizontal Asymptote: Degrees equal, so $y=1$

• x-intercepts: Set $x^2+16=0$ (no real solutions)

• y-intercept: $f(0) = \frac{16}{4} = 4$

Additional info:

• Some questions require students to graph functions, identify key features, and write equations based on graphs.

• Topics covered are foundational for Precalculus, including function analysis, graphing, and polynomial/rational properties.