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

Piecewise Functions

Definition and Graphing

Piecewise functions are defined by different expressions depending on the input value. Each 'piece' applies to a specific interval of the domain.

• Piecewise Function: A function composed of multiple sub-functions, each with its own domain.

• Graphing: Plot each piece on its respective interval, ensuring endpoints are correctly marked (open or closed circles).

• Example: Given: $f(x) = \begin{cases} -2x-5, & -4 \leq x < -2 \\ -3(x+1)^2, & -2.5 < x < 0 \\ |x-1|-5, & x \geq 0 \end{cases}$ Graph each segment on its specified interval.

Evaluating Piecewise Functions

To evaluate a piecewise function at a specific value, determine which interval the input falls into and use the corresponding formula.

• Example: For $f(x)$ as above, find $f(-3)$ and $f(1)$ by substituting into the correct piece.

Function Transformations

Types of Transformations

Transformations alter the graph of a function by shifting, stretching, compressing, or reflecting it.

• Vertical Shifts: $f(x) + k$ shifts the graph up/down.

• Horizontal Shifts: $f(x-h)$ shifts the graph right/left.

• Reflections: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Stretch/Compression: $af(x)$ stretches ($|a|>1$) or compresses ($0<|a|<1$) vertically.

• Example: $f(x) = \sqrt{x+5}$ is a square root function shifted left by 5 units.

Quadratic Functions

Standard and Vertex Form

Quadratic functions are polynomials of degree 2. Their graphs are parabolas.

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

• Vertex Form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

• Axis of Symmetry: $x = h$ in vertex form, or $x = -\frac{b}{2a}$ in standard form.

• Domain: All real numbers, $(-\infty, \infty)$

• Range: Depends on $a$; if $a>0$, range is $[k, \infty)$; if $a<0$, range is $(-\infty, k]$

• Example: $f(x) = -2x^2 - 12x - 21$

Finding the Equation from a Graph

Given a graph, identify the vertex and another point to write the equation in standard form.

• Example: If the vertex is at $(2, 5)$ and another point is $(3, 2)$, use vertex form and expand.

Polynomial Functions

Degree, Leading Coefficient, and Intercepts

Polynomial functions are sums of terms with non-negative integer exponents.

• Degree: The highest exponent 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.

• y-intercept: Value of $f(0)$.

• End Behavior: Determined by degree and leading coefficient.

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

Multiplicity

• Multiplicity: If $(x-a)^k$ is a factor, $x=a$ is a root of multiplicity $k$.

• If $k$ is even, the graph touches and turns at $x=a$; if odd, it crosses the axis.

Rational Functions

Vertical and Horizontal Asymptotes

Rational functions are quotients of polynomials. Their graphs may have asymptotes and holes.

• Vertical Asymptotes (V.A.): Values of $x$ where the denominator is zero and the numerator is nonzero.

• Horizontal Asymptotes (H.A.): Determined by degrees of numerator and denominator:

  • If degree numerator < degree denominator: $y=0$

  • If degrees equal: $y=\frac{\text{leading coefficient numerator}}{\text{leading coefficient denominator}}$

  • If degree numerator > degree denominator: No H.A.; may have slant asymptote.

• Slant Asymptote (S.A.): If degree numerator is exactly one more than denominator, divide to find S.A.

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

Intercepts and Holes

• x-intercepts: Set numerator equal to zero (excluding values that make denominator zero).

• y-intercept: Evaluate $f(0)$.

• Holes: Occur at values where both numerator and denominator are zero (common factors).

Summary Table: Asymptotes and Intercepts

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

Square Root and Absolute Value Functions

Graphing and Transformations

Square root and absolute value functions have distinctive graphs and respond to transformations.

• Square Root Function: $f(x) = \sqrt{x}$; domain $x \geq 0$.

• Absolute Value Function: $f(x) = |x|$; V-shaped graph.

• Transformations: Apply shifts, stretches, and reflections as with other functions.

• Example: $f(x) = \frac{1}{2}|x-7|-4$ is an absolute value function shifted right by 7, stretched by $\frac{1}{2}$, and down by 4.

Summary of Key Concepts

• Piecewise functions require careful attention to intervals.

• Transformations affect the position and shape of graphs.

• Quadratic and polynomial functions are analyzed via degree, leading coefficient, intercepts, and end behavior.

• Rational functions involve asymptotes, intercepts, and holes.

• Square root and absolute value functions have restricted domains and unique graph shapes.

Additional info: The study notes expand on brief worksheet questions, providing definitions, formulas, and academic context for Precalculus topics such as piecewise functions, transformations, quadratics, polynomials, rational functions, and graphing techniques.