Precalculus Study Guide: Functions, Graphs, and Polynomial Analysis

Functions and Their Graphs

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• Interval Notation: Expresses the domain as intervals, e.g., $(-\infty, 2] \cup (3, \infty)$.

• Inequality Notation: Uses inequalities, e.g., $x \leq 2$ or $x > 3$.

• Number Line: Shade the regions corresponding to the domain on a number line.

Example: For $f(x) = \frac{1}{x-2}$, the domain is all real numbers except $x=2$.

Function Notation and Ordered Pairs

Functions can be represented as sets of ordered pairs $(a, f(a))$, where $a$ is an input and $f(a)$ is the corresponding output.

• Example: If $f(x) = x^2$, then $(2, 4)$ is an ordered pair on the function.

Average Rate of Change (ARC)

The average rate of change of a function $f$ from $x=a$ to $x=b$ is given by:

• $\displaystyle \text{ARC} = \frac{f(b) - f(a)}{b - a}$

• Represents the slope of the secant line between $(a, f(a))$ and $(b, f(b))$.

• Example: For $f(x) = x^2$, the ARC from $x=1$ to $x=3$ is $\frac{9-1}{3-1} = 4$.

Difference Quotient

The difference quotient is a formula that gives the average rate of change over an interval of length $h$:

• $\displaystyle \frac{f(x+h) - f(x)}{h}$

• Used as a foundation for the derivative in calculus.

• Example: For $f(x) = x^2$, the difference quotient is $\frac{(x+h)^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$.

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a function's graph.

• Vertical Shifts: $f(x) + k$ shifts up by $k$ units.

• Horizontal Shifts: $f(x - h)$ shifts right by $h$ units.

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

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Horizontal Stretch/Compression: $f(bx)$ compresses if $|b| > 1$, stretches if $0 < |b| < 1$.

Frame-by-Frame Transformation: Apply each transformation step by step, starting from the parent function.

• Example: For $f(x) = (x-2)^2 + 3$, shift right 2 units, then up 3 units.

Transforming Ordered Pairs

• Given a set of points, apply the transformation rules to each $(x, y)$ to find the new points.

• Example: If $f(x)$ is shifted up by 2, $(a, f(a))$ becomes $(a, f(a)+2)$.

Piecewise Functions

Constructing and Graphing Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Write each piece with its domain, e.g., $f(x) = \begin{cases} x^2 & x < 0 \\ 2x+1 & x \geq 0 \end{cases}$

• Graph each piece over its specified interval.

Analyzing Piecewise Functions

• Domain: Union of all intervals where the function is defined.

• Range: All possible output values.

• Intervals of Increase/Decrease/Constant: Identify where the function rises, falls, or stays flat.

• Local and Absolute Extrema: Find highest/lowest points in a region (local) or overall (absolute).

Example: For $f(x)$ as above, analyze each piece for increasing/decreasing behavior and extrema.

Constructing Piecewise Functions from Applications

• Translate a real-world scenario into a piecewise function by defining rules for different cases.

• Example: A taxi charges $\$3$ for the first mile and $\$2$ for each additional mile: $f(x) = \begin{cases} 3 & 0 < x \leq 1 \\ 3 + 2(x-1) & x > 1 \end{cases}$

Linear Functions

Equations of Lines

A linear function has the form $f(x) = mx + b$.

• Through Two Points: Find the slope $m = \frac{y_2 - y_1}{x_2 - x_1}$, then use point-slope form.

• Parallel Lines: Same slope as the given line.

• Perpendicular Lines: Slope is the negative reciprocal of the given line's slope.

• Forms of Linear Equations:

  • Slope-Intercept: $y = mx + b$

  • Point-Slope: $y - y_1 = m(x - x_1)$

  • Standard: $Ax + By = C$

Example: Find the equation of the line through $(1,2)$ and $(3,6)$: $m = 2$, so $y = 2x$.

Applications of Linear and Quadratic Functions

Cost, Revenue, and Breakeven Analysis

• Cost Function: $C(x)$, total cost to produce $x$ items.

• Revenue Function: $R(x)$, total income from selling $x$ items.

• Breakeven Point: Where $C(x) = R(x)$.

• Supply and Demand: Functions relating price and quantity supplied/demanded.

Example: If $C(x) = 5x + 100$ and $R(x) = 8x$, breakeven at $5x + 100 = 8x \implies x = 33.33$.

Quadratic Functions

Standard and General Forms

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

• Standard Form: $f(x) = a(x-h)^2 + k$

• Completing the Square (CTS): Method to convert general to standard form.

Example: $f(x) = x^2 + 4x + 1$ becomes $f(x) = (x+2)^2 - 3$ after completing the square.

Graphing Quadratics

• Vertex: $(h, k)$ in standard form.

• Axis of symmetry: $x = h$.

• Intercepts: Set $x=0$ for y-intercept, $f(x)=0$ for x-intercepts.

• Use calculator to graph and find vertex, intercepts, and breakeven points in applications.

Polynomial Functions

Finding Polynomials from Zeros and Multiplicities

• Given zeros $r_1, r_2, \ldots$ and their multiplicities, construct $P(x) = a(x - r_1)^{m_1}(x - r_2)^{m_2}\ldots$

• Example: Zeros at $x=1$ (multiplicity 2), $x=-3$ (multiplicity 1): $P(x) = a(x-1)^2(x+3)$

Analyzing Graphs of Polynomials

• Use zeros and their multiplicities to determine where the graph crosses or touches the x-axis.

• End behavior depends on degree and leading coefficient:

Degree	Leading Coefficient	End Behavior
Even	Positive	Both ends up
Even	Negative	Both ends down
Odd	Positive	Left down, right up
Odd	Negative	Left up, right down

Example: Given a graph, identify zeros, their multiplicities, and reconstruct $P(x)$ up to a constant factor.

Rational Functions (Extra Credit)

Graphing Rational Functions

• General form: $R(x) = \frac{P(x)}{Q(x)}$

• Horizontal Asymptotes: Compare degrees $n$ (numerator) and $m$ (denominator):

  • If $n < m$, asymptote at $y=0$.

  • If $n = m$, asymptote at $y=\frac{\text{leading coefficient of }P}{\text{leading coefficient of }Q}$.

• Vertical Asymptotes: Set $Q(x)=0$ and solve for $x$.

• X-intercepts: Set $P(x)=0$ and solve for $x$.

• Y-intercept: Evaluate $R(0)$ if defined.

• Sketch the graph using intercepts, asymptotes, and test points.

Example: $R(x) = \frac{2x^2+1}{x^2-4}$ has vertical asymptotes at $x=2$ and $x=-2$, horizontal asymptote at $y=2$.

Additional info: Some context and examples were inferred based on standard Precalculus curriculum and the provided outline.