Precalculus Exam Study Guide: Functions, Equations, and Graphs

Algebraic Equations and Factoring

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically written in the form $ax^2 + bx + c = 0$. Solutions can be found using factoring, completing the square, or the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Example: Solve $3x^2 - 10x - 5 = 0$ using the quadratic formula.

Polynomial Factoring

Factoring polynomials involves expressing them as products of lower-degree polynomials. Techniques include grouping, using special products, and long division.

• Grouping: Rearranging terms to factor by grouping common factors.

• Long Division: Used for dividing polynomials and finding factors.

• Example: Factor $x^3 + 6x^2 + 5x - 12$ by long division.

Functions and Their Properties

Function Notation and Evaluation

A function $f(x)$ assigns each input $x$ to a unique output. Evaluating a function means substituting a value for $x$.

• Example: Evaluate $f(0)$ and $f(4)$ for a given piecewise function.

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Example: Find the domain of $f(x) = -2\sqrt{x-2}$.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} \frac{1}{2}x + 1, & x \leq 0 \\ x^2 - 1, & x > 2 \end{cases}$

Graphing and Parent Functions

Graphing Functions

Graphing involves plotting points $(x, f(x))$ and analyzing features such as intercepts, asymptotes, and transformations.

• Parent Functions: Basic functions such as $y = x^2$, $y = \tan x$, $y = \ln x$, $y = \sqrt{x}$.

• Transformations: Shifts, stretches, and reflections applied to parent functions.

• Example: Identify the parent function from a given graph.

Amplitude and Period of Trigonometric Functions

For functions like $y = a \cos(bx) + c$, the amplitude is $|a|$ and the period is $\frac{2\pi}{|b|}$.

• Example: Find amplitude and period for $y = 2\cos\left(\frac{2\pi}{3}x\right) + 1$.

Trigonometric Functions and Identities

Evaluating Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles. Common functions include sine, cosine, tangent, cosecant, secant, and cotangent.

• Example: Evaluate $\sin\left(\frac{11\pi}{6}\right)$.

Trigonometric Table

Values of trigonometric functions for specific angles are often summarized in tables.

Angle	$\sin(\theta)$	$\tan(\theta)$	$\cos(\theta)$	$\sec(\theta)$	$\cot(\theta)$
0	0	0	1	1	Undefined
$\frac{\pi}{2}$	1	Undefined	0	Undefined	0
$\pi$	0	0	-1	-1	Undefined
$\frac{3\pi}{2}$	-1	Undefined	0	Undefined	0
$2\pi$	0	0	1	1	Undefined

Additional info: Table entries inferred for standard angles.

Linear Equations and Systems

Equations of Lines

The equation of a line can be written in slope-intercept form $y = mx + b$ or point-slope form $y - y_1 = m(x - x_1)$. Lines perpendicular to a given line have slopes that are negative reciprocals.

• Example: Find the equation of a line passing through $(-1, 4)$ and perpendicular to $y = \frac{1}{3}x - 5$.

Solving Systems of Equations

Systems of equations can be solved by substitution, elimination, or graphing. The solution is the point(s) where the equations intersect.

• Example: Solve $3x - y = 1$ and $2x + y = 1$.

Function Transformations and Operations

Function Operations

Functions can be added, subtracted, multiplied, divided, and composed. The difference quotient is a key concept for understanding rates of change.

• Difference Quotient: For $g(x) = x^2 - 3x$, $g(x + h) - g(x)$ simplifies to $2xh + h^2 - 3h$.

Set Notation and Intervals

Interval Notation

Intervals describe subsets of the real numbers. Use parentheses for open intervals and brackets for closed intervals.

• Example: $(-\infty, -2) \cup (-2, 7)$

Exponent Rules and Simplification

Exponent Properties

Exponent rules are used to simplify expressions involving powers.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Example: Simplify $\frac{\sqrt{x^2}x^4}{3y}$ and $2x^2/y^2$.

Factoring and Simplification of Expressions

Factoring by Grouping

Factoring by grouping is a method used when a polynomial has four or more terms. Group terms to factor common elements.

• Example: Factor $x^3 - 3x^2 - 4x + 12$ by grouping.

Function Identification from Graphs

Piecewise and Absolute Value Functions

Some graphs represent piecewise or absolute value functions. Recognizing these from their shape is important for function identification.

• Example: Identify $y = |x| + 2$ from a graph.

Summary Table: Parent Functions

Function	Equation	Graph Shape
Quadratic	$y = x^2$	Parabola
Absolute Value	$y = |x|$	V-shape
Square Root	$y = \sqrt{x}$	Half-parabola (right)
Logarithmic	$y = \ln x$	Increasing, vertical asymptote at $x=0$
Tangent	$y = \tan x$	Repeating vertical asymptotes

Additional info: Table entries inferred for standard parent functions.