Precalculus Study Guide: Algebraic and Polynomial Functions (Exam 1 Objectives)

Chapter 1: Fundamental Concepts of Algebra and Polynomial Functions

Intercepts and Symmetry of Equations

Understanding the intercepts and symmetry of equations in two variables is foundational for graphing and analyzing functions.

• Intercepts: The x-intercept is where the graph crosses the x-axis ($y=0$), and the y-intercept is where it crosses the y-axis ($x=0$).

• Symmetry: A relation is symmetric with respect to the x-axis, y-axis, or origin if substituting $-x$ or $-y$ yields the same equation.

• Example: The equation $y = x^2$ is symmetric about the y-axis.

Functions and Relations

Functions assign each input exactly one output. Relations may not be functions if an input maps to multiple outputs.

• Definition: A function is a relation where each element of the domain is paired with exactly one element of the range.

• Vertical Line Test: If a vertical line crosses the graph more than once, it is not a function.

• Example: $y = x^2$ passes the vertical line test; $x^2 + y^2 = 1$ does not.

Operations and Composition of Functions

Functions can be combined using addition, subtraction, multiplication, division, and composition.

• Operations: $(f+g)(x) = f(x) + g(x)$, $(f-g)(x) = f(x) - g(x)$, $(fg)(x) = f(x)g(x)$, $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$ (if $g(x) \neq 0$).

• Composition: $(f \circ g)(x) = f(g(x))$.

• Example: If $f(x) = x+1$ and $g(x) = 2x$, then $(f \circ g)(x) = f(2x) = 2x+1$.

Domain and Range

The domain is the set of possible input values; the range is the set of possible output values.

• Finding Domain: Exclude values that cause division by zero or negative square roots.

• Example: For $f(x) = \frac{1}{x-2}$, domain is $x \neq 2$.

Graphing Functions and Piecewise Functions

Graphing by hand is essential for understanding function behavior, including basic and piecewise-defined functions.

• Basic Functions: Identity, square, cube, square root, cube root, absolute value, reciprocal, and greatest integer.

• Piecewise Functions: Defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x & x \geq 0 \end{cases}$

Quadratic Functions and Equations

Quadratic functions are polynomials of degree 2 and have a characteristic parabolic graph.

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

• Vertex: $x = -\frac{b}{2a}$

• Factoring: Expressing as a product of binomials, e.g., $x^2 - 5x + 6 = (x-2)(x-3)$

• Solving: Set $f(x) = 0$ and solve for $x$.

Polynomial Functions

Polynomials are sums of powers of $x$ with real coefficients. Their degree, leading term, and zeros determine their behavior.

• Degree: Highest power of $x$.

• Leading Term: Term with the highest degree.

• Zeros: Values where $f(x) = 0$.

• Factor Theorem: If $f(a) = 0$, then $(x-a)$ is a factor.

• Complex Conjugate Theorem: If $a+bi$ is a zero, so is $a-bi$ (for real coefficients).

Application Problems and Optimization

Quadratic and polynomial functions are used to model real-world scenarios and optimize values.

• Maximum/Minimum: The vertex of a quadratic function gives the maximum or minimum value.

• Example: $f(x) = -2x^2 + 4x + 1$ has a maximum at $x = 1$.

Solving Equations and Inequalities

Solving equations and inequalities involves finding values that satisfy the given conditions.

• Quadratic Equations: Use factoring, completing the square, or quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Polynomial Inequalities: Find intervals where the polynomial is positive or negative.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on $[a, b]$ and $f(a)$ and $f(b)$ have opposite signs, there is at least one zero between $a$ and $b$.

• Application: Used to prove existence of roots.

Rational Functions and Rational Roots

Rational functions are ratios of polynomials. The Rational Root Theorem helps find possible rational zeros.

• Rational Root Theorem: Possible rational zeros are $\pm \frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

• Example: For $f(x) = 2x^3 - 3x^2 + x - 5$, possible rational zeros are $\pm1, \pm5, \pm\frac{1}{2}, \pm\frac{5}{2}$.

Graphing and Analyzing Functions

Graphing by hand and analyzing function properties, such as domain, range, intervals of increase/decrease, and concavity, are essential skills.

• Concavity: A function is concave up where its second derivative is positive, concave down where negative.

• Inflection Points: Where concavity changes.

Inverse Functions

An inverse function reverses the roles of inputs and outputs. A function must be one-to-one to have an inverse.

• Horizontal Line Test: If a horizontal line crosses the graph more than once, the function is not one-to-one.

• Finding Inverse: Solve $y = f(x)$ for $x$ in terms of $y$.

Summary Table: Key Properties of Functions

Property	Definition	Example
Intercepts	Where graph crosses axes	$y = x^2$ at (0,0)
Symmetry	Invariant under reflection	$y = x^2$ (y-axis)
Domain	Allowed input values	$x \neq 2$ for $f(x) = \frac{1}{x-2}$
Range	Possible output values	$y \geq 0$ for $y = x^2$
Zeros	Where $f(x) = 0$	$x = 2, 3$ for $f(x) = (x-2)(x-3)$
Inverse	Reverses input/output	$f^{-1}(x)$

Additional info:

• Some objectives reference the use of the Factor Theorem, Rational Root Theorem, and Complex Conjugate Theorem, which are standard in precalculus polynomial analysis.

• Graphing by hand and analyzing basic functions is emphasized, including transformations and piecewise functions.