Precalculus Exam Review Guide: Polynomial, Rational, Exponential, Logarithmic, and Trigonometric Functions

Polynomial Functions

Sketching and Analyzing Polynomial Functions

Polynomial functions are algebraic expressions involving powers of x with real coefficients. Understanding their behavior is essential for graphing and solving equations.

• End Behavior: Determined by the degree and leading coefficient of the polynomial. For large values of |x|, the highest degree term dominates.

• x-intercepts: Points where the polynomial equals zero; found by solving $P(x) = 0$.

• Multiplicity of Zeros: The number of times a particular root occurs. If a zero has even multiplicity, the graph touches the x-axis; if odd, it crosses.

• Symmetry: Check for even (symmetric about y-axis) or odd (symmetric about origin) functions.

• Turning Points: Points where the graph changes direction. A polynomial of degree n has at most n-1 turning points.

• Approximating Roots: Use the "appropriate" solution for $y = 0$.

• Intermediate Value Theorem: If $f(a)$ and $f(b)$ have opposite signs, there is at least one root between a and b.

Example: Sketch the graph of $f(x) = x^3 - 4x^2 + x + 6$ by finding intercepts, end behavior, and turning points.

Polynomial Equations and Zeros

Finding Zeros and Analyzing Behavior

• Number of Possible Zeros: A polynomial of degree n has at most n real zeros.

• Number of Turning Points: At most n-1 for degree n.

• Rational Zeros Theorem: Possible rational zeros are factors of the constant term divided by factors of the leading coefficient.

• Graphing: Use zeros, multiplicity, and end behavior to sketch the graph.

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

Rational Functions

Graphs and Asymptotes

Rational functions are quotients of polynomials. Their graphs can have vertical, horizontal, or oblique asymptotes.

• Intercepts: x-intercepts occur where the numerator is zero; y-intercept at $x=0$.

• Vertical Asymptotes: Occur where the denominator is zero and the numerator is not zero.

• Horizontal Asymptotes: Determined by comparing degrees of numerator and denominator.

• Oblique Asymptotes: If degree of numerator is one more than denominator, use polynomial division.

• Transformations: Shifts, stretches, and reflections can be applied to basic rational functions.

Example: For $f(x) = \frac{2x+1}{x-3}$, vertical asymptote at $x=3$, horizontal asymptote at $y=2$.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a^x$, where a > 0 and a ≠ 1. They model growth and decay.

• Basic Shape: Rapid increase or decrease depending on base.

• Domain and Range: Domain is all real numbers; range is positive real numbers.

• Transformations: Shifts, stretches, and reflections affect the graph.

Example: $f(x) = 2^x$ increases rapidly as x increases.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions, with the form $f(x) = \log_a x$.

• Definition: $\log_a x = y$ means $a^y = x$.

• Domain: $x > 0$; Range: all real numbers.

• Properties: $\log_a(xy) = \log_a x + \log_a y$, $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$, $\log_a(x^r) = r\log_a x$.

• Solving Equations: Use properties to combine or expand logarithmic expressions and solve for x.

Example: Solve $\log_2(x-1) = 3$; $x-1 = 8$, so $x = 9$.

Trigonometric Functions and Identities

Negative Angle and Pythagorean Identities

Trigonometric identities are equations involving trigonometric functions that hold for all values in their domains.

• Negative Angle Identities: $\sin(-x) = -\sin(x)$, $\cos(-x) = \cos(x)$, $\tan(-x) = -\tan(x)$.

• Pythagorean Identities: $\sin^2 x + \cos^2 x = 1$, $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$.

Example: Use $\sin^2 x + \cos^2 x = 1$ to find $\sin x$ if $\cos x = \frac{3}{5}$.

Solving Triangles

Right triangle trigonometry involves finding unknown sides or angles using trigonometric ratios.

• SOH CAH TOA: Mnemonic for sine, cosine, and tangent ratios.

• Applications: Use angles of elevation/depression and regression for real-world problems.

• Signs in Quadrants: Trigonometric functions have different signs depending on the quadrant.

• Reference Angles: Used to find exact values of trigonometric functions for multiples of $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$.

Example: Find the length of the side opposite a $30^\circ$ angle in a right triangle with hypotenuse 10: $\sin 30^\circ = \frac{1}{2}$, so side = $10 \times \frac{1}{2} = 5$.

Analytic Trigonometry

Trigonometric Identities

Analytic trigonometry focuses on proving and applying identities to simplify expressions and solve equations.

• Sum/Difference Formulas: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Double-Angle Formulas: $\sin 2x = 2\sin x \cos x$, $\cos 2x = \cos^2 x - \sin^2 x$

• Product-to-Sum Formulas: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Example: Simplify $\sin x \cos x$ using double-angle formula: $\sin x \cos x = \frac{1}{2} \sin 2x$.

Trigonometric Equations

Solving trigonometric equations involves finding all solutions within a given interval.

• General Solution: For $\sin x = k$, $x = \arcsin k + 2\pi n$ or $x = \pi - \arcsin k + 2\pi n$ for integer n.

• Interval Solutions: Find all solutions in a specified interval, such as $[0, 2\pi)$.

Example: Solve $\sin x = \frac{1}{2}$ for $x$ in $[0, 2\pi)$: $x = \frac{\pi}{6}, \frac{5\pi}{6}$.

Summary Table: Key Properties of Functions

Function Type	General Form	Key Features
Polynomial	$a_nx^n + \ldots + a_0$	Degree, zeros, end behavior, turning points
Rational	$\frac{P(x)}{Q(x)}$	Vertical/horizontal/oblique asymptotes, intercepts
Exponential	$a^x$	Rapid growth/decay, domain: $\mathbb{R}$, range: $a>0$
Logarithmic	$\log_a x$	Inverse of exponential, domain: $x>0$
Trigonometric	$\sin x, \cos x, \tan x$	Periodic, identities, equations, applications in triangles

Additional info: These notes synthesize the main topics from the provided exam review guide, expanding brief points into full academic explanations and including key formulas and examples for Precalculus students.