Precalculus Semester 1 Exam Review (Chapters 1–6): Comprehensive Study Notes

Precalculus Semester 1 Exam Review

Overview

This study guide covers essential topics from Chapters 1–6 of a typical Precalculus course, including graphs, functions, linear and quadratic equations, polynomial and rational functions, exponential and logarithmic functions, and trigonometric functions. Each section provides definitions, key properties, and example problems to reinforce understanding.

Graphs and Functions

Converting Angles and Graphing Functions

• Radians and Degrees: Angles can be measured in degrees or radians. To convert degrees to radians, use the formula: $\text{Radians} = \text{Degrees} \times \frac{\pi}{180}$ Example: Convert $18^\circ$ to radians: $18^\circ \times \frac{\pi}{180} = \frac{\pi}{10}$

• Graphing Functions: To graph $y = -4\sin(2x + \frac{\pi}{3})$, identify amplitude, period, phase shift, and vertical shift.

  • Amplitude: $|A|$ where $A$ is the coefficient of sine or cosine.

  • Period: $\frac{2\pi}{|B|}$ for $y = A\sin(Bx + C)$.

  • Phase Shift: $-\frac{C}{B}$.

Linear, Quadratic, Polynomial, and Rational Functions

Key Properties and Transformations

• Quadratic Formula: For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Domain and Range: The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Even and Odd Functions:

  • Even: $f(-x) = f(x)$ (symmetric about the y-axis)

  • Odd: $f(-x) = -f(x)$ (symmetric about the origin)

• Transformations: Shifts, stretches, and reflections can be applied to parent functions to obtain new graphs.

Exponential and Logarithmic Functions

Properties and Applications

• Exponential Functions: $f(x) = a^x$ where $a > 0$, $a \neq 1$.

• Logarithmic Functions: $\log_a(x)$ is the inverse of $a^x$.

  • Change of Base Formula: $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

  • Properties:

    • $\log_a(xy) = \log_a(x) + \log_a(y)$

    • $\log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)$

    • $\log_a(x^k) = k\log_a(x)$

• Solving Logarithmic Equations: Use properties of logarithms to combine or expand expressions and solve for the variable.

Trigonometric Functions

Amplitude, Period, and Phase Shift

• General Form: $y = A\sin(Bx + C) + D$ or $y = A\cos(Bx + C) + D$

• Amplitude: $|A|$

• Period: $\frac{2\pi}{|B|}$

• Phase Shift: $-\frac{C}{B}$

• Vertical Shift: $D$

• Example: For $y = 2\sin(x + \frac{\pi}{3})$, amplitude is 2, period is $2\pi$, phase shift is $-\frac{\pi}{3}$.

Applications and Problem Solving

Word Problems and Modeling

• Average Rate of Change: For $f(x)$ over $[a, b]$, $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Projectile Motion: Height as a function of time can be modeled by $h(t) = -16t^2 + v_0t + h_0$ (in feet, with $t$ in seconds).

• Mixture and Optimization Problems: Set up equations based on the problem context and solve for the unknowns.

Sample Table: Properties of Functions

Function Type	General Form	Domain	Range	Key Features
Linear	$y = mx + b$	All real numbers	All real numbers	Straight line, constant rate of change
Quadratic	$y = ax^2 + bx + c$	All real numbers	$y \geq k$ or $y \leq k$	Parabola, vertex, axis of symmetry
Exponential	$y = a^x$	All real numbers	$y > 0$ (if $a > 0$)	Rapid growth or decay
Logarithmic	$y = \log_a(x)$	$x > 0$	All real numbers	Inverse of exponential
Trigonometric	$y = \sin x, \cos x$	All real numbers	$[-1, 1]$	Periodic, amplitude, period

Additional info:

• Some questions involve graphing, finding intercepts, and analyzing symmetry, which are foundational skills in Precalculus.

• Application problems (e.g., mixture, projectile motion, and optimization) require translating word problems into mathematical equations.