Precalculus Exam Solutions and Study Guide

Algebraic Expressions and Simplification

Simplifying Radicals and Rational Expressions

Understanding how to simplify radicals and rational expressions is foundational in algebra and precalculus. This involves reducing expressions to their simplest form and rationalizing denominators when necessary.

• Simplifying Radicals: To simplify $\sqrt{25/3}$, separate the numerator and denominator: $\sqrt{25}/\sqrt{3} = 5/\sqrt{3}$. Rationalize by multiplying numerator and denominator by $\sqrt{3}$: $5\sqrt{3}/3$.

• Rationalizing Denominators: For $3/\sqrt{5}$, multiply by $\sqrt{5}/\sqrt{5}$ to get $3\sqrt{5}/5$.

• Multiplying and Simplifying Polynomials: Expand $(x+1)(x^2+3x+4)$ using the distributive property.

• Factoring: Factor quadratics and higher-degree polynomials by grouping, using formulas, or recognizing patterns (e.g., difference of squares, sum/difference of cubes).

Example: Factor $x^2-9x+6x+6 = (x-9)(x+6)$.

Functions and Their Properties

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• For $f(x) = \sqrt{x-7}$, the domain is $x \geq 7$ (since the expression under the square root must be non-negative), and the range is $y \geq 0$.

Vertical and Horizontal Lines

• The equation of a vertical line passing through $(-7, 10)$ is $x = -7$.

• The midpoint of a segment with endpoints $(2, 4)$ and $(-13, 14)$ is $\left(\frac{2+(-13)}{2}, \frac{4+14}{2}\right) = (-5.5, 9)$.

Systems of Equations and Matrices

Solving Systems

Systems of equations can be solved by substitution, elimination, or using matrices.

• Augmented Matrix: For the system $2x + y = 2$ $3x - y = 4$ $x - 3y = -6$ The augmented matrix is:

  2	1	2
  3	-1	4
  1	-3	-6

Word Problems with Systems

Translate real-world scenarios into systems of equations, then solve using algebraic or matrix methods.

Example: If a package contains $x$ 12-blade, $y$ 24-blade, and $z$ 36-blade packs, and you know the total number of blades, set up equations based on the information given.

Polynomials and Factoring

Factoring Techniques

• Sum and Difference of Cubes: $x^3 + 8 = (x+2)(x^2-2x+4)$

• Quadratic Factoring: $2x^2 - x - 18 = (2x+3)(x-6)$

Conic Sections

Circles

The standard form of a circle is $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• For $x^2 - 4x + y^2 + 2y = 1$, complete the square to get $(x-2)^2 + (y+1)^2 = 9$. The center is $(2, -1)$ and the radius is $3$.

Functions: Evaluation and Composition

• Given $g(x) = \sqrt{x-4}$ for $x \geq 4$, evaluate $g(4) = 0$, $g(5) = 1$, $g(8) = 2$.

Linear Equations and Slope

Finding Equations of Lines

• Given two points, use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ and point-slope form $y - y_1 = m(x - x_1)$ to find the equation.

• For perpendicular lines, the slopes are negative reciprocals.

Example: The line perpendicular to $y = \frac{3}{4}x$ has slope $-\frac{4}{3}$.

Quadratic and Absolute Value Functions

Graph Transformations

• Transformations include shifting, reflecting, and stretching graphs. For $y = -2|x-1|$, reflect $|x|$ over the x-axis, stretch by 2, and shift right by 1.

• For $y = \sqrt{x-4}$, shift the basic $\sqrt{x}$ graph right by 4 units.

Applications and Word Problems

• Set up equations based on the problem context (e.g., area, cost, or motion problems).

• For motion under gravity: $s(t) = s_0 + v_0 t - 16t^2$ (feet, with $-16$ as the acceleration due to gravity).

Example: If a ball is dropped from 80 ft, $s(t) = 80 - 16t^2$.

Summary Table: Key Algebraic Forms

Form	Example	Key Feature
Standard Form (Quadratic)	$ax^2 + bx + c$	Used for factoring, finding vertex
Factored Form	$(x-r_1)(x-r_2)$	Roots are $r_1$, $r_2$
Circle	$(x-h)^2 + (y-k)^2 = r^2$	Center $(h, k)$, radius $r$
Line	$y = mx + b$	Slope $m$, y-intercept $b$

Additional info: This guide covers core Precalculus topics including algebraic manipulation, functions, systems, matrices, conic sections, and applications. All solutions are based on standard Precalculus curriculum and exam practice.