College Algebra: Practice Questions and Key Concepts

Systems of Equations and Solutions

Solving Systems of Linear Equations

Systems of equations involve finding values for variables that satisfy multiple equations simultaneously. In College Algebra, these are often linear or simple nonlinear systems.

• Key Point: A system may have one solution, no solution, or infinitely many solutions.

• Example: Solve for x and y given: $y = \frac{1}{2}x$, $y = -2$, $x + y = 1$, $x - y = 1$ Method: Substitute and solve using substitution or elimination.

Algebraic Simplification

Combining Rational Expressions

Rational expressions can be simplified by finding a common denominator and combining terms.

• Key Point: Always factor denominators and numerators when possible to simplify.

• Example: Simplify $\frac{b}{b+1} + \frac{b}{b-1}$ Solution: Find a common denominator and combine.

Quadratic Equations

Solving Quadratic Equations

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

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

• Sum of Solutions: For $ax^2 + bx + c = 0$, the sum is $-\frac{b}{a}$.

• Example: Solve $2x^2 - 3x + 4 = 0$.

Functions and Their Properties

Odd and Even Functions

A function $f(x)$ is even if $f(-x) = f(x)$ for all $x$ in its domain, and odd if $f(-x) = -f(x)$.

• Example: $f(x) = x^3 - 7x$ is odd because $f(-x) = -f(x)$.

Function Composition

Given two functions $f$ and $g$, the composition $f(g(x))$ means applying $g$ first, then $f$.

• Example: If $f(x) = 2x^2 + 1$ and $g(x) = x - 1$, then $f(g(2)) = f(1) = 2(1)^2 + 1 = 3$.

Domain of a Function

The domain of a function is the set of all input values for which the function is defined.

• Example: For $f(x) = \frac{1}{x^2 - 9}$, the domain excludes $x = 3$ and $x = -3$ because the denominator is zero at these points.

Inverse Functions

The inverse of a function $f(x)$, denoted $f^{-1}(x)$, reverses the effect of $f(x)$. To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$.

• Example: If $f(y) = \frac{2y - 3}{4}$, solve for $y$ in terms of $x$ to find $f^{-1}(x)$.

Polynomials

Factoring and Roots

Factoring polynomials helps in finding their roots (solutions where the polynomial equals zero).

• Example: Factor $2x^2 + 3x - 5$ and find its roots using the quadratic formula.

Graph Transformations

Shifts and Stretches

Transformations change the position or shape of a graph. Common transformations include:

• Vertical Shift: $f(x) + k$ shifts up/down by $k$ units.

• Horizontal Shift: $f(x - h)$ shifts right by $h$ units.

• Vertical Stretch/Shrink: $a f(x)$ stretches if $|a| > 1$, shrinks if $0 < |a| < 1$.

• Example: $2|x - 4| - 5$ is a vertical stretch by 2, right shift by 4, and down shift by 5.

Lines and Their Equations

Point-Slope and Parallel Lines

The equation of a line can be written in point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line.

• Parallel Lines: Two lines are parallel if they have the same slope.

• Example: Find the equation of the line passing through $(2, -3)$ and parallel to $2x - y = 5$.

Summary Table: Function Properties

Property	Definition	Example
Even Function	$f(-x) = f(x)$	$f(x) = x^2$
Odd Function	$f(-x) = -f(x)$	$f(x) = x^3$
Domain	Set of all valid $x$	$f(x) = \frac{1}{x}$, domain: $x \neq 0$
Inverse	Solves $y = f(x)$ for $x$	$f(x) = 2x + 3 \implies f^{-1}(x) = \frac{x - 3}{2}$

Additional info:

• Some questions involve finding the sum of solutions, which uses Vieta's formulas for quadratics.

• Graph transformations are a key topic in understanding how algebraic changes affect the visual representation of functions.