College Algebra Study Guide: Test 1 - Modules 1 - 3

Graphs and Graphing Utilities

Plotting Points and Graphing Equations

The rectangular coordinate system (also called the Cartesian plane) is used to graph equations and visualize relationships between variables.

• Plotting Points: Each point is represented by an ordered pair (x, y).

• Graphing Equations: Substitute values for x to find corresponding y-values and plot the points.

• 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).

• Viewing Rectangle: The window settings on a graphing utility determine which portion of the graph is visible.

Example: Graph $y = \frac{2}{3}x - 3$ for $x = -3, -2, -1, 0, 1, 2, 3$.

Additional info: When graphing equations with fractions, choose integer values for x to simplify calculations.

Linear Equations and Rational Equations

Solving Linear and Rational Equations

Linear equations have the form $ax + b = c$. Rational equations contain variables in the denominator.

• Identity: An equation true for all values of the variable.

• Conditional Equation: True for some values.

• Inconsistent Equation: No solution exists.

• Solving Rational Equations: Multiply both sides by the least common denominator (LCD) to clear fractions.

Example: Solve $\frac{16}{2x} = \frac{13}{7x}$.

Additional info: Always check for restricted values that make denominators zero.

Models and Applications

Using Linear Equations to Solve Word Problems

Linear equations can model real-world situations, such as finance, geometry, and science.

• Strategy: Define variables, write equations, solve, and interpret the answer.

• Solving Formulas: Rearranging formulas to isolate a specified variable.

Example: If 75% of a number is added to the number and the result is 280, find the number.

• Let $x$ be the number: $x + 0.75x = 280 \Rightarrow 1.75x = 280 \Rightarrow x = 160$

Complex Numbers

Operations with Complex Numbers

Complex numbers are of the form $a + bi$, where $i = \sqrt{-1}$.

• Add/Subtract: Combine real and imaginary parts.

• Multiply: Use distributive property; $i^2 = -1$.

• Divide: Multiply numerator and denominator by the conjugate.

• Square Roots of Negative Numbers: Express as multiples of $i$.

Example: $\sqrt{-25} = 5i$.

Quadratic Equations

Methods for Solving Quadratic Equations

Quadratic equations have the form $ax^2 + bx + c = 0$.

• Factoring: Set equation to zero and factor.

• Square Root Property: If $x^2 = k$, then $x = \pm \sqrt{k}$.

• Completing the Square: Rearrange and add to both sides to form a perfect square.

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

• Discriminant: $b^2 - 4ac$ determines the number and type of solutions.

Example: Solve $x^2 - 6x + 1 = 0$ using the quadratic formula.

Other Types of Equations

Solving Polynomial, Radical, and Absolute Value Equations

Equations may involve higher-degree polynomials, radicals, or absolute values.

• Polynomial Equations: Factor and set each factor to zero.

• Radical Equations: Isolate the radical and square both sides.

• Absolute Value Equations: Set up two equations: one for the positive case, one for the negative.

Example: Solve $|x - 3| = 5$; solutions are $x = 8$ and $x = -2$.

Linear Inequalities and Absolute Value Inequalities

Solving and Expressing Solutions to Inequalities

Linear inequalities are solved similarly to equations, but solution sets are often intervals.

• Interval Notation: Use parentheses for open intervals, brackets for closed.

• Compound Inequalities: Solve and express as intersection or union of intervals.

• Absolute Value Inequalities: Set up two inequalities and solve.

Example: Solve $|x - 2| < 5$; solution is $-3 < x < 7$.

Basics of Functions and Their Graphs

Understanding Functions and Their Properties

A function is a relation where each input has exactly one output.

• Domain: Set of possible input values (x).

• Range: Set of possible output values (y).

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

• Intercepts: Points where the graph crosses axes.

Example: $f(x) = x^2 + 5x + 2$; domain is all real numbers.

More on Functions and Their Graphs

Increasing, Decreasing, Even, Odd, and Piecewise Functions

Functions can be classified by their behavior and symmetry.

• Increasing/Decreasing: A function increases if y-values rise as x increases; decreases if y-values fall.

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

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

• Piecewise Function: Defined by different expressions for different intervals of x.

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$; used to measure average rate of change.

Example: $f(x) = x^3$ is an odd function.

Linear Functions and Slope

Slope and Equations of Lines

The slope of a line measures its steepness and direction.

• Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• General Form: $Ax + By + C = 0$

• Horizontal Line: $y = c$; slope is 0.

• Vertical Line: $x = c$; slope is undefined.

Example: Line through $(-1, 5)$ with x-intercept $-4$: $y = \frac{5}{3}x + \frac{20}{3}$

More on Slope

Parallel and Perpendicular Lines; Rate of Change

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Parallel: $m_1 = m_2$

• Perpendicular: $m_1 = -\frac{1}{m_2}$

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

Example: Average rate of change of $f(x) = x^2 + 4x$ from $x = 2$ to $x = 6$ is $7$.

Transformations of Functions

Shifts, Reflections, and Stretching/Shrinking

Transformations alter the appearance of a function's graph.

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

• Horizontal Shift: $f(x - h)$ shifts right/left.

• Reflection: $-f(x)$ reflects over x-axis; $f(-x)$ reflects over y-axis.

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

• Horizontal Stretch/Shrink: $f(bx)$ shrinks if $|b| > 1$, stretches if $0 < |b| < 1$.

Example: $f(x) = x^2$; $f(x) = 2(x - 3)^2 + 1$ is shifted right 3 units, stretched vertically by 2, and up by 1.

Combinations of Functions; Composite Functions

Algebra of Functions and Composition

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

• Domain: The set of x-values for which the function is defined.

• Composite Function: $f(g(x))$; apply $g$ first, then $f$.

• Domain of Composite: Values of x for which $g(x)$ is in the domain of $f$.

Example: If $f(x) = x^2 - 4$, $g(x) = \sqrt{x}$, then $f(g(x)) = (\sqrt{x})^2 - 4 = x - 4$; domain is $x \geq 0$.

Inverse Functions

Finding and Verifying Inverses

An inverse function reverses the effect of the original function.

• One-to-One Function: Each output corresponds to one input.

• Horizontal Line Test: If any horizontal line crosses the graph more than once, the function does not have an inverse.

• Finding Inverse: Swap x and y, solve for y.

Example: $f(x) = 2x + 8$; inverse is $f^{-1}(x) = \frac{x - 8}{2}$.

Distance and Midpoint Formulas; Circles

Geometry in the Coordinate Plane

Distance and midpoint formulas are used to analyze geometric figures.

• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Midpoint Formula: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Equation of a Circle: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.

Example: Circle with center $(-8, 5)$ and radius $11$:$(x + 8)^2 + (y - 5)^2 = 121$.

Table: Categories of Functions and Their Domains

Category	Domain
Polynomial	All real numbers
Rational	Exclude values where denominator = 0
Radical (even root)	Expression under radical $\geq 0$
Composite	Values where inner and outer functions are defined

Additional info: For composite functions, check both the domain of the inner function and the domain of the outer function evaluated at the inner function's output.