College Algebra: Graphs, Functions, and Models – Study Notes

Chapter 1: Graphs, Functions, and Models

Introduction to Graphing

Graphing is a foundational skill in algebra, allowing us to visualize equations, functions, and data. The Cartesian coordinate system is used to plot points, graph equations, and analyze relationships between variables.

• Cartesian Coordinate System: Consists of two perpendicular axes (x-axis and y-axis) intersecting at the origin (0, 0). The plane is divided into four quadrants.

• Plotting Points: An ordered pair (x, y) locates a point by moving x units left/right and y units up/down from the origin.

• Solutions of Equations: An ordered pair (x, y) is a solution to an equation in two variables if substituting x and y yields a true statement.

• Graph of an Equation: The set of all solutions plotted in the coordinate plane forms the graph of the equation.

• x-Intercept: The point where the graph crosses the x-axis (y = 0).

• y-Intercept: The point where the graph crosses the y-axis (x = 0).

• Graphing Nonlinear Equations: For equations not in the form Ax + By = C, such as quadratics, plot several points and connect them smoothly.

• Distance Formula: The distance between points (x₁, y₁) and (x₂, y₂) is given by:

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

• Midpoint Formula: The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is:

$\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Equation of a Circle: A circle with center (h, k) and radius r has the equation:

$(x - h)^2 + (y - k)^2 = r^2$

Solving Linear Inequalities

Linear inequalities are statements involving <, >, ≤, or ≥. The solution set consists of all values making the inequality true.

• Addition Principle: Adding the same value to both sides preserves the inequality.

• Multiplication Principle: Multiplying both sides by a positive number preserves the inequality; multiplying by a negative reverses the sign.

• Compound Inequalities: Combine two inequalities with 'and' (conjunction) or 'or' (disjunction).

Chapter 2: More on Functions

Increasing, Decreasing, and Constant Functions

Functions can be classified by their behavior over intervals:

• Increasing: $f(a) < f(b)$ for $a < b$ in interval I.

• Decreasing: $f(a) > f(b)$ for $a < b$ in interval I.

• Constant: $f(a) = f(b)$ for all $a, b$ in interval I.

• Relative Maximum/Minimum: A function value $f(c)$ is a relative maximum if it is greater than all nearby values; a relative minimum if it is less than all nearby values.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• Evaluate by determining which interval the input belongs to and using the corresponding formula.

Greatest Integer Function

The greatest integer function, $f(x) = \lfloor x \rfloor$, returns the largest integer less than or equal to x. Its graph is a step function.

Functions and Graphs

Definition of a Function

A function is a correspondence between a domain and a range such that each input has exactly one output.

• Relation: Any correspondence between two sets; not all relations are functions.

• Function Notation: $f(x)$ denotes the output for input x.

Graphing Functions

To graph a function, plot points (x, f(x)) and connect them smoothly. The vertical-line test determines if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

Domain and Range

The domain is the set of all possible inputs (x-values); the range is the set of all possible outputs (y-values). For rational and radical functions, exclude values that make the denominator zero or the radicand negative.

Linear Functions, Slope, and Applications

Linear Functions

A linear function has the form $f(x) = mx + b$, where m is the slope and b is the y-intercept.

• Slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$ measures the steepness of a line.

• Horizontal Lines: $y = b$ (slope 0).

• Vertical Lines: $x = a$ (not a function; slope undefined).

Equations of Lines

• Slope-Intercept Form: $y = mx + b$

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

• Parallel Lines: Same slope, different y-intercepts.

• Perpendicular Lines: Slopes are negative reciprocals.

Algebra of Functions

Operations with Functions

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x)g(x)$

• Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Difference Quotient

The difference quotient is used to compute the average rate of change of a function:

$\frac{f(x + h) - f(x)}{h}$

Composition and Decomposition of Functions

Composition of Functions

The composition $f \circ g$ is defined as $f(g(x))$. The domain consists of all x in the domain of g such that $g(x)$ is in the domain of f.

Decomposition:

Expressing a function as a composition of two or more simpler functions.

Symmetry, Even and Odd Functions

Symmetry

• x-axis: Replace y with –y; if equivalent, symmetric about x-axis.

• y-axis: Replace x with –x; if equivalent, symmetric about y-axis.

• Origin: Replace x with –x and y with –y; if equivalent, symmetric about the origin.

Even and Odd Functions

• Even: $f(-x) = f(x)$ for all x in the domain (symmetric about y-axis).

• Odd: $f(-x) = -f(x)$ for all x in the domain (symmetric about the origin).

Transformations of Functions

Translations

• Vertical: $y = f(x) + b$ shifts up by b; $y = f(x) - b$ shifts down by b.

• Horizontal: $y = f(x - d)$ shifts right by d; $y = f(x + d)$ shifts left by d.

Reflections

• Across x-axis: $y = -f(x)$

• Across y-axis: $y = f(-x)$

Stretching and Shrinking

• Vertical Stretch: $y = a f(x)$, $|a| > 1$

• Vertical Shrink: $0 < |a| < 1$

• Horizontal Stretch/Shrink: $y = f(cx)$, $|c| > 1$ shrinks, $0 < |c| < 1$ stretches

Variation

Direct Variation

y varies directly as x if $y = kx$ for some constant k.

Inverse Variation

y varies inversely as x if $y = \frac{k}{x}$ for some constant k.

Combined and Joint Variation

• Joint Variation: $y = kxz$

• Combined Variation: $y = \frac{kxz}{w^n}$

Additional info:

Some sections reference applications, modeling, and problem-solving strategies, which are essential for interpreting and applying algebraic concepts in real-world contexts.