Business Calculus Study Notes: Functions, Graphs, and Applications

Chapter 1: Functions and Graphs

1.1 Functions

Understanding functions is foundational in business calculus, as they model relationships between variables such as cost, revenue, and profit. This section introduces the concept of functions, their graphical representation, and their applications in business contexts.

• Graphs of Equations: The graph of an equation in two variables (x and y) is a visual representation of all solutions to the equation. For example, the graph of $y = x^2 - 4$ is a parabola.

• Functions Specified by Equations: An equation specifies a function if each input (x) produces exactly one output (y). If an input yields multiple outputs, it is not a function.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

• Domain of a Function: The domain is the set of all possible input values (x) for which the function is defined.

• Profit-Loss Analysis: In business, profit is calculated as revenue minus cost: $P(x) = R(x) - C(x)$.

• Price-Demand Analysis: The price-demand function $p(x)$ relates the price of a product to the quantity demanded. Revenue is given by $R(x) = x \cdot p(x)$.

Example: For the price-demand function $p(x) = 2000 - 60x$ (where $1 \leq x \leq 25$), the revenue function is $R(x) = x(2000 - 60x)$.

Tables: Cost and Profit Data

Tabular data is often used to analyze cost and profit in business scenarios.

x (millions)	C(x) (million $)
1	175
5	260
8	305
12	395

x (millions)	P(x) (million $)
1	-86
3
6
9
12
15

1.2 A Library of Elementary Functions

Elementary functions are the building blocks for modeling business scenarios. Their graphs and properties are essential for understanding more complex functions.

• Identity Function: $f(x) = x$

• Square Function: $f(x) = x^2$

• Cube Function: $f(x) = x^3$

• Square Root Function: $f(x) = \sqrt{x}$

• Cube Root Function: $f(x) = \sqrt[3]{x}$

• Absolute Value Function: $f(x) = |x|$

Example: The domain of $f(x) = \sqrt{x}$ is $x \geq 0$, and its range is $y \geq 0$.

Transformations of Functions

Transformations allow us to shift, stretch, shrink, or reflect the graphs of functions, which is useful for modeling changes in business variables.

• Vertical Shift: $y = f(x) + k$ shifts the graph up or down by $k$ units.

• Horizontal Shift: $y = f(x - h)$ shifts the graph left or right by $h$ units.

• Reflection: $y = -f(x)$ reflects the graph across the x-axis.

• Vertical Stretch/Shrink: $y = Af(x)$ stretches if $|A| > 1$, shrinks if $0 < |A| < 1$.

Example: The graph of $y = \sqrt{x} + 5$ is the graph of $y = \sqrt{x}$ shifted up by 5 units.

Piecewise-Defined Functions

Piecewise-defined functions are used to model situations where a rule changes depending on the input value. These are common in business for tiered pricing or cost structures.

• Example: $f(x) = \begin{cases} -2x + 4, & 0 \leq x \leq 2 \\ x - 1, & x > 2 \end{cases}$

1.3 Linear and Quadratic Functions

Linear Functions, Equations, and Inequalities

Linear functions model relationships with constant rates of change, such as cost per item or revenue per sale. Their graphs are straight lines.

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

• Intercepts: The x-intercept is where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.

• Forms of Linear Equations:

Form	Equation	Notes
Standard	$Ax + By = C$	A and B not both 0
Slope-intercept	$y = mx + b$	Slope: m, y-intercept: b
Point-slope	$y - y_1 = m(x - x_1)$	Slope: m, point: $(x_1, y_1)$
Horizontal line	$y = b$	Slope: 0
Vertical line	$x = a$	Slope: undefined

Quadratic Functions, Equations, and Inequalities

Quadratic functions model scenarios with variable rates of change, such as profit maximization or cost analysis. Their graphs are parabolas.

• Standard Form: $f(x) = ax^2 + bx + c$, $a \neq 0$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$

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

• Domain: All real numbers

• Range: $(-\infty, k]$ if $a < 0$ or $[k, \infty)$ if $a > 0$

Example: For $f(x) = -0.25x^2 - 2x + 2$, the vertex form is $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Applications: Supply, Demand, and Profit Maximization

Business calculus uses linear and quadratic functions to model supply and demand, equilibrium price, and profit maximization.

• Equilibrium Price: The price at which supply equals demand.

• Profit Function: $P(x) = R(x) - C(x)$, where $R(x)$ is revenue and $C(x)$ is cost.

• Maximizing Profit: Find the vertex of the profit function to determine the production level that yields maximum profit.

Example: For $C(x) = 156 + 19.7x$ and $R(x) = x(94.8 - 5x)$, the profit function is $P(x) = x(94.8 - 5x) - (156 + 19.7x)$.

Linear and Quadratic Regression

Regression analysis is used to find the best-fit line or curve for a set of data points, which helps in predicting future values and making business decisions.

• Linear Regression: Finds the best linear model $y = ax + b$ for a set of data.

• Quadratic Regression: Finds the best quadratic model $y = ax^2 + bx + c$ for a set of data.

Weight (carats)	Price ($)
0.5	1,677
0.6	2,353
0.7	2,718
0.8	3,218
0.9	3,982
1.0	4,510

Additional info: These notes cover the essential concepts and applications of functions, graphs, and their transformations, as well as linear and quadratic modeling in business calculus. The included tables and images reinforce the connection between mathematical models and real-world business data.