📘Intermediate Algebra Unit 3 – Graphs and Functions

Key Concepts and Definitions

• Functions defined as a relation between a set of inputs and a set of outputs with exactly one output for each input
• Domain refers to the set of all possible input values (usually x-values) for a function
• Range represents the set of all possible output values (usually y-values) that a function can produce
• Independent variable (usually x) serves as the input value of a function and can be freely chosen
• Dependent variable (usually y) represents the output value of a function and depends on the value of the independent variable
  • For example, in the function $y = 2x + 1$, y is the dependent variable because its value depends on the value of x
• Function notation $f(x)$ used to represent the output value of a function for a given input value x
• Vertical line test determines if a relation is a function by checking if any vertical line intersects the graph more than once (if so, it's not a function)

Types of Graphs and Functions

• Linear functions represented by straight lines on a graph with a constant rate of change (slope)
  • General form of a linear function: $y = mx + b$, where m is the slope and b is the y-intercept
• Quadratic functions represented by parabolas on a graph with a degree of 2
  • General form of a quadratic function: $y = ax^2 + bx + c$, where a, b, and c are constants and a ≠ 0
• Exponential functions involve a constant base raised to a variable power, resulting in rapid growth or decay
  • General form of an exponential function: $y = a \cdot b^x$, where a is the initial value and b is the growth or decay factor
• Logarithmic functions are the inverse of exponential functions and represent the power to which a base must be raised to get a certain value
• Rational functions are the quotient of two polynomial functions, often resulting in asymptotes and holes in the graph
• Absolute value functions consist of two linear pieces joined at a vertex, forming a V-shape on the graph
• Piecewise functions defined by different rules for different intervals of the domain, resulting in a graph with distinct sections

Graphing Techniques

• Plotting points involves finding ordered pairs (x, y) that satisfy the function and placing them on the coordinate plane
• Intercepts refer to the points where a graph crosses the x-axis (x-intercept) or y-axis (y-intercept)
  • To find x-intercepts, set y = 0 and solve for x; to find y-intercepts, set x = 0 and solve for y
• Symmetry can be used to quickly graph functions by reflecting points across the x-axis, y-axis, or origin
• Transformations alter the shape, position, or orientation of a graph without changing its fundamental characteristics
  • Translations shift the graph horizontally or vertically, while reflections flip the graph across an axis or line
• Asymptotes are lines that a graph approaches but never touches, often occurring in rational and logarithmic functions
  • Vertical asymptotes occur when the denominator of a rational function equals zero
  • Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity
• Sketching involves creating a rough graph by hand using key features like intercepts, symmetry, and asymptotes to capture the general shape and behavior of the function

Function Operations and Transformations

• Addition of functions $f(x) + g(x)$ results in a new function where the y-values of the original functions are added together for each x-value
• Subtraction of functions $f(x) - g(x)$ creates a new function where the y-values of g(x) are subtracted from the y-values of f(x) for each x-value
• Multiplication of functions $f(x) \cdot g(x)$ produces a new function where the y-values of the original functions are multiplied together for each x-value
• Division of functions $\frac{f(x)}{g(x)}$ results in a new function where the y-values of f(x) are divided by the y-values of g(x) for each x-value (as long as g(x) ≠ 0)
• Composition of functions $f(g(x))$ involves substituting the function g(x) as the input for the function f(x)
  • For example, if $f(x) = x^2$ and $g(x) = x + 1$, then $f(g(x)) = (x + 1)^2$
• Transformations alter the shape, position, or orientation of a graph without changing its fundamental characteristics
  • Vertical translations shift the graph up or down by adding or subtracting a constant to the function: $y = f(x) \pm k$
  • Horizontal translations shift the graph left or right by adding or subtracting a constant to the input: $y = f(x \pm h)$
  • Vertical stretches or compressions multiply the function by a constant: $y = a \cdot f(x)$, where |a| > 1 stretches and 0 < |a| < 1 compresses
  • Horizontal stretches or compressions multiply the input by a constant: $y = f(b \cdot x)$, where 0 < |b| < 1 stretches and |b| > 1 compresses

Real-World Applications

• Linear functions model situations with a constant rate of change, such as converting between temperature scales or calculating the cost of renting equipment over time
• Quadratic functions describe the path of projectiles, the height of bouncing balls, and the profit of businesses based on production and sales
• Exponential functions represent population growth, compound interest, and radioactive decay
  • For example, the value of an investment with compound interest can be modeled by $A = P(1 + r)^t$, where A is the final amount, P is the principal, r is the interest rate, and t is the number of compounding periods
• Logarithmic functions are used to measure the intensity of earthquakes (Richter scale), the acidity of solutions (pH scale), and the loudness of sound (decibel scale)
• Rational functions model the concentration of drugs in the bloodstream over time and the efficiency of chemical reactions based on the concentration of reactants
• Absolute value functions can represent the distance between two points on a number line or the profit/loss of a business based on production costs and revenue
• Piecewise functions describe situations where different rules apply under different conditions, such as tax brackets, shipping rates, or mobile phone plans

Common Challenges and Solutions

• Identifying the domain and range of a function by considering any restrictions on the input and output values
  • For example, a square root function has a domain limited to non-negative inputs, and a rational function has a domain that excludes values that make the denominator zero
• Determining the equations of asymptotes for rational and logarithmic functions by setting the denominator equal to zero (vertical) or evaluating limits (horizontal)
• Applying function transformations in the correct order: horizontal and vertical translations, stretches/compressions, and reflections
  • Remember the mnemonic "THOR" (Translations, Horizontal stretches/compressions, Output (vertical) stretches/compressions, Reflections)
• Solving equations and inequalities involving functions by applying inverse operations and considering the domain and range
  • When solving inequalities, be aware that multiplying or dividing by a negative number reverses the inequality sign
• Interpreting the meaning of function parameters in context, such as the slope and y-intercept of a linear function or the growth factor and initial value of an exponential function
• Recognizing and avoiding common errors, such as misinterpreting function notation, confusing the order of operations, or misapplying transformations
  • Pay close attention to the placement of negative signs, parentheses, and function arguments to avoid algebraic mistakes

Practice Problems and Examples

• Determine if the relation {(1, 2), (3, 4), (3, 5), (4, 6)} is a function and explain why or why not
  • This relation is not a function because the input value 3 is paired with two different output values (4 and 5), violating the definition of a function
• Graph the function $y = -2x^2 + 4x + 3$ by finding the vertex, intercepts, and at least two additional points
  • Vertex: (-1, 5); x-intercepts: (-0.58, 0) and (2.58, 0); y-intercept: (0, 3); additional points: (-2, -5) and (1, 3)
• Find the domain and range of the function $f(x) = \sqrt{x - 1}$
  • Domain: $x \geq 1$ (x must be greater than or equal to 1 to avoid negative values under the square root)
  • Range: $y \geq 0$ (the square root always produces non-negative output values)
• Given $f(x) = 2x + 1$ and $g(x) = x^2 - 3$, find $(f \circ g)(x)$ and $(g \circ f)(x)$
  • $(f \circ g)(x) = f(g(x)) = f(x^2 - 3) = 2(x^2 - 3) + 1 = 2x^2 - 5$
  • $(g \circ f)(x) = g(f(x)) = g(2x + 1) = (2x + 1)^2 - 3 = 4x^2 + 4x - 2$
• A population of bacteria doubles every 4 hours. If there are initially 500 bacteria, write an exponential function to model the population growth and determine the population after 24 hours.
  • Exponential growth function: $P(t) = 500 \cdot 2^{\frac{t}{4}}$, where t is the time in hours
  • After 24 hours: $P(24) = 500 \cdot 2^{\frac{24}{4}} = 500 \cdot 2^6 = 32,000$ bacteria

Connections to Other Math Topics

• Coordinate geometry uses the concepts of functions and graphs to analyze shapes, lines, and curves in the plane
  • The equation of a circle with center (h, k) and radius r is $(x - h)^2 + (y - k)^2 = r^2$, which can be graphed using function transformations
• Trigonometry involves the study of trigonometric functions (sine, cosine, tangent) and their graphs, which have periodic behavior and specific transformations
  • The sine function, $y = \sin(x)$, has a period of $2\pi$, an amplitude of 1, and a vertical shift of 0
• Calculus builds upon the foundation of functions and graphs to study rates of change (derivatives) and accumulation (integrals)
  • The derivative of a function $f(x)$ represents the slope of the tangent line at any point on the graph of $f(x)$
  • The definite integral of a function $f(x)$ over an interval $[a, b]$ gives the area between the graph of $f(x)$ and the x-axis from x = a to x = b
• Sequences and series are special types of functions where the domain is limited to positive integers
  • Arithmetic sequences have a constant difference between consecutive terms, while geometric sequences have a constant ratio
  • The Fibonacci sequence, defined by $F_n = F_{n-1} + F_{n-2}$ with $F_1 = F_2 = 1$, has connections to golden ratios and natural phenomena
• Probability and statistics use functions to model the likelihood of events and the distribution of data
  • The normal distribution, represented by the bell curve, is a continuous probability distribution with a symmetric shape and specific properties
  • Linear regression involves finding the line of best fit for a set of data points, which can be used to make predictions and analyze trends