📊Mathematical Modeling Unit 2 – Functions and Graphs

Key Concepts

• Functions map input values from the domain to output values in the range, establishing a relationship between two sets of numbers
• Function notation $f(x)$ represents the output value of a function $f$ for a given input value $x$
• Determine if a relation is a function using the vertical line test, which states that a vertical line drawn through the graph of a relation should intersect the graph at most once for it to be a function
• Identify the domain (set of all possible input values) and range (set of all possible output values) of a function
  • The domain is typically represented using set notation or interval notation (e.g., $\{x \mid x \in \mathbb{R}, x \geq 0\}$ or $[0, \infty)$)
  • The range is determined by evaluating the function for all values in the domain
• Understand the concept of function composition, where the output of one function becomes the input of another function, denoted as $(f \circ g)(x) = f(g(x))$
• Recognize one-to-one functions, which have a unique output value for each input value and pass the horizontal line test
• Determine the inverse of a function, which "undoes" the original function, such that $(f^{-1} \circ f)(x) = x$ and $(f \circ f^{-1})(x) = x$

Types of Functions

• Linear functions have the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept, and produce straight-line graphs
• Quadratic functions have the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and produce parabolic graphs
  • The vertex of a parabola is the point where the function reaches its minimum or maximum value
  • The axis of symmetry is the vertical line that passes through the vertex, dividing the parabola into two symmetrical halves
• Polynomial functions have the form $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$, where $a_i$ are constants and $n$ is a non-negative integer
• Rational functions are the ratio of two polynomial functions, expressed as $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomial functions
• Exponential functions have the form $f(x) = a \cdot b^x$, where $a$ is the initial value and $b$ is the growth or decay factor
• Logarithmic functions have the form $f(x) = \log_b(x)$, where $b$ is the base, and are the inverse of exponential functions
• Trigonometric functions (sine, cosine, tangent) relate angles to the ratios of sides in a right triangle and produce periodic graphs

Graphing Techniques

• Plotting points is the most basic method of graphing a function, which involves evaluating the function for various input values and plotting the corresponding $(x, y)$ coordinates on a Cartesian plane
• Determine the x- and y-intercepts of a function by setting $y = 0$ and $x = 0$, respectively, and solving for the other variable
• Identify the domain and range of a function from its graph by observing the x-values and y-values covered by the graph
• Use the slope-intercept form $y = mx + b$ to easily graph linear functions by plotting the y-intercept $(0, b)$ and using the slope $m$ to find additional points
• Graph quadratic functions by finding the vertex, axis of symmetry, and y-intercept, then plotting additional points on either side of the axis of symmetry
• Sketch polynomial functions by finding the x-intercepts (roots), y-intercept, and analyzing the end behavior (the behavior of the function as $x$ approaches positive or negative infinity)
• Graph rational functions by identifying the vertical and horizontal asymptotes, x- and y-intercepts, and plotting additional points
• Use transformations to graph exponential, logarithmic, and trigonometric functions based on their parent functions

Function Transformations

• Translations shift the graph of a function horizontally or vertically without changing its shape
  • $f(x) + k$ shifts the graph of $f(x)$ vertically by $k$ units (up for $k > 0$, down for $k < 0$)
  • $f(x - h)$ shifts the graph of $f(x)$ horizontally by $h$ units (right for $h > 0$, left for $h < 0$)
• Reflections flip the graph of a function across the x-axis or y-axis
  • $-f(x)$ reflects the graph of $f(x)$ across the x-axis
  • $f(-x)$ reflects the graph of $f(x)$ across the y-axis
• Stretches and compressions change the shape of the graph vertically or horizontally
  • $a \cdot f(x)$ stretches the graph of $f(x)$ vertically by a factor of $|a|$ for $|a| > 1$ and compresses it for $0 < |a| < 1$
  • $f(b \cdot x)$ compresses the graph of $f(x)$ horizontally by a factor of $|b|$ for $|b| > 1$ and stretches it for $0 < |b| < 1$
• Combinations of transformations can be applied to a function in the order: reflections, stretches/compressions, and translations

Real-World Applications

• Linear functions model situations with constant rates of change, such as converting between temperature scales (Celsius to Fahrenheit) or calculating the cost of renting equipment per day
• Quadratic functions describe the motion of objects under the influence of gravity, such as the height of a ball thrown upward or the path of a projectile
• Exponential functions represent growth or decay processes, including population growth, radioactive decay, or compound interest
• Logarithmic functions are used to measure the magnitude of earthquakes (Richter scale), the intensity of sound (decibels), and the acidity of a solution (pH scale)
• Trigonometric functions model periodic phenomena, such as sound waves, tides, and the motion of a pendulum
• Rational functions can represent the concentration of a drug in the bloodstream over time or the efficiency of a manufacturing process as a function of production rate

Common Challenges

• Distinguishing between relations and functions, especially when given a graph or a set of ordered pairs
• Determining the domain and range of a function from its equation, particularly for rational, logarithmic, and square root functions
• Identifying the correct parent function and applying transformations in the appropriate order when graphing transformed functions
• Remembering the proper notation for function composition and inverses, and understanding the conditions for the existence of an inverse function
• Sketching graphs of polynomial and rational functions, especially identifying the end behavior and asymptotes
• Interpreting the meaning of function parameters in real-world applications and translating verbal descriptions into mathematical functions
• Recognizing the appropriate type of function to model a given real-world situation and fitting the data to the function

Practice Problems

1. Determine if the relation $\{(2, 3), (4, 5), (2, 6), (7, 8)\}$ is a function.
2. Find the domain and range of the function $f(x) = \sqrt{x - 3}$.
3. Given $f(x) = 2x + 1$ and $g(x) = x^2 - 4$, find $(f \circ g)(x)$ and $(g \circ f)(x)$.
4. Graph the function $f(x) = -3(x + 2)^2 + 4$ by applying transformations to the parent quadratic function.
5. Sketch the graph of the rational function $f(x) = \frac{x + 1}{x - 2}$, identifying the x- and y-intercepts and asymptotes.
6. A population of bacteria doubles every 4 hours. If the initial population is 1000, write an exponential function that models the population growth and determine the population after 24 hours.
7. The height $h$ (in meters) of a projectile launched with an initial velocity of 50 m/s at an angle of 60° is given by $h(t) = -4.9t^2 + 43.3t$, where $t$ is the time in seconds. Find the maximum height reached by the projectile and the time at which it hits the ground.

Additional Resources

• Khan Academy's Functions course covers the fundamentals of functions, including notation, domain and range, and graphing
• Paul's Online Math Notes offers a comprehensive guide to Functions, with explanations, examples, and practice problems
• Desmos is an online graphing calculator that allows users to explore and manipulate function graphs interactively
• OpenStax's College Algebra textbook has chapters dedicated to various types of functions, their properties, and applications
• 3Blue1Brown's YouTube video series Essence of Calculus provides an intuitive introduction to the concepts of functions and their role in calculus
• MIT OpenCourseWare's Single Variable Calculus course includes lectures and problem sets on functions, limits, derivatives, and integrals
• The Wolfram Functions Site is a comprehensive resource for exploring the properties and graphs of various mathematical functions