🔢Lower Division Math Foundations Unit 3 – Functions: Types and Graphs

What Are Functions?

• Functions are rules that assign each input value to exactly one output value
• The input values make up the domain, while the output values make up the range
• Functions can be represented using equations, graphs, or tables
• One-to-one functions have a unique output for each input and pass the horizontal line test
• Many-to-one functions have multiple inputs that can produce the same output (not one-to-one)
• Vertical line test determines if a graph represents a function - if any vertical line intersects the graph more than once, it is not a function
• Piecewise functions are defined using different rules over different intervals of the domain

Types of Functions

• Linear functions have the form $y = mx + b$ and produce a straight line graph
• Quadratic functions are degree 2 polynomials in the form $y = ax^2 + bx + c$ and create parabolic graphs
• Cubic functions are degree 3 polynomials and have a general form of $y = ax^3 + bx^2 + cx + d$
  • Their graphs can have one or two turning points and up to three real roots
• Exponential functions have a constant base raised to a variable power, such as $y = a^x$
  • They increase or decrease at a rate proportional to their current value
• Logarithmic functions are the inverse of exponential functions and have the form $y = \log_b(x)$
• Trigonometric functions (sine, cosine, tangent) relate angles to side lengths in right triangles
• Absolute value functions output the distance of the input from zero, written as $y = |x|$

Function Notation and Terminology

• Function notation $f(x)$ is read as "f of x" and indicates the output when $x$ is the input
• The input variable is also called the independent variable, often represented by $x$
• The output variable, or dependent variable, is frequently represented by $y$ or $f(x)$
• Evaluating a function means finding the output value for a given input
  • To evaluate $f(3)$, substitute 3 for $x$ in the function's equation and simplify
• The domain is the set of all possible input values for a function
• The range is the set of all possible output values a function can produce
• Interval notation [a, b] represents all real numbers between a and b, including endpoints

Graphing Basics

• The Cartesian coordinate plane consists of a horizontal x-axis and a vertical y-axis
• The origin (0, 0) is the point where the axes intersect
• Points are plotted as ordered pairs (x, y), where x is the horizontal coordinate and y is the vertical coordinate
• To graph a function, plot points by evaluating the function at various x-values and connecting the points with a smooth curve
• The x-intercept is the point where a graph crosses the x-axis, and the y-coordinate is always 0
• The y-intercept is the point where a graph crosses the y-axis, and the x-coordinate is always 0
• Symmetry in graphs can be described as even (symmetric about the y-axis) or odd (180° rotational symmetry about the origin)

Common Function Graphs

• Linear function graphs are straight lines with a constant slope
• Quadratic function graphs are parabolas that open upward (positive leading coefficient) or downward (negative leading coefficient)
  • The vertex is the turning point of a parabola, either a maximum or minimum point
• Exponential function graphs increase or decrease rapidly, approaching but never touching the x-axis
• Logarithmic function graphs are the reflections of exponential graphs across the line $y = x$
• Sine and cosine graphs are periodic, repeating wave patterns with a constant amplitude and period
• The absolute value graph is a V-shape with the vertex at the origin and symmetric about the y-axis

Transformations of Functions

• Transformations alter the position, shape, or orientation of a function's graph
• Vertical shifts move the graph up or down by adding or subtracting a constant term
  • $y = f(x) + k$ shifts the graph of $f(x)$ up by $k$ units
  • $y = f(x) - k$ shifts the graph of $f(x)$ down by $k$ units
• Horizontal shifts move the graph left or right by adding or subtracting a constant term inside the function
  • $y = f(x - h)$ shifts the graph of $f(x)$ right by $h$ units
  • $y = f(x + h)$ shifts the graph of $f(x)$ left by $h$ units
• Reflections flip the graph across the x-axis or y-axis
  • $y = -f(x)$ reflects the graph of $f(x)$ across the x-axis
  • $y = f(-x)$ reflects the graph of $f(x)$ across the y-axis
• Vertical stretches and compressions multiply the function's output by a constant factor
  • $y = af(x)$, where $|a| > 1$, vertically stretches the graph of $f(x)$
  • $y = af(x)$, where $0 < |a| < 1$, vertically compresses the graph of $f(x)$

Analyzing Function Behavior

• Increasing functions have outputs that increase as inputs increase, resulting in a graph that rises from left to right
• Decreasing functions have outputs that decrease as inputs increase, resulting in a graph that falls from left to right
• Local maxima and minima are the highest and lowest points, respectively, within a specific interval of the domain
• Absolute (global) maximum and minimum are the highest and lowest points, respectively, across the entire domain
• Asymptotes are lines that a graph approaches but never touches
  • Vertical asymptotes occur at domain values where the function is undefined (denominator equals zero)
  • Horizontal asymptotes describe the graph's long-term behavior as x approaches positive or negative infinity
• Continuity means a function has no breaks, gaps, or jumps in its graph
  • Continuous functions have a well-defined output for every input in their domain

Real-World Applications

• Linear functions can model situations with constant rates of change (uniform motion, cost per unit)
• Quadratic functions can represent the height of objects in freefall or the path of projectiles
• Exponential functions describe growth or decay processes (population growth, radioactive decay)
• Logarithmic functions are used to measure the intensity of earthquakes (Richter scale) and sound (decibels)
• Trigonometric functions model periodic phenomena (ocean tides, sound waves)
• Absolute value functions can represent the distance between two points on a number line
• Piecewise functions can describe situations with different rules for different intervals (tax brackets, shipping rates)