Glossary

3.4 Graphing Functions and Their Properties

3 min read•august 12, 2024

Graphing functions is a powerful way to visualize mathematical relationships. By on a coordinate plane, we can see how variables interact and analyze key properties like , symmetry, and .

Understanding these properties helps us interpret and predict function behavior. We can identify or trends, find symmetry, and determine asymptotes. This knowledge is crucial for solving problems and modeling real-world situations using functions.

Coordinate System and Intercepts

Understanding the Cartesian Plane

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  • Cartesian coordinate system consists of two perpendicular number lines intersecting at the origin (0, 0)
  • Horizontal axis represents x-values, vertical axis represents y-values
  • Points on the plane expressed as ordered pairs (x, y)
  • Plane divided into four quadrants, numbered counterclockwise from upper right
  • Quadrant I: (+, +), Quadrant II: (-, +), Quadrant III: (-, -), Quadrant IV: (+, -)
  • Used to graph functions and visualize mathematical relationships

Identifying Intercepts

  • x-intercept occurs where a graph crosses the x-axis
  • Found by setting y = 0 in the function equation and solving for x
  • Represents the roots or zeros of a function
  • y-intercept occurs where a graph crosses the y-axis
  • Found by setting x = 0 in the function equation and solving for y
  • Represents the initial value or starting point of many functions
  • [f(x)](https://www.fiveableKeyTerm:f(x))=mx+b[f(x)](https://www.fiveableKeyTerm:f(x)) = mx + b has y-intercept at (0, b)
  • f(x)=ax2+bx+cf(x) = ax^2 + bx + c has y-intercept at (0, c)

Function Types and Behavior

Monotonicity and Constancy

  • Increasing function grows larger as x increases
  • Formally defined as f(x1)<f(x2)f(x_1) < f(x_2) for all x1<x2x_1 < x_2 in the
  • Decreasing function grows smaller as x increases
  • Formally defined as f(x1)>f(x2)f(x_1) > f(x_2) for all x1<x2x_1 < x_2 in the domain
  • Constant function maintains the same y-value for all x in its domain
  • Expressed as f(x)=kf(x) = k, where k is a fixed real number
  • Graph of a constant function appears as a horizontal line

Symmetry in Functions

  • symmetric about the y-axis
  • Satisfies the condition f(−x)=f(x)f(-x) = f(x) for all x in the domain
  • Graph remains unchanged when reflected over the y-axis
  • symmetric about the origin
  • Satisfies the condition f(−x)=−f(x)f(-x) = -f(x) for all x in the domain
  • Graph remains unchanged when rotated 180 degrees around the origin
  • Functions can be neither even nor odd (asymmetric)

Analyzing End Behavior

  • End behavior describes how a function behaves as x approaches positive or negative infinity
  • Expressed using limit notation: lim⁡x→∞f(x)\lim_{x \to \infty} f(x) and lim⁡x→−∞f(x)\lim_{x \to -\infty} f(x)
  • Polynomial functions' end behavior determined by the leading term's degree and coefficient
  • Rational functions' end behavior influenced by the degrees of numerator and denominator
  • Exponential functions approach horizontal asymptotes as x approaches infinity in one direction

Function Properties

Exploring Symmetry and Asymptotes

  • Symmetry in functions includes point symmetry, line symmetry, and rotational symmetry
  • Point symmetry occurs when a graph remains unchanged after a 180-degree rotation (odd functions)
  • Line symmetry occurs when a graph remains unchanged after over a line (even functions)
  • represents a line that a graph approaches but never reaches
  • Vertical asymptotes occur where a function is undefined (denominator equals zero in rational functions)
  • Horizontal asymptotes describe end behavior as x approaches infinity
  • Slant asymptotes appear in rational functions where numerator degree exceeds denominator degree by 1

Analyzing Continuity and Transformations

  • Continuity describes a function with no breaks, gaps, or jumps in its graph
  • Continuous function can be drawn without lifting the pencil from the paper
  • Three conditions for continuity at a point a: f(a) is defined, lim⁡x→af(x)\lim_{x \to a} f(x) exists, and lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)
  • Discontinuities include removable, jump, and infinite discontinuities
  • Transformation of functions alters the graph of a parent function
  • Vertical shifts move the graph up or down: f(x)+kf(x) + k
  • Horizontal shifts move the graph left or right: f(x+h)f(x + h)
  • Vertical stretches or compressions: af(x)af(x), where |a| > 1 stretches and 0 < |a| < 1 compresses
  • Horizontal stretches or compressions: f(bx)f(bx), where 0 < |b| < 1 stretches and |b| > 1 compresses
  • Reflections over x-axis: −f(x)-f(x), over y-axis: f(−x)f(-x)

Key Terms to Review (23)

Asymptote: An asymptote is a line that a graph approaches but never actually touches or intersects. It can provide important information about the behavior of a function as it reaches extreme values, guiding the understanding of limits and the function's overall shape. Recognizing asymptotes helps in sketching graphs accurately, indicating where the function diverges or stabilizes.
Decreasing: In mathematics, a function is described as decreasing when, as the input values increase, the output values decrease. This property indicates that the function is moving downward as you move from left to right on a graph, showing that higher input values yield lower output values. Decreasing functions are significant in understanding the overall behavior of graphs, particularly in identifying trends and analyzing rates of change.
Domain: The domain of a function is the complete set of possible values of the independent variable, or input, for which the function is defined. Understanding the domain is crucial as it determines where the function can operate without issues, such as division by zero or taking the square root of negative numbers. The concept of domain applies across various mathematical areas, influencing how we interpret quantifiers, perform operations with sets, and graph functions effectively.
End Behavior: End behavior describes the behavior of a function's graph as the input values approach positive or negative infinity. Understanding end behavior is crucial because it helps identify how a function behaves at its extremes, providing insights into the overall shape and characteristics of the graph. This concept is particularly significant when analyzing polynomial functions, rational functions, and others, as it influences how graphs intersect the x-axis and where they approach as they move outward.
Even Function: An even function is a type of mathematical function that exhibits symmetry about the y-axis. This means that for any input value $$x$$, the output is the same when the input is the negative of that value; in other words, $$f(-x) = f(x)$$ for all $$x$$ in the domain of the function. The property of being even leads to unique characteristics when graphing, as it results in a mirror image on either side of the y-axis.
Exponential Function: An exponential function is a mathematical function of the form $$f(x) = a \cdot b^{x}$$ where 'a' is a constant, 'b' is the base of the exponential and is greater than zero, and 'x' is the exponent. Exponential functions are characterized by their rapid growth or decay, depending on the value of the base 'b'. These functions have unique properties such as a horizontal asymptote and specific behavior in their graphs, which distinguish them from linear or polynomial functions.
F(x): The notation f(x) represents a function named 'f' evaluated at the input 'x'. This is a foundational concept in mathematics, as it allows us to express relationships between variables in a clear and structured way. The function f takes an input value and produces an output value, which can be visualized and analyzed through graphing and other properties of functions.
G(x): The notation g(x) represents a function where 'g' is the name of the function and 'x' is the input variable. This term is used to describe how the function takes an input value, processes it through a specific rule or equation, and produces an output. Understanding g(x) is crucial for analyzing the behavior of functions, as it allows you to investigate properties such as continuity, limits, and transformations.
Horizontal Asymptote: A horizontal asymptote is a horizontal line that a graph approaches as the input value (x) approaches infinity or negative infinity. This concept helps in understanding the long-term behavior of functions, especially rational functions, by indicating the value that the function approaches but may never actually reach.
Increasing: Increasing refers to a characteristic of a function where, as the input values (or x-values) increase, the output values (or y-values) also increase. This property is essential when analyzing the behavior of functions, as it helps to identify intervals where a function is rising and can indicate trends within the graph. Understanding when a function is increasing allows for better interpretation of its overall behavior and can help in determining local and global extrema.
Intercepts: Intercepts are the points where a graph crosses the axes on a coordinate plane. Specifically, the x-intercept is where the graph touches or crosses the x-axis, and the y-intercept is where it touches or crosses the y-axis. These points are crucial for understanding the behavior of functions, helping to visualize and analyze key features such as zeros, trends, and function transformations.
Linear function: A linear function is a type of function that creates a straight line when graphed on a coordinate plane. This function can be expressed in the form $$f(x) = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. The characteristics of a linear function include constant rates of change, and its graph showcases key properties like symmetry, slope, and intercepts.
Local Maximum: A local maximum is a point on a function's graph where the function value is greater than the values of the function at nearby points. This term is crucial for understanding the overall shape of the graph and is linked to the behavior of functions in relation to their peaks and valleys. Recognizing local maxima helps in identifying trends and critical points within the graph of a function, which can indicate where the function reaches its highest value in a specific interval.
Odd Function: An odd function is a type of mathematical function that satisfies the property f(-x) = -f(x) for all x in its domain. This means that if you take any input x and replace it with its negative counterpart, the output will also be the negative of the original output. Odd functions exhibit symmetry about the origin, which visually reflects their unique characteristics when graphed.
Plotting Points: Plotting points involves marking specific coordinates on a Cartesian plane to represent the relationship between two variables. Each point is defined by an ordered pair, usually written as $(x, y)$, where 'x' indicates the horizontal position and 'y' indicates the vertical position. Understanding how to plot points is essential for graphing functions and analyzing their properties, as it allows one to visualize trends, intercepts, and overall behavior of the function being studied.
Quadratic function: A quadratic function is a polynomial function of degree two, typically expressed in the standard form $$f(x) = ax^2 + bx + c$$, where $$a$$, $$b$$, and $$c$$ are constants and $$a \neq 0$$. The graph of a quadratic function is a parabola, which opens either upward or downward depending on the sign of the coefficient $$a$$. Understanding the characteristics of quadratic functions is essential for graphing them accurately and analyzing their properties.
Range: In mathematics, the range refers to the set of all possible output values of a function, derived from its input values. It is crucial to understanding how functions behave, as it helps determine which values can actually be produced. The range is often influenced by the nature of the function and the domain from which inputs are selected, making it a key concept when exploring relationships and mappings in various contexts.
Reflection: Reflection is a transformation that produces a mirror image of a geometric figure across a specific line known as the line of reflection. This concept is crucial in understanding the properties of functions, as it relates to symmetry, even and odd functions, and the graphical representation of mathematical relationships.
Slope: Slope is a measure of the steepness or incline of a line on a graph, represented as the ratio of the vertical change to the horizontal change between two points. It helps in understanding how a function behaves and can indicate whether a function is increasing or decreasing. Slope is fundamental in identifying linear relationships and analyzing the rates of change in various contexts.
Stretch: Stretch refers to the transformation of a graph that alters its vertical or horizontal dimensions, making it wider or narrower without changing its overall shape. This transformation impacts the behavior of the function, particularly how steep or flat it appears, which is crucial for understanding the properties of the graph and the function it represents.
Translation: Translation refers to the shifting of a graph or geometric figure in a specific direction without changing its shape, size, or orientation. This transformation is essential for understanding how functions behave when altered, as it helps illustrate how the entire graph moves horizontally or vertically based on changes in the function's equation. Recognizing translation is key to analyzing the properties of functions and their graphs.
Using a graphing calculator: Using a graphing calculator refers to the process of employing a handheld electronic device that can plot graphs, solve equations, and analyze mathematical functions. These calculators help visualize the behavior of functions, allowing users to better understand properties such as intercepts, asymptotes, and transformations. By utilizing this technology, students can enhance their comprehension of mathematical concepts and explore complex functions in a more interactive way.
Vertex: The vertex is a significant point on a graph, often representing the highest or lowest point of a parabola in the context of quadratic functions. It serves as a crucial feature that helps in determining the overall shape and position of the graph, providing insights into the behavior of the function. The vertex is not just a point but a key factor in analyzing properties like maximum or minimum values, axis of symmetry, and intercepts.
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