3.4 Graphing Functions and Their Properties

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

Understanding these properties helps us interpret and predict function behavior. We can identify increasing or decreasing 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
• Linear function $[f(x)](https://www.fiveableKeyTerm:f(x)) = mx + b$ has y-intercept at (0, b)
• Quadratic function $f(x) = ax^2 + bx + c$ has y-intercept at (0, c)

Function Types and Behavior

Monotonicity and Constancy

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

Symmetry in Functions

• Even function symmetric about the y-axis
• Satisfies the condition $f(-x) = f(x)$ for all x in the domain
• Graph remains unchanged when reflected over the y-axis
• Odd function symmetric about the origin
• Satisfies the condition $f(-x) = -f(x)$ 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: $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$
• 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 reflection over a line (even functions)
• Asymptote 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, $\lim_{x \to a} f(x)$ exists, and $\lim_{x \to a} f(x) = f(a)$
• 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: $f(x) + k$
• Horizontal shifts move the graph left or right: $f(x + h)$
• Vertical stretches or compressions: $af(x)$, where |a| > 1 stretches and 0 < |a| < 1 compresses
• Horizontal stretches or compressions: $f(bx)$, where 0 < |b| < 1 stretches and |b| > 1 compresses
• Reflections over x-axis: $-f(x)$, over y-axis: $f(-x)$