Functions - Graphing Functions and their Characteristics

Domain and Range: The domain is the set of all possible input values ($x$-values) for which the function is defined, visually represented as the horizontal 'spread' of the graph. The range is the set of all output values ($y$-values), represented as the vertical 'spread'. For example, the graph of $f(x) = \sqrt{x}$ starts at $(0,0)$ and extends infinitely to the right and upwards, meaning domain $x \geq 0$ and range $y \geq 0$.

Vertical Line Test: A visual method used to determine if a relation is a function. If any vertical line drawn through the graph intersects the curve at more than one point, the relation is not a function because one input $x$ maps to multiple outputs $y$.

Intercepts: The $y$-intercept is the point where the graph crosses the vertical axis (calculated by setting $x = 0$). The $x$-intercepts, also known as roots or zeros, are the points where the graph crosses the horizontal axis (calculated by setting $f(x) = 0$). Visually, these are the 'anchor points' of the sketch.

Asymptotes: These are lines that the graph approaches but never actually reaches. Vertical asymptotes often occur where the denominator of a rational function is zero, creating a 'break' in the graph. Horizontal asymptotes describe the end behavior of the function as $x$ approaches $\infty$ or $-\infty$, appearing as a flat boundary line.

Function Transformations: Shifts and stretches change the position and shape of the parent graph. Adding $k$ to the function ($f(x) + k$) shifts the graph vertically, while subtracting $h$ from the input ($f(x - h)$) shifts it horizontally. A negative sign outside the function ($-f(x)$) reflects the graph across the $x$-axis, flipping it upside down.

Quadratic Characteristics: The graph of a quadratic function $f(x) = ax^2 + bx + c$ is a parabola. It features a vertex, which is the maximum or minimum point where the graph turns. An imaginary vertical line called the axis of symmetry passes through the vertex, dividing the parabola into two identical mirror images.

Inverse Functions and Symmetry: The graph of an inverse function $f^{-1}(x)$ is a reflection of the original function $f(x)$ across the diagonal line $y = x$. If a point $(a, b)$ lies on the original graph, the point $(b, a)$ will lie on the graph of the inverse.

Increasing and Decreasing Intervals: A function is increasing on an interval if, as you move from left to right, the $y$-values go up (the slope is positive). It is decreasing if the $y$-values go down (the slope is negative). Local maximums and minimums occur where the function switches between these states.