Functions - Linear, quadratic, exponential, and logarithmic functions

Linear functions represent a constant rate of change and form a straight-line graph. Visually, the slope $m$ determines the steepness—a positive slope rises from left to right, while a negative slope falls. The $y$-intercept $c$ is the point $(0, c)$ where the line crosses the vertical axis, and the 'rise over run' staircase pattern describes how to move between points on the line.

Quadratic functions are defined by a squared variable and form a U-shaped curve called a parabola. The graph is perfectly symmetrical about a vertical line called the axis of symmetry. If the leading coefficient $a > 0$, the parabola opens upwards like a cup (minimum point); if $a < 0$, it opens downwards like an umbrella (maximum point).

The vertex of a quadratic function is the highest or lowest point on the graph. In vertex form $y = a(x-h)^2 + k$, the point $(h, k)$ explicitly identifies this turning point. Visually, changing $h$ slides the parabola horizontally, while changing $k$ shifts it vertically.

Exponential functions involve a constant base raised to a variable power, such as $y = ab^x$. These functions model rapid growth (when $b > 1$) or decay (when $0 < b < 1$). The graph features a horizontal asymptote, usually the $x$-axis ($y=0$), which the curve approaches but never touches, creating a characteristic L-shaped curve.

Logarithmic functions are the inverse of exponential functions, written as $y = \log_b(x)$. Graphically, a log function is a reflection of an exponential function across the diagonal line $y = x$. It features a vertical asymptote at $x = 0$ and passes through the point $(1, 0)$, growing very slowly as $x$ increases.

The domain of a function is the set of all possible $x$-values (inputs), while the range is the set of all possible $y$-values (outputs). For linear functions, both are typically all real numbers. For quadratics, the range is restricted by the vertex $y$-coordinate. For the parent exponential function $y = b^x$, the range is $y > 0$, while for the logarithmic function $y = \log_b(x)$, the domain is $x > 0$.

Intercepts are where the graph crosses the axes. To find the $y$-intercept, set $x = 0$ and solve for $y$. To find the $x$-intercepts (also known as roots, zeros, or solutions), set $y = 0$ and solve for $x$. For a quadratic, there can be zero, one, or two $x$-intercepts depending on the discriminant.