Fundamental Functions in Precalculus: Quadratic, Absolute Value, and Linear Functions

Basic Functions in Precalculus

Quadratic Function

The quadratic function is one of the most important types of polynomial functions in precalculus. It is commonly used to model parabolic shapes and appears frequently in algebra and calculus.

• Definition: A quadratic function is any function of the form $f(x) = ax^2 + bx + c$, where $a \neq 0$.

• Standard Example: $f(x) = x^2$

• Graph: The graph of $f(x) = x^2$ is a parabola opening upwards with its vertex at the origin $(0, 0)$.

• Key Properties:

  • Vertex: $(0, 0)$

  • Axis of symmetry: $x = 0$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

• Example: For $x = 3$, $f(3) = 3^2 = 9$.

Absolute Value Function

The absolute value function is a piecewise-defined function that measures the distance of a number from zero on the real number line. It is widely used in mathematics to express magnitude without regard to direction.

• Definition: $y = |x|$

• Piecewise Form:

  • $|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

• Graph: The graph of $y = |x|$ is a "V" shape with its vertex at the origin $(0, 0)$.

• Key Properties:

  • Vertex: $(0, 0)$

  • Domain: $(-\infty, \infty)$

  • Range: $[0, \infty)$

  • Symmetry: Even function (symmetric about the $y$-axis)

• Example: For $x = -4$, $y = |-4| = 4$.

Linear Function

The linear function is the simplest type of polynomial function and represents a straight line in the coordinate plane. Linear functions are fundamental in algebra and are used to model constant rates of change.

• Definition: $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

• Example from materials: $y = -\frac{1}{2}x + 2$

• Graph: The graph is a straight line with slope $-\frac{1}{2}$ and $y$-intercept $2$.

• Key Properties:

  • Slope: $-\frac{1}{2}$ (the line decreases as $x$ increases)

  • $y$-intercept: $2$ (the point $(0, 2)$)

  • Domain: $(-\infty, \infty)$

  • Range: $(-\infty, \infty)$

• Example: For $x = 4$, $y = -\frac{1}{2}(4) + 2 = -2 + 2 = 0$.

Comparison Table: Key Properties of Functions

Function	Equation	Graph Shape	Domain	Range	Vertex / Intercept
Quadratic	$y = x^2$	Parabola	$(-\infty, \infty)$	$[0, \infty)$	Vertex: $(0, 0)$
Absolute Value	$y = |x|$	"V" shape	$(-\infty, \infty)$	$[0, \infty)$	Vertex: $(0, 0)$
Linear	$y = -\frac{1}{2}x + 2$	Straight line	$(-\infty, \infty)$	$(-\infty, \infty)$	$y$-intercept: $(0, 2)$

Additional info: These functions are foundational in precalculus and are used to introduce concepts such as transformations, graphing, and function analysis.