Linear and Nonlinear Functions: Key Concepts in College Algebra

Linear Functions and Constant Functions

Definition and General Form

A linear function is a function represented by the equation $f(x) = mx + b$, where m and b are constants.

• m is the slope (rate of change) of the line.

• b is the y-intercept, the point where the line crosses the y-axis.

Example:

• $f(x) = -5 + 2x$

• $f(x) = \frac{1}{2} - 3x$ (Here, $m = -3$, $b = \frac{1}{2}$)

Constant Functions

If the value of m is zero, then $f(x) = b$ is a constant function. Its graph is a horizontal line, and the rate of change is zero.

• Example: $f(x) = 5$

• m = 0, so the function does not change as x changes.

Slope of a Linear Function

Definition of Slope

The slope (from the French word "monter," meaning "to ascend") of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive slope: Line rises from left to right ($m > 0$).

• Negative slope: Line falls from left to right ($m < 0$).

• Zero slope: Horizontal line ($m = 0$).

• Undefined slope: Vertical line (not a function).

Example: Find the slope between points (2,1) and (-3,4):

• $m = \frac{1-4}{2-(-3)} = \frac{-3}{5}$

Graphical Interpretation

• Rise over run: Slope is the ratio of vertical change to horizontal change between two points.

• Graph of a function: Each x-value has only one y-value (passes the vertical line test).

Zeros (x-intercepts) of a Function

The zero or x-intercept of a function is the value of x where $f(x) = 0$.

• If a graph crosses the x-axis at $(a, 0)$, then $a$ is a zero of the function.

• The y-intercept is where the graph crosses the y-axis ($x = 0$).

Example: For $f(x) = 2x - 4$:

• Set $f(x) = 0$: $2x - 4 = 0 \implies x = 2$ (x-intercept: (2,0))

• y-intercept: $f(0) = -4$ (0, -4)

Linear vs. Nonlinear Functions

Identifying Linearity

• Linear functions have a constant rate of change (slope).

• Nonlinear functions have a variable rate of change; their graphs are not straight lines.

Example Table:

x	g(x)
-3	1
-1	-1
1	-3
3	-5

Check if the change in y over the change in x is constant to determine linearity.

• Example: $h(x) = x^2 + 1$ is nonlinear because the output changes by different amounts as x increases.

Basic Linear and Nonlinear Functions

Below are some fundamental functions, their graphs, domains, and ranges.

Function	Graph Description	Domain	Range
$f(x) = x$	Diagonal line through origin	$(-\infty, \infty)$	$(-\infty, \infty)$
$f(x) = x^2$	Parabola opening upward	$(-\infty, \infty)$	$[0, \infty)$
$f(x) = x^3$	S-shaped curve through origin	$(-\infty, \infty)$	$(-\infty, \infty)$
$f(x) = \sqrt{x}$	Curve starting at (0,0), increasing	$[0, \infty)$	$[0, \infty)$
$f(x) = |x|$	V-shaped graph	$(-\infty, \infty)$	$[0, \infty)$

Increasing and Decreasing Functions

A function $f$ is increasing on an interval if, as $x$ increases, $f(x)$ increases. It is decreasing if, as $x$ increases, $f(x)$ decreases.

• Use interval notation to describe where a function is increasing or decreasing.

• Endpoints are not included in open intervals.

Example: For a function that increases on $(-2, 2)$ and decreases on $(-\infty, -2) \cup (2, \infty)$, the intervals are determined by analyzing the graph from left to right.

Average Rate of Change

The average rate of change of a function $f(x)$ from $a$ to $b$ is:

$\frac{f(b) - f(a)}{b - a}$

• This is the slope of the secant line connecting $(a, f(a))$ and $(b, f(b))$.

• For nonlinear functions, this gives the average change over the interval.

Example: A car's speed changes from 30 mph at 2 seconds to 24 mph at 5 seconds. The average rate of change is:

$\frac{24 - 30}{5 - 2} = \frac{-6}{3} = -2$ mph per second

Application Example: Torricelli's Law

A cylindrical tank contains 100 gallons of water. The amount of water remaining after $x$ minutes is $A(x) = 100 \left(1 - \frac{x}{5}\right)^2$.

• Calculate the average rate of change of $A$ from 1 to 1.5 minutes and from 2 to 2.5 minutes.

• Interpret the results and compare the two rates.

Key Point: The average rate of change may differ over different intervals for nonlinear functions.

*Additional info: The notes include both symbolic and graphical representations, as well as practical examples, to reinforce understanding of linear and nonlinear functions, their properties, and applications in College Algebra.*