Precalculus Study Notes: Functions, Graphs, and Rates of Change

Functions and Their Representations

Definition of a Function

A function is a relation that assigns to each element in a set called the domain exactly one element in a set called the range.

• Notation: If $f$ is a function, $f(x)$ denotes the output when the input is $x$.

• Example: $f(x) = 2x + 3$ assigns to each $x$ the value $2x + 3$.

Ways to Represent Functions

• Algebraic: Using an equation, e.g., $y = x^2$.

• Graphical: Plotting points $(x, f(x))$ on a coordinate plane.

• Tabular: Listing input-output pairs in a table.

• Verbal: Describing the relationship in words.

Linear Functions

A linear function has the form $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope ($m$): Measures the rate of change; calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Y-intercept ($b$): The value of $f(x)$ when $x = 0$.

• Example: $f(x) = -4x + 3$ is a linear function with slope $-4$ and y-intercept $3$.

Graphing and Analyzing Functions

Graphing Linear Functions

• Plot the y-intercept $(0, b)$.

• Use the slope $m$ to find another point: from $(0, b)$, move up/down $m$ units and right 1 unit.

• Draw a straight line through the points.

Interpreting Graphs

• Increasing Function: The graph rises as $x$ increases.

• Decreasing Function: The graph falls as $x$ increases.

• Constant Function: The graph is a horizontal line; $f(x) = c$ for all $x$.

• Example: $f(x) = 2$ is constant; $f(x) = x$ is increasing; $f(x) = -x$ is decreasing.

Key Features of Graphs

• Intercepts: Points where the graph crosses the axes.

• Domain: All possible input values ($x$-values).

• Range: All possible output values ($y$-values).

• Symmetry: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

Rates of Change

Average Rate of Change

The average rate of change of a function $f$ from $x = a$ to $x = b$ is given by:

$\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

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

• Example: For $f(x) = x^2$, the average rate of change from $x = 1$ to $x = 3$ is $\frac{9 - 1}{3 - 1} = 4$.

Instantaneous Rate of Change

The instantaneous rate of change at a point is the slope of the tangent line at that point. In calculus, this is the derivative, but in precalculus, we estimate it by considering values very close to the point of interest.

• For linear functions, the instantaneous rate of change is constant and equal to the slope $m$.

• For nonlinear functions, it varies depending on $x$.

Transformations of Functions

Types of Transformations

• Vertical Shifts: $f(x) + k$ shifts the graph up by $k$ units if $k > 0$, down if $k < 0$.

• Horizontal Shifts: $f(x - h)$ shifts the graph right by $h$ units if $h > 0$, left if $h < 0$.

• Reflections: $-f(x)$ reflects the graph over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if $|a| > 1$, compresses if $0 < |a| < 1$.

• Horizontal Stretch/Compression: $f(bx)$ compresses horizontally if $|b| > 1$, stretches if $0 < |b| < 1$.

Example of Transformation

• Given $f(x) = x^2$, $g(x) = (x - 2)^2 + 3$ is $f(x)$ shifted right by 2 units and up by 3 units.

Special Types of Functions

Piecewise Functions

A piecewise function is defined by different expressions for different intervals of the domain.

• Example:

  • $f(x) = x$ if $x < 0$

  • $f(x) = x^2$ if $x \geq 0$

Absolute Value Function

The absolute value function is defined as $f(x) = |x|$.

• Graph is V-shaped, with vertex at the origin.

• Can be written as a piecewise function:

  • $f(x) = x$ if $x \geq 0$

  • $f(x) = -x$ if $x < 0$

Tables: Summary of Function Types and Properties

Function Type	General Form	Graph Shape	Rate of Change
Linear	$f(x) = mx + b$	Straight line	Constant ($m$)
Quadratic	$f(x) = ax^2 + bx + c$	Parabola	Variable
Absolute Value	$f(x) = |x|$	V-shape	Piecewise constant
Piecewise	Varies	Varies	Varies

Additional info:

• Some content was inferred from context due to unclear handwriting and fragmented notes. Standard definitions and examples were added for completeness.

• Key terms such as "rate of change," "domain," and "range" were expanded for clarity.