Functions and Their Graphs: Core Concepts and Transformations in Precalculus

Functions and Graphs

Relations and Functions

A relation is any set of ordered pairs. A function is a special type of relation in which each input (x-value) is paired with exactly one output (y-value). The domain of a function is the set of all possible input values, and the range is the set of all possible output values.

• Set Notation: Domains and ranges are often written in set notation, e.g., {x | x ≥ 0}.

• Example: The relation {(−2, 0), (1, 1), (2, −1), (1, 3)} is not a function because the input 1 is paired with two different outputs (1 and 3).

Increasing, Decreasing, and Constant Intervals

The behavior of a function can be described by intervals where it is increasing, decreasing, or constant:

• Increasing: A function is increasing on an interval if, as x increases, f(x) also increases.

• Decreasing: A function is decreasing on an interval if, as x increases, f(x) decreases.

• Constant: A function is constant on an interval if f(x) remains the same as x increases.

Even and Odd Functions

Functions can be classified based on their symmetry:

• Even Function: $f(-x) = f(x)$ for all x in the domain. Even functions are symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all x in the domain. Odd functions have 180° rotational symmetry about the origin.

• Neither: If a function does not satisfy either property, it is neither even nor odd.

• Example: $f(x) = x^3 - x$ is an odd function because $f(-x) = -x^3 + x = -f(x)$.

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain. To evaluate a piecewise function, determine which interval the input belongs to and use the corresponding formula.

• Example: $f(x) = \begin{cases} 3, & x < -1 \\ -x^2, & x \geq -1 \end{cases}$

• To find $f(-5)$, use the first piece: $f(-5) = 3$.

Analyzing Graphs of Functions

Key Features of Graphs

When analyzing the graph of a function, identify the following:

• Domain: All x-values for which the function is defined.

• Range: All y-values the function attains.

• x-intercepts: Points where the graph crosses the x-axis (set y = 0).

• y-intercepts: Points where the graph crosses the y-axis (set x = 0).

• Intervals of Increase/Decrease: Where the function rises or falls as x increases.

• Relative Maxima/Minima: Local highest or lowest points on the graph.

• Even/Odd/Neither: Symmetry properties as described above.

Graphing Piecewise Functions

To graph a piecewise function, plot each piece on its specified interval, paying attention to open or closed endpoints.

• Example: $f(x) = \begin{cases} \frac{1}{2}x + 2, & x \leq -1 \\ x + 3, & -1 < x < 1 \\ -x + 1, & x \geq 1 \end{cases}$

Linear Functions and Slope

Slope and Rate of Change

The slope of a line measures its steepness and is calculated as:

• $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Parallel lines have the same slope.

• Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$.

Forms of Linear Equations

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Horizontal Line: $y = b$ (slope = 0)

• Vertical Line: $x = k$ (slope undefined)

Finding Intercepts

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

Average Rate of Change

Definition and Application

The average rate of change of a function between $x_1$ and $x_2$ is the slope of the secant line connecting the points $(x_1, f(x_1))$ and $(x_2, f(x_2))$:

• $\text{Average Rate of Change} = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

• Secant Line: The line passing through these two points.

• Example: If a man's height is 57 inches at age 13 and 76 inches at age 18, the average rate of change is $\frac{76 - 57}{18 - 13} = 3.8$ inches per year.

Difference Quotient

Definition

The difference quotient is a formula that gives the average rate of change of a function over an EP interval of length $h$:

• $\frac{f(x + h) - f(x)}{h}$, for $h \neq 0$

• This is foundational for calculus, as it leads to the concept of the derivative.

Transformations of Functions

Reflections

• Reflection about the x-axis: $y = -f(x)$ reflects the graph over the x-axis (all y-values become their opposites).

• Reflection about the y-axis: $y = f(-x)$ reflects the graph over the y-axis (all x-values become their opposites).

Vertical and Horizontal Shifts

• Vertical Shift Up: $y = f(x) + c$ shifts the graph up by $c$ units.

• Vertical Shift Down: $y = f(x) - c$ shifts the graph down by $c$ units.

• Horizontal Shift Left: $y = f(x + c)$ shifts the graph left by $c$ units.

• Horizontal Shift Right: $y = f(x - c)$ shifts the graph right by $c$ units.

Vertical Stretching and Shrinking

• Vertical Stretch: $y = c f(x)$, $c > 1$ stretches the graph vertically (y-values are multiplied by $c$).

• Vertical Shrink: $y = c f(x)$, $0 < c < 1$ shrinks the graph vertically (y-values are multiplied by $c$).

Horizontal Stretching and Shrinking

• Horizontal Shrink: $y = f(c x)$, $c > 1$ shrinks the graph horizontally (x-values are divided by $c$).

• Horizontal Stretch: $y = f(c x)$, $0 < c < 1$ stretches the graph horizontally (x-values are divided by $c$).

Summary Table of Transformations

The following table summarizes the main types of function transformations:

To Graph:	Draw the Graph of f and:	Changes in the Equation of y = f(x)
Vertical shifts	Raise/lower the graph by c units	c is added/subtracted to f(x)
Horizontal shifts	Shift left/right by c units	x is replaced with x + c or x - c
Reflection about x-axis	Reflect about x-axis	f(x) is multiplied by -1
Reflection about y-axis	Reflect about y-axis	x is replaced with -x
Vertical stretching/shrinking	Multiply y-coordinates by c	f(x) is multiplied by c
Horizontal stretching/shrinking	Divide x-coordinates by c	x is replaced with c x

Examples of Transformations

• Given: $f(x) = x^2$

• Transformed: $g(x) = -(x + 2)^2 - 8$ (reflect over x-axis, shift left 2, down 8)

• Given: $f(x) = \sqrt{x}$

• Transformed: $g(x) = 2\sqrt{x - 1} + 3$ (shift right 1, stretch vertically by 2, up 3)

• Given: $f(x) = |x|$

• Transformed: $g(x) = |3 - x| - 1$ (reflect over y = 1.5, shift down 1)