Exam 3

Functions and Graphs

Basic Function Graphs

Understanding the graphical representation of functions is a foundational skill in precalculus. The most common basic functions include linear, quadratic, cubic, absolute value, and square root functions. Each has a distinct graph and set of properties.

• Linear Function: The graph of $f(x) = x$ is a straight line passing through the origin with a slope of 1.

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

• Cubic Function: The graph of $f(x) = x^3$ passes through the origin and has point symmetry about the origin.

• Absolute Value Function: The graph of $f(x) = |x|$ forms a 'V' shape, with the vertex at the origin.

• Square Root Function: The graph of $f(x) = \sqrt{x}$ starts at the origin and increases slowly to the right.

Example: The graph of $f(x) = \sqrt{x}$ is only defined for $x \geq 0$ and passes through points (0,0), (1,1), (4,2).

Graph Transformations

Transformations allow us to shift, stretch, compress, or reflect the graphs of functions. The most common transformations are:

• 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 the graph vertically if $|a| > 1$, compresses if $0 < |a| < 1$.

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

Example: The graph of $f(x) = \sqrt{x}$ reflected over the x-axis is $f(x) = -\sqrt{x}$, which opens downward.

Graphing Practice

Many precalculus questions involve matching equations to their graphs or sketching graphs based on transformations. Practice involves:

• Identifying the parent function.

• Applying transformations step by step.

• Plotting key points and noting changes in shape or position.

Example: Sketch the graph of $f(x) = \sqrt{x}$ and then $f(x) = -\sqrt{x}$ on the same axes to compare their shapes.

Summary Table: Common Parent Functions and Their Graphs

Function	Equation	Graph Shape	Domain	Range
Linear	$f(x) = x$	Straight line	$(-\infty, \infty)$	$(-\infty, \infty)$
Quadratic	$f(x) = x^2$	Parabola	$(-\infty, \infty)$	$[0, \infty)$
Cubic	$f(x) = x^3$	S-shaped curve	$(-\infty, \infty)$	$(-\infty, \infty)$
Absolute Value	$f(x) = |x|$	V-shape	$(-\infty, \infty)$	$[0, \infty)$
Square Root	$f(x) = \sqrt{x}$	Half parabola (rightward)	$[0, \infty)$	$[0, \infty)$

Additional info: The original images contained blank coordinate grids and partial function notation, suggesting graphing and transformation practice. The above notes expand on these concepts for clarity and completeness.