Precalculus Study Guide: Functions, Graphs, and Applications

Functions and Their Graphs

Domain of a Function

The domain of a function is the set of all possible input values (usually x-values) for which the function is defined.

• Key Point: For rational functions, exclude values that make the denominator zero.

• Example: For $f(x) = \frac{3}{x-3.4}$, the domain is all real numbers except $x = 3.4$.

Difference Quotient

The difference quotient is a fundamental concept for understanding rates of change and the basis of derivatives in calculus.

• Definition: The difference quotient for a function $f(x)$ is $\frac{f(x+h) - f(x)}{h}$.

• Example: For $f(x) = x^2 - 4$, compute $f(x+h)$, subtract $f(x)$, and divide by $h$.

Analyzing Graphs of Functions

Graphs provide visual insight into the behavior of functions, including domain, range, intervals of increase/decrease, and symmetry.

• Domain and Range: The domain is the set of x-values shown; the range is the set of y-values the graph attains.

• Intercepts: Points where the graph crosses the axes.

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

• Even/Odd Functions: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• Example: Use the provided graph to answer questions about these properties.

Linear and Quadratic Functions

Piecewise Functions

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

• Key Point: Analyze each piece separately for domain, intercepts, and graphing.

• Example: $f(x) = \begin{cases} 2x-5, & x < 1 \\ x^2+4, & x \geq 1 \end{cases}$

Transformations of Functions

Transformations include shifting, stretching, compressing, and reflecting graphs of functions.

• Shifting: Moves the graph horizontally or vertically.

• Stretching/Compressing: Changes the steepness or width of the graph.

• Reflecting: Flips the graph over an axis.

• Example: $g(x) = 2|x-4|$ is a vertical stretch and horizontal shift of the basic absolute value function.

Graphing Linear Functions

Linear functions have the form $g(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Slope: Rate of change; rise over run.

• Y-intercept: The value of $y$ when $x = 0$.

• Example: $g(x) = -2x - 3$ has a slope of -2 and y-intercept of -3.

Quadratic Functions and Their Properties

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and graph as parabolas.

• Vertex: The highest or lowest point of the parabola.

• Axis of Symmetry: Vertical line through the vertex, $x = -\frac{b}{2a}$.

• Intercepts: Points where the graph crosses the axes.

• Direction: Opens upward if $a > 0$, downward if $a < 0$.

• Example: $f(x) = x^2 - 5x$

Quadratic Formula

The quadratic formula solves $ax^2 + bx + c = 0$ for $x$.

• Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Application: Use to find zeros (roots) of quadratic functions.

Applications of Functions

Linear Models and Break-Even Analysis

Linear functions can model real-world scenarios such as cost, revenue, and break-even points.

• Key Point: Set up equations based on given information and solve for unknowns.

• Example: If the cost per item is $1.50 and total cost for 60 items is $100, set up a linear equation to find the break-even point.

Quadratic Applications: Projectile Motion

Quadratic functions model the height of objects in projectile motion.

• Formula: $h(t) = -16t^2 + vt + s$, where $v$ is initial velocity and $s$ is initial height.

• Key Point: Solve for $t$ when $h(t) = 0$ (object hits the ground) or $h(t) =$ specific value.

• Example: A ball thrown upward with initial velocity; find when it hits the ground and when it is above a certain height.

Summary Table: Function Properties

Function Type	General Form	Key Properties	Example
Linear	$y = mx + b$	Slope, y-intercept, straight line	$y = -2x - 3$
Quadratic	$y = ax^2 + bx + c$	Vertex, axis of symmetry, parabola	$y = x^2 - 5x$
Piecewise	Defined by intervals	Multiple expressions, domain split	$f(x) = \begin{cases} 2x-5, x<1 \\ x^2+4, x\geq1 \end{cases}$

*Additional info: The study guide covers foundational Precalculus topics including functions, their domains and ranges, graphing, transformations, linear and quadratic models, and applications. These are essential for success in early chapters of a Precalculus course.*