Unit 1 Review: Reasoning with Functions and Linear Models

Functions and Their Graphs

Identifying Functions from Graphs

Understanding the graphical representation of functions is essential in College Algebra. Each graph shows how the output (y) changes with respect to the input (x).

• Key Point: A function assigns exactly one output to each input.

• Key Point: The vertical line test can be used to determine if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

• Example: A graph that passes the vertical line test is a function, such as $y = x^2$.

Domain of Functions

Determining the Domain

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

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

• Key Point: For square root functions, the radicand must be non-negative.

• Example: The domain of $f(x) = \frac{1}{x-5}$ is all real numbers except $x = 5$.

Linear Functions and Equations

Finding the Equation of a Line

Linear functions can be written in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Key Point: To find the equation of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$, first calculate the slope: $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Key Point: Substitute one point and the slope into the equation to solve for $b$.

• Example: Through points (1,2) and (3,6): $m = \frac{6-2}{3-1} = 2$, so $y = 2x$.

Average Rate of Change

Calculating Average Rate of Change

The average rate of change of a function $f(x)$ between $x=a$ and $x=b$ is given by:

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

• Key Point: This measures how much the function changes per unit increase in $x$.

• Example: If $f(2) = 5$ and $f(5) = 11$, the average rate of change is $\frac{11-5}{5-2} = 2$.

Linear vs. Exponential Functions

Identifying and Writing Formulas

Functions can be classified as linear or exponential based on their growth patterns.

• Linear Function: Increases or decreases by a constant amount. Formula: $f(x) = mx + b$.

• Exponential Function: Increases or decreases by a constant ratio. Formula: $f(x) = ab^x$.

• Example: If a table shows outputs increasing by the same amount, it is linear; if by the same factor, it is exponential.

Function Notation and Operations

Evaluating and Combining Functions

Function notation allows us to express and evaluate functions efficiently.

• Key Point: $f(a)$ means the value of function $f$ at $x=a$.

• Key Point: Functions can be added, subtracted, multiplied, or composed: $(f \circ g)(x) = f(g(x))$.

• Example: If $f(x) = 2x+1$ and $g(x) = x^2$, then $(f \circ g)(x) = f(g(x)) = 2x^2 + 1$.

Systems of Equations

Solving Systems Algebraically

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Key Point: Common methods include substitution and elimination.

• Key Point: The solution can be interpreted graphically as the intersection point(s) of the equations.

• Example: Solve $x + y = 10$ and $x - y = 2$ by adding to eliminate $y$.

Word Problems and Applications

Modeling with Functions and Systems

Many real-world problems can be modeled using functions and systems of equations.

• Key Point: Translate the problem into mathematical equations using variables.

• Key Point: Use tables, graphs, and equations to represent relationships.

• Example: If a rideshare fare is modeled by $f(x) = 2.5x + 1$, where $x$ is miles, then for 4 miles, $f(4) = 2.5 \times 4 + 1 = 11$.

Simplifying Algebraic Expressions

Properties and Techniques

Simplifying expressions involves combining like terms and applying exponent rules.

• Key Point: Product of powers: $a^m \cdot a^n = a^{m+n}$

• Key Point: Power of a power: $(a^m)^n = a^{mn}$

• Example: $x^2 \cdot x^3 = x^{2+3} = x^5$

Tables: Linear and Exponential Functions

Comparing Function Types

Tables can help distinguish between linear and exponential functions by examining the pattern of change.

x	Linear Function Example	Exponential Function Example
0	2	3
1	5	6
2	8	12
3	11	24

Additional info: Table entries inferred for illustration; actual tables in the file may differ.

Application: Rideshare Fare Modeling

Linear Models in Real Life

Linear functions are often used to model costs that increase at a constant rate, such as rideshare fares.

• Key Point: The cost per mile is the slope of the linear function.

• Key Point: The base fare is the y-intercept.

• Example: If the fare for 1 mile is $5.30 and for 2 miles is $6.85, the cost per mile is $6.85 - 5.30 = 1.55$ per mile.

Summary

This review covers essential topics in College Algebra, including functions, domains, linear and exponential models, systems of equations, and applications. Mastery of these concepts is foundational for further study in mathematics and its applications.