Graphs of Functions

Definition and Understanding of a Complete Graph

A complete graph of a function is a visual representation that shows all relevant features of the function, including its domain, range, intercepts, and any asymptotes or discontinuities. In College Algebra, understanding how to interpret and construct complete graphs is essential for analyzing functions.

• Key Point: A complete graph should accurately reflect the behavior of the function over its entire domain.

• Key Point: Features such as intercepts, maxima, minima, and asymptotes should be clearly indicated.

• Example: The graph of shows intercepts and turning points that are important for understanding the function's behavior.

Graphing with a Calculator

Techniques for Graphing Functions

Graphing calculators, such as the TI-84, are powerful tools for visualizing functions. They allow students to input equations and view their graphs using different window settings to explore various aspects of the function.

• Standard Window: The default viewing window on most calculators, typically set to , , , .

• Custom Window: Adjusting the window settings (e.g., , , , ) can help reveal more details about the graph, such as intercepts or extreme values.

• Key Point: Choosing an appropriate window is crucial for accurately displaying the function's important features.

• Example: Graph using both standard and custom windows to compare which provides a better view of the graph's behavior.

Modeling Real-World Problems with Functions

Cost-Benefit Model for Pollution Removal

Functions can be used to model real-world scenarios, such as the cost of removing pollution from drinking water. Understanding how to interpret and analyze these models is a key skill in College Algebra.

• Model Equation: The cost (in dollars) of removing of pollution is given by:

• Key Point: The denominator indicates that as approaches 100, the cost increases dramatically, reflecting the difficulty of removing all pollution.

• Application: Use the restriction on to determine the valid domain for the function (i.e., ).

• Example: Find the cost when by substituting into the formula:

dollars

• Interpretation: The point on the graph represents the cost to remove 90% of the pollution.

Data Modeling and Scatterplots

Projecting and Graphing Real-World Data

College Algebra often involves modeling data and creating scatterplots to visualize trends. The following table shows projected numbers of U.S. adults with diabetes (in thousands) for selected years.

• Key Point: To analyze trends, align the data so that the x-values represent years since 2015 (e.g., 2020 becomes 5, 2025 becomes 10, etc.).

• Scatterplot: Enter the aligned data into the calculator and use the scatterplot function to visualize the relationship between year and number of adults with diabetes.

• Application: Scatterplots help identify patterns, trends, and possible models (such as linear or exponential growth).

• Example: Creating a scatterplot on the TI-84 involves entering the x-values (years since 2015) in L1 and the y-values (thousands of adults) in L2, then using the STAT PLOT feature.

Calculator Skills for College Algebra

Using the TI-84 for Graphs and Data Analysis

Mastery of the TI-84 calculator is essential for success in College Algebra. Key skills include graphing functions, adjusting window settings, using the TABLE feature to find specific values, and creating scatterplots from data.

• Graphing Functions: Enter the equation in the Y= menu and adjust the window as needed.

• Finding Values: Use the TABLE feature to find the output for a given input (e.g., cost when ).

• Creating Scatterplots: Enter data into lists and use STAT PLOT to visualize the data.

• Key Point: Accurate calculator use supports deeper understanding of algebraic concepts and real-world applications.