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Which of the following are graphs of functions? Justify your answer for those graphs that are functions as well as those that are not. (graphs in pic below)

Which of the following are graphs of functions? Justify your answer for those graphs that are functions as well as those that are not. (graphs in pic below)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Which of the following are graphs of functions? Justify your answer for those graphs that are functions as well as those that are not. (graphs in pic below)

### Types of Graphs and Their Characteristics Understanding the different types of graphs and their characteristics is essential in the study of mathematics. Below are examples of common graph forms and a brief description of each. #### Diagram Description 1. **Panel a:** - The graph displays a parabola that opens upwards. - This shape is representative of a quadratic function \( f(x) = ax^2 + bx + c \) where \( a > 0 \). 2. **Panel b:** - This graph shows a curve characteristic of a cubic function. - It demonstrates the shape of \( f(x) = x^3 \), which has a peculiar S-like structure. 3. **Panel c:** - The graph illustrates an absolute value function which forms a V-shape. - It represents \( f(x) = |x| \), which has a distinct cusp at the origin. 4. **Panel d:** - This graph is that of a linear function, showing two intersecting straight lines. - It indicates a system of linear equations or inequalities. ### Graph Analysis - **Quadratic Function (Panel a):** - The vertex of the parabola is at the minimum point. - This type of graph reflects symmetrical behavior around its axis of symmetry and demonstrates how variable changes are squared. - **Cubic Function (Panel b):** - The graph passes through the origin and shows two inflection points. - With one real root at the origin and a change in concavity, the cubic function's graph illustrates polynomial behavior of degree 3. - **Absolute Value Function (Panel c):** - The V-shape graph is characterized by a clear vertex at the origin indicating the absolute minimum. - Both arms of the graph exhibit linear behavior indicating the positive range output of the absolute value. - **Linear Function (Panel d):** - The straight lines crossing indicates a solution set for a system of equations. - The slopes of the lines dictate the angles at which they intersect. These various graph types are foundational in mathematics, from algebra to advanced calculus, offering visual insight into the behavior of different types of functions and their corresponding equations. Understanding these graphs is crucial for solving equations, analyzing function behavior, and interpreting mathematical data effectively.
Transcribed Image Text:### Types of Graphs and Their Characteristics Understanding the different types of graphs and their characteristics is essential in the study of mathematics. Below are examples of common graph forms and a brief description of each. #### Diagram Description 1. **Panel a:** - The graph displays a parabola that opens upwards. - This shape is representative of a quadratic function \( f(x) = ax^2 + bx + c \) where \( a > 0 \). 2. **Panel b:** - This graph shows a curve characteristic of a cubic function. - It demonstrates the shape of \( f(x) = x^3 \), which has a peculiar S-like structure. 3. **Panel c:** - The graph illustrates an absolute value function which forms a V-shape. - It represents \( f(x) = |x| \), which has a distinct cusp at the origin. 4. **Panel d:** - This graph is that of a linear function, showing two intersecting straight lines. - It indicates a system of linear equations or inequalities. ### Graph Analysis - **Quadratic Function (Panel a):** - The vertex of the parabola is at the minimum point. - This type of graph reflects symmetrical behavior around its axis of symmetry and demonstrates how variable changes are squared. - **Cubic Function (Panel b):** - The graph passes through the origin and shows two inflection points. - With one real root at the origin and a change in concavity, the cubic function's graph illustrates polynomial behavior of degree 3. - **Absolute Value Function (Panel c):** - The V-shape graph is characterized by a clear vertex at the origin indicating the absolute minimum. - Both arms of the graph exhibit linear behavior indicating the positive range output of the absolute value. - **Linear Function (Panel d):** - The straight lines crossing indicates a solution set for a system of equations. - The slopes of the lines dictate the angles at which they intersect. These various graph types are foundational in mathematics, from algebra to advanced calculus, offering visual insight into the behavior of different types of functions and their corresponding equations. Understanding these graphs is crucial for solving equations, analyzing function behavior, and interpreting mathematical data effectively.
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