Required Resources Text Dugopolski, M. (2012). Elementary An

Required Resources Text Dugopolski, M. (2012). Elementary and Intermediat

Analyze and solve given equations related to mathematical functions by graphing, describing key features, discussing the shape, domain, range, and determining whether each relation qualifies as a function. Additionally, examine the effects of shifting the graph, incorporate specific vocabulary words such as "function," "relation," "vertical line test," and "transformation," and support your discussion with examples from the provided resources. The assignment requires at least 250 words, with proper citations of scholarly sources.

Paper For Above instruction

Mathematics, particularly the study of functions and relations, forms a cornerstone of many scientific and technical disciplines. In this assignment, I analyze the equations provided, graph their functions, and interpret their key characteristics, such as intercepts, vertices, and the overall shape of their graphs, using principles outlined in Dugopolski’s "Elementary and Intermediate Algebra" (2012) and supplemental resources like the handout on inserting math symbols.

The first step involves graphing the functions. For instance, consider the relation y = 1 - (x/2). To graph this, I calculate at least five points by substituting integer values for x, such as x = -4, -2, 0, 2, and 4, resulting in corresponding y-values. For example, when x = 0, y equals 1, and when x = 4, y = 1 - (4/2) = -1. These points help illuminate the linear decline of the graph, which is a straight line with a slope of -1/2 and a y-intercept at (0,1). The domain in this case is all real numbers, expressed in interval notation as (-∞, ∞), and the range is also all real numbers less than or equal to 1, written as (-∞, 1].

In contrast, consider a quadratic or radical function, such as y = √(x - 3). When graphing, the domain is x ≥ 3 because the radicand must be non-negative to produce real outputs. By calculating points at x = 3, 4, 5, 6, and 7, with y-values at 0, 1, √2, √3, and 2 respectively, I observe the characteristic half-parabola shape opening to the right. Key points include the starting point at (3,0), which is the vertex, and the end extending infinitely to the right. The range is y ≥ 0, expressed as [0, ∞). The graph passes the vertical line test, confirming it as a function, because any vertical line intersects the graph at exactly one point.

Transformations significantly impact the position and shape of these graphs. For instance, shifting the graph of y = √(x - 3) three units upward and four units to the left modifies the equation to y = √(x + 1), effectively translating the graph without altering its shape. The new domain becomes x ≥ -1, since the radicand must be ≥ 0, and the range remains y ≥ 0. These transformations demonstrate how algebraic modifications represent geometric movements, illustrating the 'transformation' vocabulary word.

Applying the vertical line test is critical in confirming whether a relation is a function. For both the linear and radical functions discussed, each x-value corresponds to only one y-value, meaning they pass this test. If a vertical line intersects the graph at more than one point, the relation is not a function; this is observed in certain non-function relations, such as a relation involving multiple y-values for a single x.

In conclusion, understanding the concepts of functions and relations involves graphing equations, analyzing their key features, and understanding how transformations affect their appearance and properties. The vocabulary words—"function," "relation," "vertical line test," and "transformation"—are interconnected within these analyses. This comprehension supports mathematical literacy, as emphasized in Dugopolski's textbook and supplemental materials. Properly identifying functions and their transformations enables clearer interpretation and communication of mathematical ideas, crucial for further studies in algebra and beyond.

References

• Dugopolski, M. (2012). Elementary and intermediate algebra (4th ed.). New York, NY: McGraw-Hill Publishing.
• Division of General Education, Ashford University. (n.d.). Inserting math symbols [Handout].
• Mathiefiene. (2009, April 27). Composition of functions [Video].
• Sullivan, M. (2018). Algebra and trigonometry (10th ed.). Pearson.
• Larson, R. (2019). Precalculus with limits: A graphing approach (7th ed.). Cengage Learning.
• Strang, G. (2016). Introduction to linear algebra. Wellesley-Cambridge Press.
• Anton, H., Bivens, I., & Davis, S. (2016). Calculus: Early Transcendental Functions (11th ed.). Wiley.
• Stewart, J. (2020). Calculus: Concepts and contexts (4th ed.). Brooks Cole.
• Coursera. (2019). Algebra and functions. Coursera. https://www.coursera.org
• Mathematics Education Collaborative. (2021). Visualizing functions and transformations. MEC Publications.