Using the concepts of continuity and a limit, describe your graph (a) from Discussion 1 (see attached). Is your function continuous? Why or why not? When the curve is followed from left to right, how could it be described using limits?
2. This discussion re-visits the concept of a function. You will explore the online graphing tool Desmos and graph algebraic functions. Access the attached Desmos User Guide and review all sections except for Regressions. Use Desmos (https://www.desmos.com/calculator ) to graph three assigned functions, as follows: Start by graphing either y = 1/x or y = 1/x^2 and modify that function to create the graphs of three other algebraic functions with the following properties: a. one vertical asymptote (no horizontal asymptote) b. one horizontal and two vertical asymptotes c. one horizontal asymptote (no vertical asymptote). Ensure that the entire function 2c is in the positive halfplane. ANSWER 1 a. Function with one vertical asymptote (no horizontal asymptote): Starting with 𝑦 = 𝑥 2 and modifying it by introducing a polynomial that the power is greater than 2. 1 So, we have 𝑦 = 𝑥 2 + 𝑥 2 b. Function with one horizontal and two vertical asymptotes: Starting with 𝑦 = subtracting 9 from 𝑥 2 in the denominator. 1 So, we have 𝑦 = 𝑥 2 −9 1 and modifying it by 𝑥2 1 c. Function with one horizontal asymptote (no vertical asymptote): Starting with 𝑦 = 𝑥 2 and modifying it by adding 1 to the denominator and introducing x to the numerator and adding one to the whole function because we want the entire function in the positive halfplane. 𝑥 So, we have 𝑦 = 2 + 1 𝑥 +1 3. Using the graph of each function, state the domain and range, and write the equations of the horizontal and vertical asymptotes. ANSWER 1 a. 𝑦 = 𝑥 2 + 𝑥2 From the graph, it can be seen that the domain for the function is (−∞, 0) ∪ (0, +∞) and the range [2, +∞). The vertical asymptote is 𝑥 = 0 and there is no horizontal asymptote because as y grows larger it tends to ∞. 1 b. 𝑦 = 𝑥 2 −9 From the graph, it can be seen that the domain for the function is (−∞, −3) ∪ (−3,3) ∪ (3, +∞) and the range (−∞, 0) ∪ (0, +∞). The vertical asymptotes are 𝑥 = −3 and 𝑥 = 3 and the horizontal asymptote is 𝑦 = 0. 𝑥 c. 𝑦 = 𝑥 2 +1 + 1 From the graph, it can be seen that the domain for the function is (−∞, ∞) and the range [0.5, 1.5]. There is no vertical asymptote because as x grows larger it tends to ∞ and the horizontal asymptote is 𝑦 = 1. 4. In five to ten sentences, explain your rationale for determining the domain, the range, and the asymptotes from the graph. What ideas have you studied previously that were useful in the analysis? ANSWER For the domain, by examining the graph horizontally, I can determine the x-values for which the function is defined. The domain covers all x-values where the graph exists without any discontinuities or undefined points. Also, for range, I vertically scan the graph and this allows me to know the range of y-values that the function can have. The range extends from the lowest point of the graph to the highest, covering all possible y-values. Finally, for the asymptotes, the vertical asymptote(s) are point(s) (vertically) where the function gets infinitely close to but never touches and horizontal asymptote(s) are point(s) (horizontally) where the function gets infinitely close to but never touches. Ideas that I have studied previous that were useful in this analysis are function properties such as domain and range restrictions, asymptotic behavior and also rational functions particularly the behavior near asymptotes and how to determine their existence. For example, if a function has a horizontal asymptote, the degree of the numerator is always less than the degree of the denominator, and if the degree of the numerator is greater than the denominator, then the function doesn’t have a horizontal asymptote.
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