follow the seven - step strategy to graph the following rational function. f(x)=\\frac{4x^{2}}{x^{2}+4} to…

follow the seven - step strategy to graph the following rational function. f(x)=\\frac{4x^{2}}{x^{2}+4} to graph the function, find determine the symmetry of the graph of f. choose the correct answer below. y - axis symmetry origin symmetry neither y - axis symmetry nor origin symmetry what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is (type an integer or a simplified fraction.) b. there is no y - intercept. what are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is/are (type an integer or a simplified fraction. use a comma to separate answers as needed.) b. there is no x - intercept. find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the equation(s) of the vertical asymptote(s) is/are (type an equation. use a comma to separate answers as needed.) b. there is no vertical asymptote. find the horizontal asymptote(s). type the equation(s) of the horizontal asymptote(s). select the correct choice below and fill in any answer boxes within your choice. a. the equation(s) of the horizontal asymptote(s) is/are (type an equation. use a comma to separate answers as needed.) b. there is no horizontal asymptote. plot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of x. x: - 5, - 3, 1, 8 f(x)=\\frac{4x^{2}}{x^{2}+4} (simplify your answers.) use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below. a. b. c. d.

follow the seven - step strategy to graph the following rational function. f(x)=\\frac{4x^{2}}{x^{2}+4} to graph the function, find determine the symmetry of the graph of f. choose the correct answer below. y - axis symmetry origin symmetry neither y - axis symmetry nor origin symmetry what is the y - intercept? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the y - intercept is (type an integer or a simplified fraction.) b. there is no y - intercept. what are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the x - intercept(s) is/are (type an integer or a simplified fraction. use a comma to separate answers as needed.) b. there is no x - intercept. find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the equation(s) of the vertical asymptote(s) is/are (type an equation. use a comma to separate answers as needed.) b. there is no vertical asymptote. find the horizontal asymptote(s). type the equation(s) of the horizontal asymptote(s). select the correct choice below and fill in any answer boxes within your choice. a. the equation(s) of the horizontal asymptote(s) is/are (type an equation. use a comma to separate answers as needed.) b. there is no horizontal asymptote. plot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of x. x: - 5, - 3, 1, 8 f(x)=\\frac{4x^{2}}{x^{2}+4} (simplify your answers.) use the information obtained in the previous steps to graph the function between and beyond the vertical asymptotes. choose the correct graph below. a. b. c. d.

Answer

Explanation:

Step1: Check for symmetry

Replace (x) with (-x) in (f(x)=\frac{4x^{2}}{x^{2}+4}). We get (f(-x)=\frac{4(-x)^{2}}{(-x)^{2}+4}=\frac{4x^{2}}{x^{2}+4}=f(x)). So, the function has y - axis symmetry.

Step2: Find y - intercept

Set (x = 0) in (y=\frac{4x^{2}}{x^{2}+4}). Then (y=\frac{4\times0^{2}}{0^{2}+4}=0). So the y - intercept is (0).

Step3: Find x - intercept

Set (y = 0) in (y=\frac{4x^{2}}{x^{2}+4}). Then (4x^{2}=0), which gives (x = 0). So the x - intercept is (0).

Step4: Find vertical asymptotes

Set the denominator (x^{2}+4 = 0). But (x^{2}+4>0) for all real (x) since (x^{2}\geq0) for all real (x), so there is no vertical asymptote.

Step5: Find horizontal asymptotes

Since the degree of the numerator (n = 2) and the degree of the denominator (m=2) and the leading - coefficient of the numerator (a = 4) and the leading - coefficient of the denominator (b = 1), (\lim_{x\rightarrow\pm\infty}\frac{4x^{2}}{x^{2}+4}=\lim_{x\rightarrow\pm\infty}\frac{4}{1+\frac{4}{x^{2}}}=4). So the horizontal asymptote is (y = 4).

Step6: Evaluate the function at given points

When (x=-5), (f(-5)=\frac{4\times(-5)^{2}}{(-5)^{2}+4}=\frac{4\times25}{25 + 4}=\frac{100}{29}) When (x=-3), (f(-3)=\frac{4\times(-3)^{2}}{(-3)^{2}+4}=\frac{4\times9}{9 + 4}=\frac{36}{13}) When (x = 1), (f(1)=\frac{4\times1^{2}}{1^{2}+4}=\frac{4}{5}) When (x = 8), (f(8)=\frac{4\times8^{2}}{8^{2}+4}=\frac{4\times64}{64 + 4}=\frac{256}{68}=\frac{64}{17})

Answer:

Symmetry: y - axis symmetry y - intercept: A. (0) x - intercept: A. (0) Vertical asymptote: B. There is no vertical asymptote Horizontal asymptote: A. (y = 4) (f(-5)=\frac{100}{29}), (f(-3)=\frac{36}{13}), (f(1)=\frac{4}{5}), (f(8)=\frac{64}{17}) (The graph choice depends on the actual plotted points and the asymptotes and intercepts found above)