For each problem, analyze and oketch a graph of the function thablany $y$-interclepts, relative extrema (Pirst Derivative Test), points of finflection (Second Derivative Test). asymptotes for any rational functions (Cearned in Module 2 Vopic 4), Also find and label either the $x$-intercepts or at least one additional point above the highest critical value and another below the Jowest critical value. Use a graphing utility to verify your results. You must show all work to recelve full credlt. (1 point each)
1.
\[
y=\frac{x^{2}}{x^{2}+3}
\]
Answer
\(\boxed{\text{Final Answer: The y-intercept of the function } y=\frac{x^{2}}{x^{2}+3} \text{ is at the point } (0,0). \text{ The x-intercept is also at the point } (0,0). \text{ The function has a relative extrema at the point } (0,0). \text{ The points of inflection are at the points } (-1, \frac{1}{4}) \text{ and } (1, \frac{1}{4}). \text{ The function has horizontal asymptotes at } y=1 \text{ as } x \text{ approaches infinity and negative infinity.}}\)
Step 1 ：Find the y-intercept by setting x = 0 in the function. The y-intercept is the value of y when x = 0.
Step 2 ：Find the x-intercepts by setting y = 0 in the function. The x-intercepts are the values of x when y = 0.
Step 3 ：Find the relative extrema by taking the first derivative of the function and setting it equal to 0 or undefined. The relative extrema are the points where the first derivative of the function is either 0 or undefined.
Step 4 ：Find the points of inflection by taking the second derivative of the function and setting it equal to 0 or changes sign. The points of inflection are the points where the second derivative of the function is either 0 or changes sign.
Step 5 ：Find the asymptotes by analyzing the behavior of the function as x approaches infinity or negative infinity. The asymptotes are the lines that the graph of the function approaches as x approaches infinity or negative infinity.
Step 6 ：\(y = \frac{x^{2}}{x^{2} + 3}\), the y-intercept is at the point \((0,0)\).
Step 7 ：The x-intercept is also at the point \((0,0)\).
Step 8 ：The function has a relative extrema at the point \((0,0)\).
Step 9 ：The points of inflection are at the points \((-1, \frac{1}{4})\) and \((1, \frac{1}{4})\).
Step 10 ：The function has horizontal asymptotes at \(y=1\) as \(x\) approaches infinity and negative infinity.
Step 11 ：\(\boxed{\text{Final Answer: The y-intercept of the function } y=\frac{x^{2}}{x^{2}+3} \text{ is at the point } (0,0). \text{ The x-intercept is also at the point } (0,0). \text{ The function has a relative extrema at the point } (0,0). \text{ The points of inflection are at the points } (-1, \frac{1}{4}) \text{ and } (1, \frac{1}{4}). \text{ The function has horizontal asymptotes at } y=1 \text{ as } x \text{ approaches infinity and negative infinity.}}\)
