Is My Graphing Strategy and Calculations for y=(4x)/(x^2+1) Correct?

[donjt81]

I have this problem and I really need someone else to look at it and see if the answers are correct

problem: use the graphing strategy to sketch the graph of y=(4x)/(x^2+1). check for domain values, intercepts, critical values, interval where the function is increasing and where it is decreasing. Then graph it. please use sign charts.

domain: (-infinity, +infinity)

x intercept: is where y = 0 : (0, 0)
y intercept: is where x = 0 : (0, 0)

critical value
y' = (4-4x^2)/(x^2+1)^2 = 0
solving for x gives us x = +- 1
so the critical values are x = 1 and x = -1

using the sign charts
interval where function is increasing: (-1, 1)
interval where function is decreasing: (-infinity, -1) & (1, infinity)

Can someone verify these answers please.

 

[mighty2000]

it is always important to have someone else review and verify our work. I would be happy to take a look at your problem and provide some feedback.

First, let's check the domain of the function. The given function is a rational function, which means the denominator cannot equal to 0. In this case, the denominator is x^2+1, so the domain is all real numbers.

Next, let's find the x and y intercepts. You have correctly identified that the y intercept is (0,0). However, for the x intercept, we need to solve for y = 0, not x = 0. So, we have (0,0) as the y intercept and no x intercepts.

Moving on to the critical values, I see that you have correctly found the values of x where the derivative is equal to 0. However, we also need to check the behavior of the function at these points. By plugging in values slightly greater and slightly less than the critical values into the original function, we can see that the function is increasing on the interval (-1,1) and decreasing on the intervals (-infinity,-1) and (1,infinity). This information should be included in the sign charts.

Lastly, let's graph the function. We can start by plotting the y intercept at (0,0). Then, using the information from the sign charts, we can draw a curve that is increasing on the interval (-1,1) and decreasing on the intervals (-infinity,-1) and (1,infinity). The graph should approach the x axis as x approaches infinity and -infinity.

I hope this helps to verify your answers. Remember, it is always a good idea to have someone else review your work to catch any mistakes or provide additional insights. Keep up the good work!