- #1
lamerali
- 62
- 0
Sketch the following function, showing all work needed to sketch each curve.
y = [tex]\frac{1}{3 + x^2}[/tex]
The question is asking for all the work done to find x and y intercepts, vertical, horizontal and slant asymptotes; critical points and points of inflection, i have completed the question but I'm not sure i did it correctly, any guidance is appreciated. My answer is below:
x - intercept: there are no x - intercepts.
y - intercept: sub x = 0 into the function and you get y = 1/3
there are no vertical asymptotes
horizontal asymptote is y = 0 found by dividing every term in the function by x^2
critical points:
to find the critical points find the first derivative
y1 = -2x [tex]^{-3}[/tex]
= [tex]\frac{-2}{x^-3}[/tex]
y1 can never equal zero therefore there are no max or min.
to find the point of infliction find the second derivative
y11 = 6x [tex]^{-4}[/tex]
= [tex]\frac{6}{x^-4}[/tex]
y11 can never equal zero
for x [tex]\geq[/tex] 0, y11 is negative and for x [tex]\leq[/tex] 0, y11 is positive
I have sketched the graph but not added it here i just would like to check if the work i did to get the sketch of the function is correct!
Thanks A LOT!
I really appreciate it! :D
y = [tex]\frac{1}{3 + x^2}[/tex]
The question is asking for all the work done to find x and y intercepts, vertical, horizontal and slant asymptotes; critical points and points of inflection, i have completed the question but I'm not sure i did it correctly, any guidance is appreciated. My answer is below:
x - intercept: there are no x - intercepts.
y - intercept: sub x = 0 into the function and you get y = 1/3
there are no vertical asymptotes
horizontal asymptote is y = 0 found by dividing every term in the function by x^2
critical points:
to find the critical points find the first derivative
y1 = -2x [tex]^{-3}[/tex]
= [tex]\frac{-2}{x^-3}[/tex]
y1 can never equal zero therefore there are no max or min.
to find the point of infliction find the second derivative
y11 = 6x [tex]^{-4}[/tex]
= [tex]\frac{6}{x^-4}[/tex]
y11 can never equal zero
for x [tex]\geq[/tex] 0, y11 is negative and for x [tex]\leq[/tex] 0, y11 is positive
I have sketched the graph but not added it here i just would like to check if the work i did to get the sketch of the function is correct!
Thanks A LOT!
I really appreciate it! :D