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Plotting a function graph/ -(64x)/(x²-16)²

Graphing y = -(64x)/(x²-16)²

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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         -64*x   
f(x) = ----------
                2
       / 2     \ 
       \x  - 16/
f(x)=(−1)64x(x2−16)2f{\left(x \right)} = \frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} 
f = (-64*x)/(x^2 - 16)^2
The graph of the function
02468-8-6-4-2-1010-1000010000
The domain of the function
The points at which the function is not precisely defined:
x1=−4x_{1} = -4
x2=4x_{2} = 4
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
(−1)64x(x2−16)2=0\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
Numerical solution
x1=37301.7544113321x_{1} = 37301.7544113321
x2=39848.2926695335x_{2} = 39848.2926695335
x3=−42262.9925603031x_{3} = -42262.9925603031
x4=0x_{4} = 0
x5=42394.3844016916x_{5} = 42394.3844016916
x6=−39716.8799025336x_{6} = -39716.8799025336
x7=−38868.0798379301x_{7} = -38868.0798379301
x8=−38019.2265141401x_{8} = -38019.2265141401
x9=41545.7317146792x_{9} = 41545.7317146792
x10=−41414.3333237521x_{10} = -41414.3333237521
x11=−40565.6300435631x_{11} = -40565.6300435631
x12=−37170.3162915843x_{12} = -37170.3162915843
x13=40697.0353978237x_{13} = 40697.0353978237
x14=38150.655617385x_{14} = 38150.655617385
x15=38999.5005066015x_{15} = 38999.5005066015
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (-64*x)/(x^2 - 16)^2.
(−1)0⋅64(−16+02)2\frac{\left(-1\right) 0 \cdot 64}{\left(-16 + 0^{2}\right)^{2}}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
256x2(x2−16)3−64(x2−16)2=0\frac{256 x^{2}}{\left(x^{2} - 16\right)^{3}} - \frac{64}{\left(x^{2} - 16\right)^{2}} = 0
Solve this equation
Solutions are not found,
function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
256x(−6x2x2−16+3)(x2−16)3=0\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
x1=−4x_{1} = -4
x2=4x_{2} = 4

lim⁡x→−4−(256x(−6x2x2−16+3)(x2−16)3)=∞\lim_{x \to -4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty
lim⁡x→−4+(256x(−6x2x2−16+3)(x2−16)3)=∞\lim_{x \to -4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = \infty
- limits are equal, then skip the corresponding point
lim⁡x→4−(256x(−6x2x2−16+3)(x2−16)3)=−∞\lim_{x \to 4^-}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty
lim⁡x→4+(256x(−6x2x2−16+3)(x2−16)3)=−∞\lim_{x \to 4^+}\left(\frac{256 x \left(- \frac{6 x^{2}}{x^{2} - 16} + 3\right)}{\left(x^{2} - 16\right)^{3}}\right) = -\infty
- limits are equal, then skip the corresponding point

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
(−∞,0]\left(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\right)
Vertical asymptotes
Have:
x1=−4x_{1} = -4
x2=4x_{2} = 4
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((−1)64x(x2−16)2)=0\lim_{x \to -\infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞((−1)64x(x2−16)2)=0\lim_{x \to \infty}\left(\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=0y = 0
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (-64*x)/(x^2 - 16)^2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(−64(x2−16)2)=0\lim_{x \to -\infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(−64(x2−16)2)=0\lim_{x \to \infty}\left(- \frac{64}{\left(x^{2} - 16\right)^{2}}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
(−1)64x(x2−16)2=64x(x2−16)2\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = \frac{64 x}{\left(x^{2} - 16\right)^{2}}
- No
(−1)64x(x2−16)2=−64x(x2−16)2\frac{\left(-1\right) 64 x}{\left(x^{2} - 16\right)^{2}} = - \frac{64 x}{\left(x^{2} - 16\right)^{2}}
- No
so, the function
not is
neither even, nor odd
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