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Plotting a function graph/ tg(4x)+4

Graphing y = tg(4x)+4

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = tan(4*x) + 4
f(x)=tan⁡(4x)+4f{\left(x \right)} = \tan{\left(4 x \right)} + 4 
f = tan(4*x) + 4
The graph of the function
02468-8-6-4-2-1010-100100
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:
tan⁡(4x)+4=0\tan{\left(4 x \right)} + 4 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−atan⁡(4)4x_{1} = - \frac{\operatorname{atan}{\left(4 \right)}}{4}
Numerical solution
x1=67.998185799661x_{1} = 67.998185799661
x2=−50.5969368733537x_{2} = -50.5969368733537
x3=52.290222531712x_{3} = 52.290222531712
x4=−53.7385295269435x_{4} = -53.7385295269435
x5=−60.8071129975205x_{5} = -60.8071129975205
x6=46.0070372245324x_{6} = 46.0070372245324
x7=34.2260647735707x_{7} = 34.2260647735707
x8=−31.7473809518149x_{8} = -31.7473809518149
x9=16.1619070154294x_{9} = 16.1619070154294
x10=38.153055590558x_{10} = 38.153055590558
x11=23.2304904860064x_{11} = 23.2304904860064
x12=40.5092500807503x_{12} = 40.5092500807503
x13=−9.75623237668639x_{13} = -9.75623237668639
x14=96.2725196819691x_{14} = 96.2725196819691
x15=44.4362408977375x_{15} = 44.4362408977375
x16=12.2349161984422x_{16} = 12.2349161984422
x17=−18.3956121740583x_{17} = -18.3956121740583
x18=−75.729678102072x_{18} = -75.729678102072
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(4*x) + 4.
tan⁡(0⋅4)+4\tan{\left(0 \cdot 4 \right)} + 4
The result:
f(0)=4f{\left(0 \right)} = 4
The point:
(0, 4)
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
4tan⁡2(4x)+4=04 \tan^{2}{\left(4 x \right)} + 4 = 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
32(tan⁡2(4x)+1)tan⁡(4x)=032 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0

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[0, \infty\right)
Convex at the intervals
(−∞,0]\left(-\infty, 0\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(tan⁡(4x)+4)y = \lim_{x \to -\infty}\left(\tan{\left(4 x \right)} + 4\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(tan⁡(4x)+4)y = \lim_{x \to \infty}\left(\tan{\left(4 x \right)} + 4\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(4*x) + 4, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(tan⁡(4x)+4x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(4x)+4x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)} + 4}{x}\right)
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:
tan⁡(4x)+4=4−tan⁡(4x)\tan{\left(4 x \right)} + 4 = 4 - \tan{\left(4 x \right)}
- No
tan⁡(4x)+4=tan⁡(4x)−4\tan{\left(4 x \right)} + 4 = \tan{\left(4 x \right)} - 4
- No
so, the function
not is
neither even, nor odd
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