Mister Exam

Other calculators

Plotting a function graph/ -3tg(x)+6x-1,5pi+5

Graphing y = -3tg(x)+6x-1,5pi+5

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                         3*pi    
f(x) = -3*tan(x) + 6*x - ---- + 5
                          2
f(x)=((6x−3tan⁡(x))−3π2)+5f{\left(x \right)} = \left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5 
f = 6*x - 3*tan(x) - 3*pi/2 + 5
The graph of the function
-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.05-5
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:
((6x−3tan⁡(x))−3π2)+5=0\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5 = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−0.0961679027967665x_{1} = -0.0961679027967665
x2=1.18573345368751x_{2} = 1.18573345368751
x3=4.60534867249648x_{3} = 4.60534867249648
x4=−1.14219421468648x_{4} = -1.14219421468648
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to -3*tan(x) + 6*x - 3*pi/2 + 5.
(−3π2+(−3tan⁡(0)+0⋅6))+5\left(- \frac{3 \pi}{2} + \left(- 3 \tan{\left(0 \right)} + 0 \cdot 6\right)\right) + 5
The result:
f(0)=5−3π2f{\left(0 \right)} = 5 - \frac{3 \pi}{2}
The point:
(0, 5 - 3*pi/2)
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
3−3tan⁡2(x)=03 - 3 \tan^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
 -pi            
(----, 8 - 3*pi)
  4             

 pi    
(--, 2)
 4


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=−π4x_{1} = - \frac{\pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Increasing at intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \infty\right)
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
−6(tan⁡2(x)+1)tan⁡(x)=0- 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(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(-\infty, 0\right]
Convex at the intervals
[0,∞)\left[0, \infty\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→−∞(((6x−3tan⁡(x))−3π2)+5)y = \lim_{x \to -\infty}\left(\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(((6x−3tan⁡(x))−3π2)+5)y = \lim_{x \to \infty}\left(\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -3*tan(x) + 6*x - 3*pi/2 + 5, 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→−∞(((6x−3tan⁡(x))−3π2)+5x)y = x \lim_{x \to -\infty}\left(\frac{\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(((6x−3tan⁡(x))−3π2)+5x)y = x \lim_{x \to \infty}\left(\frac{\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5}{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:
((6x−3tan⁡(x))−3π2)+5=−6x+3tan⁡(x)−3π2+5\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5 = - 6 x + 3 \tan{\left(x \right)} - \frac{3 \pi}{2} + 5
- No
((6x−3tan⁡(x))−3π2)+5=6x−3tan⁡(x)−5+3π2\left(\left(6 x - 3 \tan{\left(x \right)}\right) - \frac{3 \pi}{2}\right) + 5 = 6 x - 3 \tan{\left(x \right)} - 5 + \frac{3 \pi}{2}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator