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  • Graphing y =:
  • x^4-2x^2+8
  • √(x^2-1)
  • cos(3x) cos(3x)
  • 3*2^x

  • Identical expressions

  • three *cos(x/ two)- two
  • 3 multiply by co sinus of e of (x divide by 2) minus 2
  • three multiply by co sinus of e of (x divide by two) minus two
  • 3cos(x/2)-2
  • 3cosx/2-2
  • 3*cos(x divide by 2)-2

  • Similar expressions

  • 3*cos(x/2)+2
Plotting a function graph/ 3*cos(x/2)-2

Graphing y = 3*cos(x/2)-2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
            /x\    
f(x) = 3*cos|-| - 2
            \2/
f(x)=3cos⁡(x2)−2f{\left(x \right)} = 3 \cos{\left(\frac{x}{2} \right)} - 2 
f = 3*cos(x/2) - 2
The graph of the function
02468-8-6-4-2-10105-10
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:
3cos⁡(x2)−2=03 \cos{\left(\frac{x}{2} \right)} - 2 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−2acos⁡(23)+4πx_{1} = - 2 \operatorname{acos}{\left(\frac{2}{3} \right)} + 4 \pi
x2=2acos⁡(23)x_{2} = 2 \operatorname{acos}{\left(\frac{2}{3} \right)}
Numerical solution
x1=−77.0803610272909x_{1} = -77.0803610272909
x2=−227.876808399601x_{2} = -227.876808399601
x3=−36.0169745019417x_{3} = -36.0169745019417
x4=48.5833451163008x_{4} = 48.5833451163008
x5=26.8148785698542x_{5} = 26.8148785698542
x6=−64.5139904129317x_{6} = -64.5139904129317
x7=1.68213734113586x_{7} = 1.68213734113586
x8=−61.14971573066x_{8} = -61.14971573066
x9=51.9476197985726x_{9} = 51.9476197985726
x10=−26.8148785698542x_{10} = -26.8148785698542
x11=23.4506038875825x_{11} = 23.4506038875825
x12=−14.248507955495x_{12} = -14.248507955495
x13=64.5139904129317x_{13} = 64.5139904129317
x14=98.8488275737375x_{14} = 98.8488275737375
x15=−86.2824569593784x_{15} = -86.2824569593784
x16=36.0169745019417x_{16} = 36.0169745019417
x17=73.7160863450192x_{17} = 73.7160863450192
x18=−23.4506038875825x_{18} = -23.4506038875825
x19=10.8842332732233x_{19} = 10.8842332732233
x20=89.6467316416501x_{20} = 89.6467316416501
x21=−1.68213734113586x_{21} = -1.68213734113586
x22=86.2824569593784x_{22} = 86.2824569593784
x23=−48.5833451163008x_{23} = -48.5833451163008
x24=−215.310437785242x_{24} = -215.310437785242
x25=77.0803610272909x_{25} = 77.0803610272909
x26=−98.8488275737375x_{26} = -98.8488275737375
x27=−51.9476197985726x_{27} = -51.9476197985726
x28=14.248507955495x_{28} = 14.248507955495
x29=111.415198188097x_{29} = 111.415198188097
x30=39.3812491842134x_{30} = 39.3812491842134
x31=−39.3812491842134x_{31} = -39.3812491842134
x32=−136.547939416815x_{32} = -136.547939416815
x33=−89.6467316416501x_{33} = -89.6467316416501
x34=−73.7160863450192x_{34} = -73.7160863450192
x35=−10.8842332732233x_{35} = -10.8842332732233
x36=61.14971573066x_{36} = 61.14971573066
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 3*cos(x/2) - 2.
−2+3cos⁡(02)-2 + 3 \cos{\left(\frac{0}{2} \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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
−3sin⁡(x2)2=0- \frac{3 \sin{\left(\frac{x}{2} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
The values of the extrema at the points:
(0, 1)

(2*pi, -5)


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=2πx_{1} = 2 \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[2π,∞)\left(-\infty, 0\right] \cup \left[2 \pi, \infty\right)
Increasing at intervals
[0,2π]\left[0, 2 \pi\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
−3cos⁡(x2)4=0- \frac{3 \cos{\left(\frac{x}{2} \right)}}{4} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
x2=3πx_{2} = 3 \pi

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
[π,3π]\left[\pi, 3 \pi\right]
Convex at the intervals
(−∞,π]∪[3π,∞)\left(-\infty, \pi\right] \cup \left[3 \pi, \infty\right)
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(3cos⁡(x2)−2)=⟨−5,1⟩\lim_{x \to -\infty}\left(3 \cos{\left(\frac{x}{2} \right)} - 2\right) = \left\langle -5, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−5,1⟩y = \left\langle -5, 1\right\rangle
lim⁡x→∞(3cos⁡(x2)−2)=⟨−5,1⟩\lim_{x \to \infty}\left(3 \cos{\left(\frac{x}{2} \right)} - 2\right) = \left\langle -5, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−5,1⟩y = \left\langle -5, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*cos(x/2) - 2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3cos⁡(x2)−2x)=0\lim_{x \to -\infty}\left(\frac{3 \cos{\left(\frac{x}{2} \right)} - 2}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3cos⁡(x2)−2x)=0\lim_{x \to \infty}\left(\frac{3 \cos{\left(\frac{x}{2} \right)} - 2}{x}\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:
3cos⁡(x2)−2=3cos⁡(x2)−23 \cos{\left(\frac{x}{2} \right)} - 2 = 3 \cos{\left(\frac{x}{2} \right)} - 2
- No
3cos⁡(x2)−2=2−3cos⁡(x2)3 \cos{\left(\frac{x}{2} \right)} - 2 = 2 - 3 \cos{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
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