Mister Exam

Other calculators

Plotting a function graph/ ((4+x)^2)/(4*(2+x))

Graphing y = ((4+x)^2)/(4*(2+x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
               2
        (4 + x) 
f(x) = ---------
       4*(2 + x)
f(x)=(x+4)24(x+2)f{\left(x \right)} = \frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)} 
f = (x + 4)^2/((4*(x + 2)))
The graph of the function
02468-8-6-4-2-1010-5050
The domain of the function
The points at which the function is not precisely defined:
x1=−2x_{1} = -2
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:
(x+4)24(x+2)=0\frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−4x_{1} = -4
Numerical solution
x1=−4.00000089570772x_{1} = -4.00000089570772
x2=−4.00000089770548x_{2} = -4.00000089770548
x3=−4.00000097190861x_{3} = -4.00000097190861
x4=−4.00000094290688x_{4} = -4.00000094290688
x5=−4.00000094508691x_{5} = -4.00000094508691
x6=−4.00000089514573x_{6} = -4.00000089514573
x7=−4.00000100103164x_{7} = -4.00000100103164
x8=−4.0000009508446x_{8} = -4.0000009508446
x9=−4.00000103181989x_{9} = -4.00000103181989
x10=−4.0000009562773x_{10} = -4.0000009562773
x11=−4.00000088845214x_{11} = -4.00000088845214
x12=−4.00000084113385x_{12} = -4.00000084113385
x13=−4.00000096627196x_{13} = -4.00000096627196
x14=−4.00000093764677x_{14} = -4.00000093764677
x15=−4.00000094590147x_{15} = -4.00000094590147
x16=−4.00000094867013x_{16} = -4.00000094867013
x17=−4.00000095792975x_{17} = -4.00000095792975
x18=−4.00000080224893x_{18} = -4.00000080224893
x19=−4.00000104788392x_{19} = -4.00000104788392
x20=−4.00000087626258x_{20} = -4.00000087626258
x21=−4.00000094676681x_{21} = -4.00000094676681
x22=−4.00000097904072x_{22} = -4.00000097904072
x23=−4.00000089393283x_{23} = -4.00000089393283
x24=−4.00000089724015x_{24} = -4.00000089724015
x25=−4.00000089675314x_{25} = -4.00000089675314
x26=−4.00000093807003x_{26} = -4.00000093807003
x27=−4.00000043966276x_{27} = -4.00000043966276
x28=−4.00000094768785x_{28} = -4.00000094768785
x29=−4.00000094359327x_{29} = -4.00000094359327
x30=−4.00000089025587x_{30} = -4.00000089025587
x31=−4.00000086388803x_{31} = -4.00000086388803
x32=−4.00000088130397x_{32} = -4.00000088130397
x33=−4.00000087205813x_{33} = -4.00000087205813
x34=−4.00000085661623x_{34} = -4.00000085661623
x35=−4.00000098835447x_{35} = -4.00000098835447
x36=−4.00000094163948x_{36} = -4.00000094163948
x37=−4.00000114337109x_{37} = -4.00000114337109
x38=−4.0000009404955x_{38} = -4.0000009404955
x39=−4.00000083408295x_{39} = -4.00000083408295
x40=−4.00000089258488x_{40} = -4.00000089258488
x41=−4.00000106923562x_{41} = -4.00000106923562
x42=−4.00000093996429x_{42} = -4.00000093996429
x43=−4.00000084709342x_{43} = -4.00000084709342
x44=−4.00000089107798x_{44} = -4.00000089107798
x45=−4.00000121659699x_{45} = -4.00000121659699
x46=−4.00000093945775x_{46} = -4.00000093945775
x47=−4.00000094971997x_{47} = -4.00000094971997
x48=−4.00000088272513x_{48} = -4.00000088272513
x49=−4.000000891853x_{49} = -4.000000891853
x50=−4.00000089455486x_{50} = -4.00000089455486
x51=−4.00000068401274x_{51} = -4.00000068401274
x52=−4.00000096893438x_{52} = -4.00000096893438
x53=−4.00000093685198x_{53} = -4.00000093685198
x54=−4.00000101929569x_{54} = -4.00000101929569
x55=−4.00000099416819x_{55} = -4.00000099416819
x56=−4.00000085219683x_{56} = -4.00000085219683
x57=−4.00000087426083x_{57} = -4.00000087426083
x58=−4.00000089815054x_{58} = -4.00000089815054
x59=−4.00000096387477x_{59} = -4.00000096387477
x60=−4.00000089857663x_{60} = -4.00000089857663
x61=−4.00000095475664x_{61} = -4.00000095475664
x62=−4.00000088526206x_{62} = -4.00000088526206
x63=−4.00000089937655x_{63} = -4.00000089937655
x64=−4.00000088404071x_{64} = -4.00000088404071
x65=−4.0000009443188x_{65} = -4.0000009443188
x66=−4.00000073159786x_{66} = -4.00000073159786
x67=−4.00000081523917x_{67} = -4.00000081523917
x68=−4.00000088639895x_{68} = -4.00000088639895
x69=−4.00000089898494x_{69} = -4.00000089898494
x70=−4.00000177187209x_{70} = -4.00000177187209
x71=−4.00000088938226x_{71} = -4.00000088938226
x72=−4.00000086048048x_{72} = -4.00000086048048
x73=−4.00000094225654x_{73} = -4.00000094225654
x74=−4.00000093611943x_{74} = -4.00000093611943
x75=−4.00000086962261x_{75} = -4.00000086962261
x76=−4.00000088745985x_{76} = -4.00000088745985
x77=−4.00000136040054x_{77} = -4.00000136040054
x78=−4.00000093647839x_{78} = -4.00000093647839
x79=−4.00000082561072x_{79} = -4.00000082561072
x80=−4.00000100925739x_{80} = -4.00000100925739
x81=−4.000000879764x_{81} = -4.000000879764
x82=−4.00000089327711x_{82} = -4.00000089327711
x83=−4.0000009597319x_{83} = -4.0000009597319
x84=−4.00000093724111x_{84} = -4.00000093724111
x85=−4.00000098336677x_{85} = -4.00000098336677
x86=−4.00000094105322x_{86} = -4.00000094105322
x87=−4.00000087808966x_{87} = -4.00000087808966
x88=−4.0000008962429x_{88} = -4.0000008962429
x89=−4.00000078550477x_{89} = -4.00000078550477
x90=−4.00000093851208x_{90} = -4.00000093851208
x91=−4.00000076310429x_{91} = -4.00000076310429
x92=−4.00000095205229x_{92} = -4.00000095205229
x93=−4.00000097525285x_{93} = -4.00000097525285
x94=−4.00000093897419x_{94} = -4.00000093897419
x95=−4.00000095335262x_{95} = -4.00000095335262
x96=−4.00000096170506x_{96} = -4.00000096170506
x97=−4.00000060380309x_{97} = -4.00000060380309
x98=−4x_{98} = -4
x99=−4.00000109900034x_{99} = -4.00000109900034
x100=−4.00000086691531x_{100} = -4.00000086691531
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (4 + x)^2/((4*(2 + x))).
422⋅4\frac{4^{2}}{2 \cdot 4}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 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
14(x+2)(2x+8)−(x+4)24(x+2)2=0\frac{1}{4 \left(x + 2\right)} \left(2 x + 8\right) - \frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)^{2}} = 0
Solve this equation
The roots of this equation
x1=−4x_{1} = -4
x2=0x_{2} = 0
The values of the extrema at the points:
(-4, 0)

(0, 2)


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=0x_{1} = 0
Maxima of the function at points:
x1=−4x_{1} = -4
Decreasing at intervals
(−∞,−4]∪[0,∞)\left(-\infty, -4\right] \cup \left[0, \infty\right)
Increasing at intervals
[−4,0]\left[-4, 0\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
12−x+4x+2+(x+4)22(x+2)2x+2=0\frac{\frac{1}{2} - \frac{x + 4}{x + 2} + \frac{\left(x + 4\right)^{2}}{2 \left(x + 2\right)^{2}}}{x + 2} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Vertical asymptotes
Have:
x1=−2x_{1} = -2
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((x+4)24(x+2))=−∞\lim_{x \to -\infty}\left(\frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)}\right) = -\infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡x→∞((x+4)24(x+2))=∞\lim_{x \to \infty}\left(\frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (4 + x)^2/((4*(2 + x))), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(14(x+2)(x+4)2x)=14\lim_{x \to -\infty}\left(\frac{\frac{1}{4 \left(x + 2\right)} \left(x + 4\right)^{2}}{x}\right) = \frac{1}{4}
Let's take the limit
so,
inclined asymptote equation on the left:
y=x4y = \frac{x}{4}
lim⁡x→∞(14(x+2)(x+4)2x)=14\lim_{x \to \infty}\left(\frac{\frac{1}{4 \left(x + 2\right)} \left(x + 4\right)^{2}}{x}\right) = \frac{1}{4}
Let's take the limit
so,
inclined asymptote equation on the right:
y=x4y = \frac{x}{4}
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:
(x+4)24(x+2)=(4−x)28−4x\frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)} = \frac{\left(4 - x\right)^{2}}{8 - 4 x}
- No
(x+4)24(x+2)=−(4−x)28−4x\frac{\left(x + 4\right)^{2}}{4 \left(x + 2\right)} = - \frac{\left(4 - x\right)^{2}}{8 - 4 x}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator