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Plotting a function graph/ (6-3y)/2

Graphing y = (6-3y)/2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       6 - 3*y
f(y) = -------
          2
f(y)=6−3y2f{\left(y \right)} = \frac{6 - 3 y}{2} 
f = (6 - 3*y)/2
The graph of the function
02468-8-6-4-2-1010-5050
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis Y at f = 0
so we need to solve the equation:
6−3y2=0\frac{6 - 3 y}{2} = 0
Solve this equation
The points of intersection with the axis Y:

Analytical solution
y1=2y_{1} = 2
Numerical solution
y1=2y_{1} = 2
The points of intersection with the Y axis coordinate
The graph crosses Y axis when y equals 0:
substitute y = 0 to (6 - 3*y)/2.
6−02\frac{6 - 0}{2}
The result:
f(0)=3f{\left(0 \right)} = 3
The point:
(0, 3)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddyf(y)=0\frac{d}{d y} f{\left(y \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddyf(y)=\frac{d}{d y} f{\left(y \right)} =
the first derivative
−32=0- \frac{3}{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
d2dy2f(y)=0\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dy2f(y)=\frac{d^{2}}{d y^{2}} f{\left(y \right)} =
the second derivative
0=00 = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at y->+oo and y->-oo
lim⁡y→−∞(6−3y2)=∞\lim_{y \to -\infty}\left(\frac{6 - 3 y}{2}\right) = \infty
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
lim⁡y→∞(6−3y2)=−∞\lim_{y \to \infty}\left(\frac{6 - 3 y}{2}\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 (6 - 3*y)/2, divided by y at y->+oo and y ->-oo
lim⁡y→−∞(6−3y2y)=−32\lim_{y \to -\infty}\left(\frac{6 - 3 y}{2 y}\right) = - \frac{3}{2}
Let's take the limit
so,
inclined asymptote equation on the left:
y=−3y2y = - \frac{3 y}{2}
lim⁡y→∞(6−3y2y)=−32\lim_{y \to \infty}\left(\frac{6 - 3 y}{2 y}\right) = - \frac{3}{2}
Let's take the limit
so,
inclined asymptote equation on the right:
y=−3y2y = - \frac{3 y}{2}
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-y) и f = -f(-y).
So, check:
6−3y2=3y2+3\frac{6 - 3 y}{2} = \frac{3 y}{2} + 3
- No
6−3y2=−3y2−3\frac{6 - 3 y}{2} = - \frac{3 y}{2} - 3
- No
so, the function
not is
neither even, nor odd
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