Mister Exam
Graphing y = 15-3y+4y
The solution
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f(y) = 15 - 3*y + 4*y
$$f{\left(y \right)} = 4 y + \left(15 - 3 y\right)$$
f = 4*y + 15 - 3*y
The graph of the function
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:
$$4 y + \left(15 - 3 y\right) = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -15$$
Numerical solution
$$y_{1} = -15$$
so we need to solve the equation:
$$4 y + \left(15 - 3 y\right) = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -15$$
Numerical solution
$$y_{1} = -15$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$1 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d y} f{\left(y \right)} = $$
the first derivative
$$1 = 0$$
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
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$0 = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
the second derivative
$$0 = 0$$
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
$$\lim_{y \to -\infty}\left(4 y + \left(15 - 3 y\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(4 y + \left(15 - 3 y\right)\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{y \to -\infty}\left(4 y + \left(15 - 3 y\right)\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(4 y + \left(15 - 3 y\right)\right) = \infty$$
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 15 - 3*y + 4*y, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty}\left(\frac{4 y + \left(15 - 3 y\right)}{y}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = y$$
$$\lim_{y \to \infty}\left(\frac{4 y + \left(15 - 3 y\right)}{y}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = y$$
$$\lim_{y \to -\infty}\left(\frac{4 y + \left(15 - 3 y\right)}{y}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = y$$
$$\lim_{y \to \infty}\left(\frac{4 y + \left(15 - 3 y\right)}{y}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = y$$
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:
$$4 y + \left(15 - 3 y\right) = 15 - y$$
- No
$$4 y + \left(15 - 3 y\right) = y - 15$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$4 y + \left(15 - 3 y\right) = 15 - y$$
- No
$$4 y + \left(15 - 3 y\right) = y - 15$$
- No
so, the function
not is
neither even, nor odd