Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- (x^2-4)/x
- x^2-8x+12
-
x^3-3x^2-1
- x^2-4x+6
- Derivative of:
- 4y
- 4y
- The double integral of:
- 4y
-
Identical expressions
- 4y
-
Similar expressions
- (1/4)y^2
- sqrt(y^2+4*y-12)
- 4*y-y^2
Graphing y = 4y
The solution
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:
$$4 y = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = 0$$
Numerical solution
$$y_{1} = 0$$
so we need to solve the equation:
$$4 y = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = 0$$
Numerical solution
$$y_{1} = 0$$
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
$$4 = 0$$
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
$$4 = 0$$
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
$$\lim_{y \to -\infty}\left(4 y\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(4 y\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{y \to -\infty}\left(4 y\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(4 y\right) = \infty$$
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*y, divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty} 4 = 4$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 4 y$$
$$\lim_{y \to \infty} 4 = 4$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 4 y$$
$$\lim_{y \to -\infty} 4 = 4$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = 4 y$$
$$\lim_{y \to \infty} 4 = 4$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = 4 y$$
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:
$$4 y = - 4 y$$
- No
$$4 y = 4 y$$
- Yes
so, the function
is
odd
So, check:
$$4 y = - 4 y$$
- No
$$4 y = 4 y$$
- Yes
so, the function
is
odd