Mister Exam
Graphing y = -sqrt(y+1)
The solution
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_______ f(y) = -\/ y + 1
$$f{\left(y \right)} = - \sqrt{y + 1}$$
f = -sqrt(y + 1)
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:
$$- \sqrt{y + 1} = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -1$$
Numerical solution
$$y_{1} = -1$$
so we need to solve the equation:
$$- \sqrt{y + 1} = 0$$
Solve this equation
The points of intersection with the axis Y:
Analytical solution
$$y_{1} = -1$$
Numerical solution
$$y_{1} = -1$$
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
$$- \frac{1}{2 \sqrt{y + 1}} = 0$$
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
$$- \frac{1}{2 \sqrt{y + 1}} = 0$$
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
$$\frac{1}{4 \left(y + 1\right)^{\frac{3}{2}}} = 0$$
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
$$\frac{1}{4 \left(y + 1\right)^{\frac{3}{2}}} = 0$$
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
$$\lim_{y \to -\infty}\left(- \sqrt{y + 1}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(- \sqrt{y + 1}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{y \to -\infty}\left(- \sqrt{y + 1}\right) = - \infty i$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{y \to \infty}\left(- \sqrt{y + 1}\right) = -\infty$$
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 -sqrt(y + 1), divided by y at y->+oo and y ->-oo
$$\lim_{y \to -\infty}\left(- \frac{\sqrt{y + 1}}{y}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(- \frac{\sqrt{y + 1}}{y}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{y \to -\infty}\left(- \frac{\sqrt{y + 1}}{y}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{y \to \infty}\left(- \frac{\sqrt{y + 1}}{y}\right) = 0$$
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(-y) и f = -f(-y).
So, check:
$$- \sqrt{y + 1} = - \sqrt{1 - y}$$
- No
$$- \sqrt{y + 1} = \sqrt{1 - y}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$- \sqrt{y + 1} = - \sqrt{1 - y}$$
- No
$$- \sqrt{y + 1} = \sqrt{1 - y}$$
- No
so, the function
not is
neither even, nor odd