# How do you graph y=sinx+x?

Mar 3, 2018

See below

#### Explanation:

$y = \sin x + x$

Note that $y$ is defined $\forall x \in \mathbb{R}$

Now, since $- 1 \le \sin x \le + 1$

Then, ${\lim}_{x \to - \infty} y = - \infty \mathmr{and} {\lim}_{x \to + \infty} y = + \infty$

To find the critical points of $y$ we will set the first differential $y ' = 0$ and test with the second differential $y ' '$.
(You will need to refer to Calculus here)

$y ' = \cos x + 1 = 0$

$\cos x = - 1 \to x = n \pi : \forall n \in \mathbb{Z}$

$y ' ' = - \sin x$

Now, $- \sin x = 0$ for $x = n \pi : \forall n \in \mathbb{Z}$

So, $y$ has inflection points for $x = n \pi : \forall n \in \mathbb{Z}$

And, as $\left\mid x \right\mid$ increases $y \to x$

We can see these results on the graph of $y$ below.
(If you zoom out the graph $\to y = x$)

graph{sinx+x [-65.83, 65.87, -32.9, 32.9]}