-4x - 12 if x ≤ -3 (2)/(5)x^2 if x

$\begin{cases} -4x - 12 & \text{if } x \le -3\\ \frac{2}{5}x^2 & \text{if } x > -3 \end{cases}$ Step 1 of 3: Identify the general shape and direction of the graph of this function on the interval $(-\infty, -3]$.

Transcript

00:01 The given function is y equal to cos 2x divided by 3 minus 2 sin squared x.

00:12 So if the function f of a plus m equal to f of a then m is period of the function.

00:34 So y of x plus pi equal to cos 2x plus pi divided by 3 minus 2 sin squared x plus pi which is equal to cos 2 of x divided by 3 minus 2 sin squared of x which is equal to f of x.

01:04 Hence pi is the period of the function.

01:20 Now the given function can be written as y equal to cos 2x divided by 3 minus 2 1 minus cos 2x divided by 2 which is equal to cos 2x divided by 3 minus 1 plus cos 2x which is equal to cos 2x divided by 2 plus cos 2x.

01:52 Therefore y will be vary according to cos 2x.

01:56 Hence 2 plus cos 2x here 2 plus cos 2x greater than 0.

02:04 Since cos 2x minimum value is minus 1 then y will be maximized according to cos 2x and y will be minimized according to cos 2x.

02:27 Now cos 2x maximum at x equal to n pi.

02:42 So y of n pi equal to cos 2 of n pi divided by 2 plus cos 2 n pi which is equal to 1 divided by 2 plus 1 which is equal to 1 divided by 3 which is equal to 0 .333.

03:08 Now cos 2x is minimum at x equal to n pi divided by 2.

03:25 So y of n pi equal to cos 2 n pi divided by 2 divided by 2 plus cos 2 n pi divided by 2 which is equal to minus 1.

03:40 So according to the above values the graph of the function will depend on the value of cos 2x finding the 0 of the graph...

Question

$\begin{cases} -4x - 12 & \text{if } x \le -3\\ \frac{2}{5}x^2 & \text{if } x > -3 \end{cases}$ Step 1 of 3: Identify the general shape and direction of the graph of this function on the interval $(-\infty, -3]$.

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