For each polynomial function, (a) find a function of the...

For each polynomial function, (a) find a function of the form $y=c x^{2}$ that has the same end behavior. (b) find the $x$ - and $y$ -intercept(s) of the graph. (c) find the interval(s) on which the value of the function is positive. (d) find the interval(s) on which the value of the function is negative. (e) use the information in parts ( $a$ ) $-$ (d) to sketch a graph of the function. $$f(x)=3 x^{3}-27 x$$

For each polynomial function,
(a) find a function of the form $y=c x^{2}$ that has the same end behavior.
(b) find the $x$ - and $y$ -intercept(s) of the graph.
(c) find the interval(s) on which the value of the function is positive.
(d) find the interval(s) on which the value of the function is negative.
(e) use the information in parts ( $a$ ) $-$ (d) to sketch a graph of the function.
$$f(x)=3 x^{3}-27 x$$

Key Concepts

Graphical Sketching of Polynomial Functions

Combining the information about end behavior, intercepts, and sign analysis enables one to sketch a rough graph of the polynomial function. This visualization helps in understanding the function’s overall behavior, including how it rises or falls, and where it lies relative to the axes.

Sign Analysis and Interval Testing

Determining the intervals where a polynomial is positive or negative involves testing regions between its x-intercepts. By choosing test points in each interval, one can identify the sign of the polynomial values in those intervals, which assists in understanding the overall shape and behavior of the graph.

Dominant Term and End Behavior

For any polynomial function, the behavior of the graph as x approaches infinity or negative infinity is determined by its highest-degree term. By comparing the polynomial with a simpler function (such as y = cx², when appropriate), one can capture the leading behavior which makes it easier to understand how the graph will behave in the extreme ends of the x-axis.

Intercepts of Polynomial Graphs

The intercepts are key points where the graph crosses the axes. The x-intercepts (or roots) are found by setting the polynomial equal to zero and solving for x, while the y-intercept is found by evaluating the polynomial at x = 0. These intercepts are critical in providing anchor points for graphing the function.

Transcript

00:01 You are given a following polynomial function and you are asked to do a number of things.

00:04 So for part a, you are asked to express a function, y is equal to c, x to the k, with the same m behavior of f of x.

00:14 So to do this, you first need to find m behavior of f of x.

00:18 So you know that m behavior is based on the meaning term, which in this case is 3x to 3 .3.

00:22 And so you know that this is an odd function.

00:24 This is positive.

00:25 And because of that, you know that as f of x approaches infinity, as x approaches infinity and you also know that f of x approaches negative infinity as x approaches negative infinity so to create the function you just have to have the same property so you need it to be a positive constant so i'll just say 2 and you know that it needs to be an odd function so let's just say 2x cube so for part b you are asked to find the x and the y intercepts and so to find the y intercept you just set the entire function equal to 0 so f of 0 0 0 0.

01:03 Is equal to 3 times 0 cube minus 27 to 0 which is just equal to 0.

01:09 For the x intercepts a little bit trickier.

01:11 So you need to set the entire function equal to 0 and you find x.

01:14 So you have 0 is equal to 3x cube minus 27x.

01:18 Try factoring it out.

01:19 So see if it makes it look a little easier.

01:21 So 3x times x squared minus 9.

01:25 I will get you 0 is equal to 3x times x minus 3 times x plus 3.

01:32 And so you know that any number multiplied by 0 is equal to 0.

01:36 So any of these terms, 3x or x minus 3 or x plus 3, if they're equal to 0, then entire thing is equal to 0.

01:42 So 3x is equal to 0 and x is equal to 0, x minus 3 is equal to 0.

01:48 You just bring a 3 over, so x is equal to 3, and x plus 3 is equal to 0, and x is equal to negative 3.

01:57 So these are your x intercept right here.

01:59 So you have your y intercept as 0 ,0, and your x intercepts.

02:08 You have three of them.

02:09 This case, one of them is the same thing as your y intercept, actually.

02:13 X intercepts are negative 30, 0 ,000, and 3 .0.

02:28 So next, for parts c and d, you are asked to figure out where in the function is negative and when it is positive.

02:35 And so remember once again that the function is f of x is equal to 3x cubed minus 27x.

02:46 And so you need to find intervals when it is positive intervals, interval, when it is positive or if it's negative.

03:00 So your intervals will be based off your x intercepts.

03:03 And so if you recall your x intercepts are negative 3 ,000, 0, and 3 .0...

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