Sketching the Graph of a Polynomial Function, sketch the graph of the function by (a) applying t

Sketching the Graph of a Polynomial Function, sketch the...

Key Concepts

Graphing Strategies for Polynomial Functions

Effective graphing of polynomial functions involves combining several strategies: using the leading coefficient test to understand end behavior, determining the real zeros to locate where the function intersects the x-axis, and plotting additional points to accurately capture the function's curvature. Finally, since polynomial functions are continuous, a smooth curve is drawn through these points to represent the function accurately.

Zeros of a Polynomial

Zeros of a polynomial are the x-values where the function equals zero, representing the points where the graph crosses or touches the x-axis. Finding these zeros is critical for sketching the graph because they serve as anchor points, and they help in understanding the distribution of the function's turning points.

Leading Coefficient Test

The leading coefficient test is a method used to determine the end behavior of a polynomial function. By examining the sign and degree of the highest power term, one can predict how the function behaves as x approaches positive and negative infinity. This test is particularly useful because it provides a quick glimpse into the overall shape of the graph.

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. These functions are continuous and smooth, meaning their graphs have no breaks, corners, or holes. Understanding the structure and degree of a polynomial is essential for analysis and graphing.

Transcript

00:03 In this question, we're going to make a sketch of the function negative x cubed plus 8.

00:09 We're going to use leading coefficient test to check its behavior to the left and right.

00:14 We'll plot any real zeros, and then we'll sketch enough sufficient points where we can make a continuous curve and get a good idea of what this looks like.

00:22 One thing to think about right off the bat is this is a cubic function, and when we're graphing a cubic function, which means the leading term is to the third power, it will have some type.

00:33 Of shape like this.

00:36 Now again, that couldn't be flipped around the y -axis, the x -axis in some way, but that's the general shape we should be anticipating.

00:43 Now, with the leading coefficient test, just to underline that in red, that's what we're going to do here in part a.

00:49 I'll call it the leading coefficient test, lct.

00:53 Now for this, we know that the leading coefficient, the x -cube, is negative.

00:59 So that is very important.

01:00 Our leading coefficient is a negative x -cube, and we want to determine what this thing is going to look like in its extreme behavior to the right or to the left.

01:10 Well, in the extreme behavior to the right, that x value would be positive.

01:15 So if we negate a positive number cubed, that would be a negative.

01:20 So to the far right side of this graph, we're going to see a trend downward.

01:24 To the left side of the graph, where x values are negative, we would have the negative version of a negative number cubed.

01:32 Now this would mean the negative version of a negative or the opposite of a negative, which would be positive.

01:38 So we would see that if our left of this graph, the function would be increasing to the left.

01:44 That's one clue as we graph.

01:47 Next, let's take a look at any of the zeros that exist here.

01:52 So what are some of the zeros of the function? for the zeros, we're just going to simply substitute zero in the place of the output or the function value.

02:00 And we want to find when negative x cubed plus 8 is equal to 0, or we'll simply add the x cubed to each side, which gives us a new function, which would just be x cubed equals 8.

02:15 Shouldn't say a new function, but a different version of it.

02:18 Now if we apply a cube root to each side, the cube root of x cubed is x, and the cube root of eight is two.

02:25 So this gives us hint number two that our zero of this function would be at two.

02:31 So just over here, kind of off to the side, we will, let's sketch a final graph here, and we know over at 2, we do have a 0.

02:42 Next we want to get some sufficient points.

02:45 So since we know it's kind of a cubic, we see where it goes through 0.

02:49 Why don't we start off with a y intercept? the y intercept means we'll plug a 0 in the place of x.

02:55 So if we plug a 0 in the place of x, that would give us a negative 0 cube, which is just 0.

03:01 So we know that if the x value is, i'm sorry, if the, yeah, the x value is zero, and we know the y value would be 8.

03:11 So all the way up here, we'll make it a little longer.

03:14 At 8, we do have a y intercept.

03:19 Now, just to get a couple other points, let's see what's going on.

03:23 We'll just pick a couple, maybe at like, negative 1, negative 2, kind of see what's going on to the left.

03:28 We know ultimately to the left it's going to skyrocket up.

03:31 Ultimately to the right it's going to skyrocket, or i should say, drop really far down.

03:36 So i'm just going to erase a little bit of what we started early on.

03:39 Just we have a little bit more room...

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