WEBVTT mathematics/algebra-2/eaton
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Welcome to Educator.com.
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Today, we are going to talk about polynomial functions, starting with some review,
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and then going on to discuss the topic of analyzing the graphs of polynomial functions.
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OK, a polynomial in one variable is what we are going to start out with; and a polynomial of the degree n,
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in one variable x, is an expression in this following form.
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a^n, the leading coefficient, cannot be 0, because if it was to be 0, this would drop out;
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and then you would actually have a lower-degree polynomial.
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And n is a non-negative number; these are not going to have a negative coefficient up here.
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This first coefficient is called the leading coefficient.
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A polynomial is just an expression in which the terms are monomials; and a restriction on this is that there cannot be variables in the denominator.
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If there were variables, it would not be in this form; if that were the case, you would have something else.
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You would have a rational expression, not a polynomial.
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Now, let's talk about degree, using numbers instead of just n.
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If you have something such as 3x⁴ (a polynomial) + x³ - 4x² - x + 4,
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the degree of the polynomial is the degree of the variable that is the highest power.
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So, right here, the highest power I have is 4; therefore, the degree equals 4.
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And the leading coefficient is the coefficient for the variable that is raised to the highest power.
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One thing to be aware of is that you might look at a polynomial, and it could be written like this:
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2x² - 6x⁵ + 4x³ + 9.
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And when you look at this, you have to realize that these are not written in descending order.
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So, you can't just look at the first one and say, "Oh, this is degree 2."
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Before you figure out what the degree of a polynomial is, or what the leading coefficient is, you first need to write it in descending order.
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So, that would be -6x⁵ + 4x³ + 2x² + 9.
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Here, the degree is actually 5, and the leading coefficient is -6.
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Just make sure that the terms are in descending order before you determine the degree and the leading coefficient.
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Now, polynomial functions: we just saw this form, but now we are talking about it as a function.
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A **polynomial function** of degree n is a function in this form.
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You may see this written as f(x), like this; you may also see p(x); that is sometimes used for a polynomial function.
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And just to give you an example, I am going to call it f(x) (but it could have been called p(x)) = 2x³ - 4x² + 8x - 5.
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This is a polynomial function; and with polynomial functions, we can evaluate the function for both numerical values and algebraic expressions.
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You are probably familiar with evaluating functions for numerical values.
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But evaluating them for algebraic expressions may be something new.
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So, first, just looking at a function: f(x) = 4x³ - 3x² + x - 1: if I am asked to find f(2),
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the strategy is just going to be to replace all of these x's with 2's.
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which is going to give me...2 times 2 is 4, times 2 is 8; so it is 4 times 8 (2³ is 8) minus 3 times 2 (is 6), plus 2, minus 1.
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This is 32 minus...actually, correction: this is an extra place with 2...2 times 2 is 4, because that is squared; so it is going to give me 3 times 4.
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So, I am going to end up with 4 times 8, minus 3 times 4, plus 2, minus 1.
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This is going to give me 32 - 12 + 2 - 1; simplifying that, 32 - 12 is 20, plus 2 is 22, minus 1--that is 21, so f(2) is 21.
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That is straightforward, but something you may be less familiar with is the idea of something like this: f(a + 3).
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Again, I am going to substitute this expression for the x's--the same idea, only now I have a variable in here.
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Wherever there is an x, I am going to put in a + 3 instead.
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Now, you could do the algebra and work this out to figure out the value; but the important part
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is just knowing how to set this up by substituting this algebraic expression for the variable in the function.
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OK, a point at which the graph of a function intersects with the x-axis is called a zero of the function.
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And previously, we have referred to x-intercepts: this is the same thing, only a different term.
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And the degree of the function can tell you something about the zeroes.
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If a polynomial function has an odd degree, there is at least one zero.
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For example, if I have something like f(x) = 6x³ - 5x² + 8x, the degree here is 3, and it is odd.
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So, I know that it has at least one zero.
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Now, just showing you examples of zeroes: we talked about linear functions, which graph as a line,
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and then quadratic functions (which are polynomial functions) are parabolas; you can also have
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more elaborate shapes, such as this, with a polynomial.
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And everywhere that intersects with the x-axis, those are all zeroes; so this actually has 1, 2, 3, 4, 5 zeroes.
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And let's say this is 2, 4, 6, 8, 10; -2, -4, -6, -8, -10; you could find the values for these.
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For example, one of the zeroes is at 4; so you could find the values for these zeroes.
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OK, so you always have at least one zero, if the degree of the polynomial function is odd.
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If the degree is even, it may or may not have a zero (it may or may not intersect with the x-axis).
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So, something like this might have a zero and might not.
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The degree equals 4; it is even; it may or may not have a zero.
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Thinking back to quadratic equations (their degree is 2--something like x² + 2x - 1--quadratic functions),
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recall that these parabolas sometimes have a zero; they sometimes intersect with the x-axis.
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They can have one or two zeroes (this one has two), or they may not intersect.
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We saw situations where parabolas did not intersect, and those have even degrees.
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And the same is true of a polynomial function with a greater degree that is even.
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When we talk about functions, we sometimes talk about end behavior.
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And the end behavior of a function refers to what the graph does as x gets very large or very small.
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So, instead of in the middle (which is often what we have been focusing on with graphing), now we are thinking about the ends--
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way out here when x is very positive, or way out here when x is very negative.
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And you can predict some of this end behavior by looking at the degree of the polynomial function and the sign of its leading coefficient.
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So, let's break this down into first talking about even degrees.
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And then, we will talk about odd degrees; polynomials of even degrees first.
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For example, x⁴ - 2x³ + x - 8: this has a degree of 4--it is even.
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We can divide this into two categories: those that have a leading coefficient that is positive...
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if the leading coefficient, a^n, is greater than 0, both ends of the graph go upward.
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Now, let's think about what that means.
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For example, if I have some polynomial function, and it is even (so it may or may not have a zero,
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but I am going to show it as having zeroes), both ends are up; so this is even,
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and a^n is greater than 0; both of these ends go up.
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What this is saying is that, when x is very large, over here, or x is very small, way out here, f(x) is large.
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So, end behavior--what is happening way out here at the extremes--when x is very large
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or when x is very small, you see that y is going to go up, up, up and be very large.
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So, that is an even degree; both ends are up; and it is a positive leading coefficient,
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so both ends are up; there are very large y's, or function values, at the extreme values of x.
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Now, the other possibility (that is one possibility for even) is that the leading coefficient is less than 0.
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I have a negative leading coefficient--something like -4x² + 3x - 1; and this is actually a quadratic function.
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And if you recall, this is going to be a parabola that faces downward.
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Let's look at this: here, this could have had a higher even degree, like 4 or 6 or 8; but let's just look at the parabola.
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Here, both ends face downward; and what this tells me is that, when x is large, or when x is small
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(very small or very large), f(x) is small--it has small values.
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So, at the ends here, when x is these small values or very large values, the y is going to have very small values.
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OK, so these are even degrees; and to help you remember this, just think back to quadratic equations,
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because we already talked about quadratic functions and the fact that, if their leading coefficient is positive, they face upward.
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If their leading coefficient is negative, they face downward.
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And it is the same idea here, except you might have more ups and downs in between.
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Now, if you have a polynomial with an odd degree, again, we have two possibilities.
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If I have a polynomial that has an odd degree, and the leading coefficient is positive (a is greater than 0),
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then what I am going to have is a graph that is going to go up to the right; that is this case right here.
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What this is saying is that, when x is large, y is large (or f(x) is large--either way you want to say it).
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And when x is small, y gets very small.
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So, I just think of it as "x and y are going in the same directions."
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Now, if the leading coefficient is negative, we have the opposite situation; here you would have something like this, where...
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this is odd degree, and it is an a that is negative; here I see that it is starting up here, and it is going down to the right.
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The graph goes down to the right.
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So, with even degrees, both ends face up, or both ends face down; with odd degrees, one end goes up; the other goes down.
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If it is a positive leading coefficient, it goes up to the right; if it is a negative leading coefficient, it goes down to the right.
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And you also saw that with linear equations--going up or down to the right.
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What this tells me is that, when x is very large, y is very small; and when x is very small, y is very large.
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x and y are going the opposite ways.
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OK, first looking at this polynomial: what is the degree and leading coefficient?
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I am going to start out by putting this in descending order; this is my largest degree here, so look at that first.
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Then, I have x³; I have -7x²; 3x; and finally, the constant.
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OK, so now that I have this in descending order, I can easily see that the degree equals 4.
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And the leading coefficient is going to be the coefficient for that variable, raised to the largest power.
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The leading coefficient is -12; and if I didn't put these in the right order, and just looked at it, I might have said, "Oh, the leading coefficient is 3."
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So, always put your polynomial in descending order first.
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OK, given this function, find f(3a⁴).
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This is an algebraic term, and we need to find the function value for it; so I am going to substitute 3a⁴ everywhere there is an x.
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I am going to use my rule here of raising a power to a power; that tells me that, when I raise a power to a power, I need to multiply the exponents.
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So, first, I have my 3 here; let's keep that out here; and I have to remember to raise 3 to 3, also.
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So, this is 3 times 3 is 9, times 3 is 27; so that is 27a; and that is 4 times 3, so that is a^12.
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And then here, I have 3², which is 9; a⁴...OK, that is 4 times 2, which is 8.
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And then, this just stays like this.
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Next, I have 3 times 27, which is actually 81; a^12; -2 times 9 gives me -18a⁸;
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plus 3a⁴, minus 7; and I always double-check and make sure I can't do any further simplification.
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And I can't; I don't have any like terms that I can combine.
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So again, evaluating a function for an algebraic expression just involves substituting that expression in for the variables.
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OK, again, evaluating this function for an algebraic expression means I am going to substitute in 3b - 2 wherever I see an x.
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Writing this out: this is a squared binomial...using the FOIL method, First is 3b times 3b, is 9b².
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The Outer terms--that gives me -6b; the Inner terms multiply to -6b; and the Last terms: -2 times -2 is 4.
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Now, using the distributive property, and multiplying each term within the parentheses by 2,
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is going to give me...2 times 9b² is 18b²; 2 times -6b is -12b; and this is -12b;
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I also could have just added these first, and then done the distributive property; it is going to come out the same either way.
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2 times 4 is 8; now, the distributive property again here: -3 times 3b is going to give me -9b.
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-3 times -2 is going to give me + 6; and then the 9.
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Adding like terms: -12b and -12b is -24b; and I also have some more simplifying I can do out here.
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This is 18b²; -24b and -9b is going to give me -33b; 8 and 6, plus 9 (that is 15 + 8) is 23.
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OK, so evaluating this function for 3b - 2 gives this result.
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We are simply substituting in 3b - 2 for all the x's, using the distributive property, and then simplifying by adding like terms.
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OK, in Example 4, we are given this polynomial function and asked to describe the end behavior.
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We are starting out by looking and making sure that this is written in descending order (which it is),
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because I need to find the degree and the leading coefficient.
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The degree is 6; I know that this is an even number, so it may or may not have zeroes, just as an aside.
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Now, talking about the leading coefficient and the degree in a little more detail:
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the degree being even tells me that both ends will be up, or both ends will be down.
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OK, so I know that both ends are going to go in the same direction with this polynomial.
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The leading coefficient here is 5, and that is positive; so since it is positive, both ends of the graph go upward,
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just like with a parabola with a positive leading coefficient in the quadratic function.
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This is to the sixth, so it is going to be more complex, and it may or may not have zeroes; but just schematically,
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the idea is that both ends will face up.
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Now, what does that tell me about the end behavior?
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What this tells me about end behavior is that, when x is large, way out here, y is large (or the function is large).
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When x is very small (way out here)--it has small values--y is large.
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So, at both ends (where x is very large or x is very small), the function has large values.
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And that tells me about the end behavior; and I was able to figure that out by realizing this is an even degree, and it has a positive leading coefficient.
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That concludes this lesson of Educator.com on graphing and polynomials.