Algebra 2
Have you noticed how the shapes of certain graphs can tell you a lot about the math behind them? Today, we’ll explore polynomial functions—learning how to predict their twists and turns, play some interactive games, and use real examples so you’ll be able to sketch and understand these graphs confidently in any situation.
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Chapters
Transcript
Introduction
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okay so this is lesson three one which
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is graphing polynomial functions
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our essential question is how do the key
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features of a polynomial function
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help you sketch its graph
Example 1 Standard Form
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so the first example is how can you
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write a polynomial in
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standard form and use it to identify the
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leading coefficient
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the degree and the number of terms
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okay so first of all standard form we
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need to define that
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it's the highest
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to lowest
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power of x
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so we arrange our terms with the highest
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power of x then the next highest and so
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on
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so if i were to rearrange this i would
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get 2
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x cubed that's my highest power of x
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minus 4 x plus
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nine so this polynomial is in standard
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form because it's decreasing powers of x
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so the other things we need to talk
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about are leading coefficient
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so use different colors here so the
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leading coefficient
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is the coefficient that is with your
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highest power of x
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so that will be two two is our leading
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coefficient
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in this problem okay
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our next term is the degree
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the degree is the highest power of x so
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this is a third
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degree polynomial
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and the final thing we need to identify
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is the number of terms
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so each term is
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variables or constants that are being
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multiplied together
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and they're separated with a plus or a
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minus so this is
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one term two terms three terms
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so we would say there are three terms
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so you can use the number of terms and
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the degree
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to name the polynomial so this
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example right here would be a third
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degree
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trinomial because we say so we say the
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degree first
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and then there are three terms so that's
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a trinomial
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if there were only two terms it'd be a
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binomial if there's only one term it's a
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monomial
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um and then usually above three terms we
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just say polynomial
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excuse me okay
Example 2 End Behavior
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so our next example is talking about end
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behavior of polynomials so there are
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four situations that we could have so
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we're gonna kind of
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label these and get used to the notation
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that you're gonna use to talk about in
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behavior
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so this first one is
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polynomials that are odd degree and
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a positive leading coefficient
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so that means that the degree is odd and
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the
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leading coefficient the number with it
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is positive
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so these are three the red the green the
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blue these are three examples of odd
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positive
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so the way we describe the end behavior
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is you're always going to have
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an x that goes to negative infinity and
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an x that goes to positive infinity so
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you're going to see that written for
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everyone
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what this is referring to so negative
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infinity is talking about to the left
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and this is talking about to the right
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so if you just understand that
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every single one so if i go over here
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this is also going to have
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a x goes to negative infinity and x goes
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to positive infinity
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referring to left and right okay
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so now we have to say what happens to
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the y
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as we move to the left so if you look
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down here
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um or here or here as we move to the
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left
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the graphs are going down so we would
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say y
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goes to negative infinity and then as we
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go to the right
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that's talking about these arrows up
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here they're going up to positive
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infinity
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so as we go to the right we're going to
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positive infinity
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this second graph over here is an
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example of odd
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with a negative leading coefficient so
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odd and negative so odd and negative
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you can see that it's the opposite as we
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go to the left
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our graph is going up and as we go to
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the right
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our graph is going down
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so getting used to that notation of as x
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goes to positive infinity and as x goes
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to negative infinity
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that's probably the trickiest part of
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this
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okay the last two are even so this is
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even oops
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even and positive and this one is even
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and negative so you can see with the odd
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graphs you have
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one end that's going up and one end
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that's going down
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where with the even graphs both ends are
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going the same direction
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so again we're going to have as we go to
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the left
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oops as we go to the left and as we go
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to the right
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and we're going to have as we go to the
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left
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and as we go to the right
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so for this graph on the left here we
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can see both
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to the left and the right they're
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increasing so we would say y is going to
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positive infinity
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and y is going to positive infinity and
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if we look over here
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as we go to the left and right y is
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going to negative infinity
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okay so that is end behavior
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the last thing we're going to talk about
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is how you can use
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features of a gra like tables and
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different features
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to identify what your graph is going to
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look like so this says to consider the
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polynomial function
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f of x equals negative point five x to
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the fourth plus three
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x squared plus two so this would be
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a fourth degree
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trinomial okay so again we have three
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terms
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the degree is four and i can tell that
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this
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is an even degree
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and my leading coefficient is negative
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so this is an
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even negative so before i even graph it
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before you look down here at my example
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parts
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um you will you can identify that you
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should
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this graph should be opening down both
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end behaviors are going to head down
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okay so one
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way of sketching a graph is to make a
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table so i was going to talk about
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i have a table here but how do i get a
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table
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so i was going to show you on desmos so
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if we go into desmos here so i already
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have the equation written out
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and notice you can turn off the graph
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sometimes
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we rely too heavily on the graph and
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that's not going to give us
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as exact values as if we use something
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like a table
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so you can use desmos but you don't need
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to focus on what the graph is looking
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like right now
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so up here the little
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settings okay you know so again
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if i hit right here up here in the upper
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um right hand part of where the equation
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is i'm going to hit
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that button and then you'll notice it
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has
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table and then it has a copy and it has
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a delete
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so i'm going to hit the table and you
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will notice
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that it made a table for me
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now maybe i want more values in my table
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so i'm going to add in
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negative 3 and a positive 3.
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and then it did dump those points on my
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graph
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and i can i can
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graph it if i want to but all i really
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care about right now is i'm focused on
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the table
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so again you use that little settings
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up in the upper right hand um part of
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the left side of the screen
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and then you are just hitting the table
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feature
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okay so if we go back to
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our example here so you can tell by the
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table over here
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you can tell where your graph is
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increasing or decreasing
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and when it's changing from increasing
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to decreasing you're going to have a
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turning point so here we have
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one two three turning points
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turning points are where your graph goes
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from increasing to decreasing
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or decreasing to increasing other key
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features
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of our graph these points up here we
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call
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relative maximums
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so what it means by a relative maximum
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is that it's relative it's a maximum
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relative to the area of the graph around
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it it might not be the very highest
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point of the graph altogether
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but it is the highest point relative to
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the points around it
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this point right here would be a
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relative minimum
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oops having trouble writing tiny okay so
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that's a relative minimum
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so again relative max is relative min's
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it's not so we can tell this point is
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not the lowest point of our graph
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because our graph keeps going down
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to negative infinity um these are the
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actual highest points of our graph so
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they are maximums but there's two of
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them so that's why we
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we can use the term relative maximums
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and relative minimums
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to just talk about the location of
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um the turning points of our graph okay
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let me know if you have any questionsDegree of a Polynomial
Property
The degree of a term is the sum of the exponents of its variables.
The degree of a constant is 00.
The degree of a polynomial is the highest degree of all its terms.
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial.
Examples
The polynomial 2x5−3x2+72x5−3x2+7 has a degree of 55, which is the highest power of xx.
The term −4a2b3−4a2b3 has a degree of 2+3=52+3=5. The polynomial a2b3+a4a2b3+a4 has a degree of 55, since the first term's degree is highest.
A constant like 1212 has a degree of 00. The binomial x+12x+12 has a degree of 11.
Explanation
The 'degree' is a term's power-ranking. For one term, add its variable exponents. For the whole polynomial, the highest degree from any single term determines the polynomial's degree. It's a quick way to classify its complexity.
Overview
Graphing Polynomial Functions — Grade 11 math | Pengi