WEBVTT

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[Music] Welcome to Finding Patterns

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Within Functions.

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My name is Julie Shively,

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and I am National Board Certified

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in Middle School,

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and I teach gifted math

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in Lawrenceville, Georgia.

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I have observed in my classes

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that many students enter understanding

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basic equations.

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They can find the answers,

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and they can graph a given function,

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but they don't really understand what

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causes the graphs to change form.

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They cannot describe why a function will

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shift along the horizontal

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or vertical axis,

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and they cannot predict what a graph can

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look like given just a function.

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I spend a lot of class time

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in the beginning of the year

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with the students exploring, comparing,

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and discussing different

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function tables.

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And because understanding how

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to manipulate functions is the beginning

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to predicting outputs in the functions,

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such as in rates or acceleration,

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pressure and volume.

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For example, I give students a basic

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function such as f(x)=|x|,

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and the first thing I do is I ask the

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students to tell me what that means,

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tell me what is the function,

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give me some behaviors of the function.

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And because they know what absolute

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value is, they can tell me

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that the function will be positive.

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They will say that it looks like a V

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and it's going up so the opening is up.

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Pushing them a little bit farther,

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they can tell me

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that the minimum is zero,

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so they know that y will never be less

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than zero, it will never be

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in the negative.

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So, when I ask them

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if they can visualize that and graph it,

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yes they can because of what they have

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been able to describe.

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So, that's the basic function.

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Then I give them a variation |x|+2.

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"What does that +2 do to the graph?

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Visualize what's going to happen."

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If they are not able to explain it

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at that point, that's when I ask them

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to create a table with the x and the y,

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and then they will graph it.

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And then they can see

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that what happens is the graph moves

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up to a positive 2.

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So what happens is

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that the constant moves the vertical up.

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Knowing that, I ask them, "Okay well,

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what about if I have a negative 2?"

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Given that pattern, they should be able

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to say, "Well, the graph moves then

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down to a negative 2,"

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and they can create a table,

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and sure enough that's what happens

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to the graph.

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Another variation is have everything

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inside of the absolute value:

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k(x)=|x+2|.

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Well, we know it can't go vertical,

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some of them may be able to guess,

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"Okay, it goes horizontal."

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"Well, let's check it and see."

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So, they create the graph,

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and then sure enough,

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it goes to the negative 2.

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That's when some

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of them become confused because, well,

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we have got a positive 2 in here,

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so why does it go to a negative 2?

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Why is the minimum at negative 2?

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As they are discussing it,

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then they realize that, well,

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it's at negative 2,0.

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So the y has to be zero.

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So, whatever is inside

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of the absolute value has

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to become zero, so they know then

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that is the opposite of the constant.

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So then we will try several different

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variations so they can confirm that,

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yes, that's what happens.

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So now they can see how the graph can

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move on the vertical axis

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and how it can move

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on the horizontal axis based

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on the equation.

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When there are fractions

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in the function, they have a lot

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of problems, especially when x is

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in the denominator.

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So we look at the basic function

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of f(x)=1/x.

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They know if a zero is

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in the denominator, then that means

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that the y is undefined

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or the function is undefined.

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Then I ask them, "Okay, well,

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what happens when x increases?"

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Well, then that means

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that the function decreases towards

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zero, and then the opposite is true.

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At this point,

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I do ask them to create tables.

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They do realize

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that the graphs will go towards zero,

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and they are reflections,

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they are mirror images.

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So, that's the basic function is 1/x.

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Well let's put in a variation.

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Let's go 1/x+4.

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Well, if they go back

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to the other functions

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that we have done,

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if we have got a plus 4,

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if we are adding in a constant,

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then that's going

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to move the graph vertically.

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So, if it's a plus 4,

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it's going to go up to 4.

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I have them practice with a lot

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of different functions starting

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with the basic and then going

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with variations,

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and they can realize then

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that when you change the coefficient,

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when you change the sign,

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when you change the constant,

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that's going to change the graph,

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but you always have to start

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with the basic function.

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To learn more about Finding Patterns

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Within Functions,

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please explore the additional resources

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on the Doing What Works website.

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[Music]