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Parent Functions and their Graphs | Algebra Review | Math for the ACT
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To end this lesson on parent functions and transformations.
I wanted to introduce to you five basic parent functions and some of their basic transformations
visually.
Now we will be working with each of these and many, many more parent functions throughout
the course of the year.
So you don't need to learn all the nit picky details to each of these at this point.
However, I do want you to recognize some of the generalizations that take place.
And so as you're looking at these, and there's five and I've only got four on the screen
right now, but there's five of them for you to be familiar with.
Notice that the original function or what we call the parent function is the red graph
and then the yellow and the green graphs are some sort of transformation that's taken place
for the parent function.
Now we've introduced to you the idea of an h and a k.
The h is making the function move horizontally left or right and opposite of whatever its
sign is.
The k value is affecting the transformation vertically, so it's vertically sliding it
up or down.
And then the a value will reflect it if it's negative.
And then depending on whether it's greater than one or less than one, it'll shrink it
or compress it.
So that is a good general start.
You've learned all of these on generic looking graphs and now we can see how they work on
algebraic functions.
And so take a quick visual scan of what's given to us here.
These are very important bits of information.
A couple points that I'd like to make while we've got it all here, linear function, obviously
all going straight.
A quadratic is U shaped.
So when you're trying to graph this, make your best effort to make a U out of it more
or less, whereas the absolute value is V-shaped.
Generally speaking, students will try to connect the dots on a quadratic and then inevitably
their graph ends up looking like an absolute value.
So just be very aware of that.
A cubic has kind of this curvature looking a shape and then square roots are actually
sideways quadratics, but we only have half of the quadratic and we will discuss why in
more detail when we get to square root functions.
But for right now, if you understand those basic things, you should be good.
Lastly, with these five functions, notice that the parent function goes through the
origin (0, 0).
Now this will not always be the case as we get to more advanced functions in Algebra
2.
However, for right now you can think of (0, 0) is a great starting place for your parent
functions.
Okay, so let's take a look.
I've given you sample problem number four and what I've asked you to do on sample problem
number four is to graph each of these functions.
Now they're all quadratics so they all have ought to have some sort of U shape looking
situation there.
What I would like you to do is to t-chart a few points and then after you graph the
points and draw the quadratic, check it on your calculator.
Now, generally speaking, you know you remember the old T chart, negative one, zero and one
are great places to start as far as picking x values is concerned.
And remember you have the option to pick any x value that you want and find what the corresponding
y value is.
Now I can tell you as these graphs translate or reflect or scale that different x values
are going to be needed.
So negative one, zero and one will not always give you the complete picture or the best
pictures.
So keep that in mind as well.
So go ahead and stop the video, do the best you can to graph these four functions and
come back and take a look at the solutions.
Okay, what I've done is I've gone ahead and graphed each of these and filled in the T
charts and I've spared you having to watch me do the order of operations and fill in
each of these two charts and graph it.
However, I do want to point out a few things of each graph.
So notice that I use negative one, zero and one as a starting place for each of my t-charts.
And in the first two parts, part a and part b, they gave me a clear enough picture to
graph those points and then go ahead and draw the picture.
Actually if you look closely, I've also done two and negative two which gave me fours here,
also done to a negative two which has given me seven on both of those.
And so give me a little bit better view of my, of my U shaped quadratic.
Notice here that this shifted the graph up three which is right in line with what that
K value should do.
It should vertically translate that up three units.
Now if we move down here to C and D, notice that if I graph just negative one, zero and
one, especially on part C it really didn't give me a clear picture.
Matter of fact, I could've made the mistake of it looking like a straight line.
And so that's why it's very important to understand what your final answer should look like.
Knowing that it's a quadratic, it shouldn't be a straight line.
So we're going to need additional points.
And that's where I came in with two, three and four and got a better picture.
Now if you notice on this one, it actually shifted it two, to the right, which is what
it should do because we're working the opposite.
So it's an H value and we're always doing the opposite of the H value.
So instead of to the left, we did to the right there.
Now on part D, we wanted to see if you could kind of put all this together.
We have a negative a value, so that's a negative one that's going to tell us that it's going
to reflect.
Then we also have an H and a K value, and so you can see we're coming over one down
two and then we've also reflected the graph to get the resulting transformation.
So work on that.
Also, check those in your calculator.
That'll give you great practice graphing these in your calculator.
If you have any questions about how to do that, be sure to ask in class.
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