In this lesson,
we will use the free online graphing calculator
linked to below.
To graph a function, adjust the window,
determine the local max and mins,
as well as to determine the open intervals
for which the function is increasing or decreasing.
The first step is enter the equation into the calculator
and the calculator does use function notation,
and therefore we will let y equal f of x.
So going to the calculator,
the first step is to clear the existing functions.
And we have f of x equals negative two x cubed
plus 42 x squared minus 270 x plus 11.
Remember the exponent key [indistinct] is shift six.
Now click plot graphs, by the way,
notice how we do not have a nice viewing window
because we don't see the high and low points of the graph.
We can try zooming out by clicking on the minus
in the lower right hand corner, but this doesn't help.
In this case, we will have to adjust the window manually.
Before we do this though, it can be helpful
to analyze a table of values, to know
how to adjust the window.
Let's click the table tab at the right.
You will leave the x values on automatic.
Let's start x, and let's say negative five,
and we'll leave this step at one, so x increases by one,
as we go down the table.
click go and it will analyze the y values as x increases.
Notice how the y values are decreasing.
It's a little hard to tell, but somewhere between four
and six, the y values changed from decreasing to increasing
from negative 525 to negative 539, the y value decreases,
but then from negative 539 to negative 529,
the y value increases.
So when I analyze the graph, when x is between four
and six, and y is somewhere between negative 500
and negative 550.
As we progressed down the table,
the y value continue increasing until somewhere
between eight and 10, the y values start to decrease again.
When x is eight, y is negative 485,
when x is nine, y is negative 475.
So y is increasing but then x equals 10,
y is negative 489, which is decreasing.
So we're also concerned about the graph
when x is between eight and 10
and y is somewhere between negative 450 and negative 500.
Let's use this information to adjust the window,
click window, let's have the x values go from
let's say, two to 12 so they include
the x, y is from four to six and eight to 10.
And let's change the y min to let's say negative 600
and the y max to negative 400.
let's click update and check the window.
Notice now we have a very nice view of the function,
we can see the low point on the left,
which represents a relative minimum
as well as a high point at the right,
which represents a relative maximum.
However, sometimes we do want to see the origin of the graph,
so let's adjust the window again.
Let's change the x min to negative two,
so we include x equals zero,
and let's change the y max to 100
so that y equals zero is included.
Again this is optional, but if you click update,
notice now we can also see the origin.
So we could use the previous window or this window
to determine the relative extrema,
as well as to determine if the function is increasing
or decreasing.
Let's go ahead and use this window,
let's find this low point here on the left,
to do this, we click on the calculation tab
from the dropdown menu, click minimum,
and then we're told to draw a box around the minimum.
We click, hold and drag to form a box
that includes the low point of the graph, release the mouse
and the ordered pair of the low point is given.
We have five comma negative 539, which means
we have a relative minimum of negative 539 at x equals five.
Notice the information is also recorded on the left,
and now to find the high point on the right,
which will give us a relative maximum,
go back to the dropdown menu, select maximum,
draw a box around the high point, click hold,
and drag to form the box, then release the mouse,
and we have the ordered pair for the high point.
We have a relative maximum of negative 475 at x equals nine.
Also notice how reading the graph from left to right,
the graph is decreasing from negative infinity
to x equals five then from x equals five to nine,
the graph is going uphill and therefore
the function is increasing to the right of nine
or from nine to infinity the graph is going downhill,
the function is decreasing.
Let's go back to our work and record this information.
Again we had a high point at nine comma, negative 475,
the low point at five comma, negative 539,
which means we have a local maximum of negative 475
at x equals nine, and a local minimum of negative 539
at x equals five.
And now to help us determine the intervals for which
the function is increasing or decreasing,
let's sketch a number line, plot open points
at x equals five and x equals nine.
The function will always change from increasing
or decreasing or decreasing to increasing
at the relative extrema.
And again, going back to the graph one last time
from left to right the function is decreasing, increasing,
and then decreasing.
So we have decreasing over this interval,
increasing over the middle interval
and decreasing to the right.
So using any one notation, to the left
we have negative infinity to the right we have infinity.
So the function is increasing
on the open interval from five to nine,
and the function is decreasing on the intervals from
negative infinity to five union from nine to infinity.
If we wanted to use inequalities we could say
the function is increasing, both interval where x is
greater than five and less than nine
and the function is decreasing when x is
less than five or when x is greater than nine.
I hope you found this helpful.