1. 1.4 Graphing Calculators and Computers
FUNCTIONS AND MODELS
1.4
Graphing Calculators
and Computers
In this section, we will learn about:
The advantages and disadvantages of
using graphing calculators and computers.

2. The default screen often gives an incomplete or misleading picture.
VIEWING RECTANGLE
The default screen often gives an
incomplete or misleading picture.
So, it is important to choose the viewing
rectangle with care.

3. If we choose the x-values to range from a
VIEWING RECTANGLE
If we choose the x-values to range from a
minimum value of Xmin = a to a maximum
value of Xmax = b and the y-values to range
from a minimum of Ymin = c to a maximum of
Ymax = d, then the visible portion of the
graph lies in the rectangle

4. The rectangle is shown in the figure.
VIEWING RECTANGLE
The rectangle is shown in the figure.
We refer to this rectangle as the [a, b] by [c, d]
viewing rectangle.

5. Draw the graph of the function f(x) = x2 + 3
GRAPHING
Example 1
Draw the graph of the function f(x) = x2 + 3
in each of these viewing rectangles.
[-2, 2] by [-2, 2]
[-4, 4] by [-4, 4]
[-10, 10] by [-5, 30]
[-50, 50] by [-100, 1000]

6. We select the range by setting Xmin = -2,
GRAPHING
Example 1 a
We select the range by setting Xmin = -2,
Xmax = 2,Ymin = -2 and Ymax = 2.
The resulting graph is shown.
The display window is blank.
f(x) = x2 + 3

7. A moment’s thought provides the explanation.
GRAPHING
Example 1 a
A moment’s thought provides
the explanation.
Notice that for all x, so for all x.
Thus, the range of the function is
This means that the graph of f lies entirely outside the viewing rectangle by [-2, 2] by [-2, 2].

8. USE ZOOM 0 The graphs for the viewing rectangles
GRAPHING
Example 1 b, c, d
The graphs for the viewing rectangles
in (b), (c), and (d) are shown.
Observe that we get a more complete picture in (c) and (d).
However, in (d), it is not clear that the y-intercept is 3.
USE ZOOM 0

9. Determine an appropriate viewing rectangle for the function
GRAPHING
Example 2
Determine an appropriate viewing
rectangle for the function
and use it to graph f.

10. The expression for f(x) is defined when:
GRAPHING
Example 2
The expression for f(x) is defined when:
Thus, the domain of f is the interval [-2, 2].
Also,
So, the range of f is the interval

11. We choose the viewing rectangle so that
GRAPHING
Example 2
We choose the viewing rectangle so that
the x-interval is somewhat larger than
the domain and the y-interval is larger than
the range.
Taking the viewing rectangle to be [-3, 3] by [-1, 4], we get the graph shown here.

12. Graph the function . GRAPHING Example 3
Here, the domain is , the set of all real numbers.
That doesn’t help us choose a viewing rectangle.

13. If we start with the viewing rectangle
GRAPHING
Example 3
If we start with the viewing rectangle
to [-20, 20] by [-20, 20], we get
the picture shown.
The graph appears to consist of vertical lines.
However, we know that can’t be correct.

14. So, we change the viewing rectangle to [-20, 20] by [-500, 500].
GRAPHING
Example 3
So, we change the viewing rectangle to
[-20, 20] by [-500, 500].
The resulting graph is shown.
It still doesn’t quite reveal all the main features of the function.

15. So, we try [-20, 20] by [-1000, 1000], as in the figure. GRAPHING
Example 3
So, we try [-20, 20] by [-1000, 1000],
as in the figure.
Now, we are more confident that we have arrived at an appropriate viewing rectangle.
In Chapter 4, we will be able to see that the graph shown here does indeed reveal all the main features of the function.

16. in an appropriate viewing rectangle.
GRAPHING
Example 4
Graph the function
f(x) = sin 50x
in an appropriate viewing rectangle.

17. The figure shows the graph of f produced by
GRAPHING
Example 4
The figure shows the graph of f produced by
a graphing calculator using the viewing
rectangle [-12, 12] by [-1.5, 1.5].
At first glance, the graph appears to be reasonable.

18. However, if we change the viewing
GRAPHING
Example 4
However, if we change the viewing
rectangle to the ones shown in the other
figures, the graphs look very different.
Something strange is happening.

19. To explain the big differences in appearance
GRAPHING
Example 4
To explain the big differences in appearance
of those graphs and to find an appropriate
viewing rectangle, we need to find the period
of the function y = sin 50x.

20. We know that the function y = sin x has
GRAPHING
Example 4
We know that the function y = sin x has
period and the graph of y = sin 50x
is compressed horizontally by a factor of 50.
So, the period of y = sin 50x is:
This suggests that we should deal only with small values of x to show just a few oscillations of the graph.

21. If we choose the viewing rectangle
GRAPHING
Example 4
If we choose the viewing rectangle
[-0.25, 0.25] by [-1.5, 1.5], we get
the graph shown here.

22. Now, we see what went wrong in the earlier graphs.
GRAPHING
Example 4
Now, we see what went wrong in
the earlier graphs.
The oscillations of y = sin 50x are so rapid that, when the calculator plots points and joins them, it misses most of the maximum and minimum points.
Thus, it gives a very misleading impression of the graph.

23. GRAPHING
Example 5
Graph the function

24. The figure shows the graph of f produced by
GRAPHING
Example 5
The figure shows the graph of f produced by
a graphing calculator with viewing rectangle
[-6.5, 6.5] by [-1.5, 1.5].
It looks much like the graph of y = sin x, but perhaps with some bumps attached.

25. If we zoom in to the viewing rectangle
GRAPHING
Example 5
If we zoom in to the viewing rectangle
[-0.1, 0.1] by [-0.1, 0.1], we can see much
more clearly the shape of these bumps—as
in the other figure.

26. The reason for this behavior is that the second term, , is very small
GRAPHING
Example 5
The reason for this behavior is that
the second term, , is very small
in comparison with the first term, sin x.

27. Thus, we really need two graphs
GRAPHING
Example 5
Thus, we really need two graphs
to see the true nature of this function.

28. GRAPHING
Example 6
Draw the graph of the
function

29. The figure shows the graph produced
GRAPHING
Example 6
The figure shows the graph produced
by a graphing calculator with viewing
rectangle [-9, 9] by [-9, 9].

30. In connecting successive points on the graph,
GRAPHING
Example 6
In connecting successive points on the graph,
the calculator produced a steep line segment
from the top to the bottom of the screen.
That line segment is not truly part of the graph.

31. of the function y = 1/(1 – x) is {x | x ≠ 1}.
GRAPHING
Example 6
Notice that the domain
of the function y = 1/(1 – x) is
{x | x ≠ 1}.

32. We can eliminate the extraneous near-vertical line by experimenting
GRAPHING
Example 6
We can eliminate the extraneous
near-vertical line by experimenting
with a change of scale.

33. When we change to the smaller viewing
GRAPHING
Example 6
When we change to the smaller viewing
rectangle [-4.7, 4.7] by [-4.7, 4.7] on this
particular calculator, we obtain the much
better graph in the other figure.