Key Feactures in Graphs (with TI)
Aims: By the end of this chapter, you will
be able to
(i) determine the key features of graphs, and
(ii) obtain a good graph with the use of GDC.
- The ability to set a good window in your TI will help you to obtain a good
visualization of a function.
- Say, you have a function y = 4x3 + 5 and you like to graph it
on your TI. To get a good visualization, you need to have a window that is
similar to the green window. Any of the red windows will give you an incomplete
picture of the function.
- Why is the green window is the best window? Well, the green window allows
you to see the most distinguishing feature of this function, that is the inflection
point (0,5), the x-intercept and the y-intercept. An inflection point is the
point that shows a change of curvature from large to small then to large again.
- . The main features of a good graph should show highlight the following:
- the x-intercepts and y-intercepts
- the turning points [maximum and/or minimum points]
- inflection point as the example above
- vertical asymptote(s) [values of x where the function is not defined]
- horizontal asymptote(s) [lines or curves that the graph approaches but
never really reach]
- A good sketch should also provide the coordinates of all intercepts, turning
points and inflection points. A good sketch should also identify various asymptotes
clearly with dotted lines. However, not all features in note 4 will necessary
exist in any given function.
- Investigation: Use your graphics
calculator to sketch a graph and identify all the important key features in
the following functions. (Use list in note 4 as a guide.)
- f : x |→ 2x - 6
- f : x |→ (x-3)2 + 5
- f : x |→ | 2x - 6 |
- f : x |→ | (x-3)2 + 5 |
- f : x |→ | - x2 |
- f : x |→ ln x , x > 0
- f : x |→ 2 ln (3x) - 1, x >0
- f : x |→ 1/x, x ≠0
- f : x |→ (2/x) + 3, x ≠0
f : x |→ ax + b ; a ≠0
|straight line stretching from negative to positive infinity,
x-intercept & y-intercept.
f : x |→ax2 + bx + c;
|smilling face, minimum point (when a>0) or
unhappy face, maximum point (when a< 0), x-intercept
(a) f : x |→ | ax + b | ; a ≠0
(b) f : x |→ | ax2 + bx +
c | ; a ≠0
(a) V shape, a minimum point on x-axis and the rest
above the x-axis.
(b) W shape, 2 minimum points on x-axis and the rest
above the x-axis OR parabola with one minimum point on
f : x |→ ax + b; a ≠
|the y-intercept, horizontal asymptote at y =
b, & the graph is always above y = b.
f : x |→ (ln x) + b ; x >
|the x-intercept, vertical asymptote at x = 0,
increases to infinity
f : x |→ (k / x) + b; x≠ 0
|vertical asymptote at x = 0 and horizontal asymptote
at y = b.
(a) f : x |→ ksin(ax+b) OR
f : x |→ kcos(ax+b)
(b) f : x |→ ktan(ax+b)
(a) maximum & minimum points & all x-intercepts.
(b) vertical asymptotes.