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.

  1. The ability to set a good window in your TI will help you to obtain a good visualization of a function.
  2. 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.
  3. 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.
  4. . The main features of a good graph should show highlight the following:
    1. the x-intercepts and y-intercepts
    2. the turning points [maximum and/or minimum points]
    3. inflection point as the example above
    4. vertical asymptote(s) [values of x where the function is not defined]
    5. horizontal asymptote(s) [lines or curves that the graph approaches but never really reach]
  5. 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.
  6. 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.)
    1. f : x |→ 2x - 6
    2.  

       

       

       

    3. f : x |→ (x-3)2 + 5
    4.  

       

       

       

    5. f : x |→ | 2x - 6 |

       

       

       

       

    6. f : x |→ | (x-3)2 + 5 |
    7.  

       

       

       

    8. f : x |→ | - x2 |
    9.  

       

       

       

    10. f : x |→ ln x , x > 0
    11.  

       

       

       

    12. f : x |→ 2 ln (3x) - 1, x >0
    13.  

       

       

       

    14. f : x |→ 1/x, x ≠0
    15.  

       

       

       

    16. f : x |→ (2/x) + 3, x ≠0
    17.  

       

       

       

 


Summary:

Name
Form
Key Features
Linear
f : x |→ ax + b ; a ≠0
straight line stretching from negative to positive infinity, x-intercept & y-intercept.
Quadratic
f : x |→ax2 + bx + c; a ≠0
smilling face, minimum point (when a>0) or unhappy face, maximum point (when a< 0), x-intercept & y-intercept.
Absoulute

(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 the x-axis.

Exponential
f : x |→ ax + b; a ≠ 1
the y-intercept, horizontal asymptote at y = b, & the graph is always above y = b.
Logarithmic
f : x |→ (ln x) + b ; x > 0
the x-intercept, vertical asymptote at x = 0, increases to infinity
Reciprocal
f : x |→ (k / x) + b; x≠ 0
vertical asymptote at x = 0 and horizontal asymptote at y = b.
Trigonometric

(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.