8 Graphs of Quadratic Expressions: The Parabola

Transcription

1 8 Graphs of Quadratic Epressions: The Parabola In Topic 6 we saw that the graph of a linear function such as = was a straight line. The graph of a function which is not linear therefore cannot be a straight line. Here, we look at certain kinds of quadratic (non-linear) functions for which the graph is an important geometrical curve called the PARABLA (a curve studied in depth as earl as the 3rd centur B.C. b the Greeks such as Apollonius). Parabolas are of course not the onl non-linear curves of importance - others being e.g = 1 (circle), discussed here. = 1 (ellipse), = 1 (hperbola) - but the are the onl ones The shape of the parabola will be obtained eperimentall i.e., b plotting enough points to see what the curve looks like. Special features of the parabola we desire to discover are: where it cuts the and aes, the equation of its ais of smmetr, whether it points verticall up or down, and the coordinates of its verte highest or lowest point). Parabolas occur widel in the world around us, e.g., the path of a projectile (e.g., a drop of water in a fountain, a football which has been kicked) is a parabola. 8.1 Linear Curve (Line) [Revision] = = } = 2 = 2 8 1

2 8.2 Non-Linear Curve (e.g., Parabola) Firstl, let us tabulate some pairs of values of and which satisf = 2 : = = } = 2 Now plot the points representing these number-pairs: parabola = 2 ais of smmetr verte bserve that in the quadratic function f() = 2 for the parabola, we have written instead of f() (i.e., = f()). Some similar parabolas which are clearl related to the standard one drawn above are shown below. Notice that = 2 points upwards instead of downward and that = 2 2 has the same general shape and position as = 2 ecept that its -value is alwas twice as big as the corresponding -value for = 2, i.e., it is thinner (more elongated). Because 2 must alwas be 0* no matter whether is positive or negative, the parabola = 2 can never lie below the -ais (i.e., the line = 0). *Note: 0 means greater than or equal to 0 (see 9). Thus, 2 0 means that 2 is not negative. = 2 2 =

3 = 2 = 2 = = ( 1) 2 = ( 1) 2 = = = 2 2 Eercises 8.1: (i) = (ii) = 3 2 (iii) = (iv) = ( + 1) 2 (v) = ( 2) 2 (vi) = ( 1) 2 2 Sketch all of these. 8 3

4 8.3 More General Parabolas A graph often gives important information about the function it represents. Eample: = = ( 2)( 3) This epression happens to have factors, but this won t alwas be the case. As = 0 when = 2 and = 3, this means that the parabola cuts the -ais ( = 0) at = 2 and = 3. [If the epression in does not factorize, use the formula in Topic 7, Section 4 to find when = 0.] When = 0, the value of is 6, i.e. the curve cuts the -ais at = 6. To get an accurate sketch of the curve, ou will have to plot several points. The ais of smmetr will lie between = 2 and = 3, and it is obvious that = is the ais of smmetr. Putting = 2 1 in the equation of the parabola we obtain 2 ( = 2 2) 1 2 ( ) = 1 4 the verte is the point ( 2 1 2, 1 4). ais of smmetr = = ( 2)( 3) (2 1 2, 1 4 ) verte In general, the equation of the ais of smmetr for a parabola of the form = a 2 + b + c is = b 2a 8 4

5 i.e., for = , the equation of the ais of smmetr is = ( 7) 2 1 = bserve that the equation of the parabola drawn ma be written ( = 2 1 ) 2 1 (= ) 2 4 which gives us the verte and ais of smmetr ver quickl. Large values of in = When is positive and large, is also positive and large (smbolicall: implies ). When is negative and large, is positive and large (smbolicall: implies ). Note: is the smbol for infinit. is not a number. means approaches infinit, i.e. increases beond the largest positive numbers. Similarl for. If ou have difficult in (mentall or otherwise) manipulating large numbers, think of = 100 as being large enough. Eercises 8.2: Sketch (i) = (ii) = Before leaving this elementar introduction to the parabola with a vertical ais of smmetr, we should notice that there is an analogous treatment for the parabola with a horizontal ais of smmetr. The simplest instance of this kind of parabola is that given b the equation = 2 for which the graph is verte 2 = ais of smmetr 8 5

6 8.4 Answers to Eercises 8.1: (i) (ii) = = 3 2 (iii) (iv) = ( + 1) 2 = (v) 2 (vi) = ( 1) 2 2 = ( 2) 2 (1, 2) 8 6

7 8.2: (i) (ii) (1, 5) ( 1 2, ) 8 7