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pair of linear equations in two variables like the crest of a peacock so is mathematics at the head of all
actually we will be seeing these sort of divisions so frequently that wed like a quicker and more efficient way of doing them luckily there is
if the ratios of the polynomial ax33bx23cxd are in ap prove that 2b3-3abca2d0ans let px ax3 3bx2 3cx
1 determine the intercepts if there are any recall that the y-intercept is specified by0 f 0 and we determine the x-intercepts by setting the
if alpha amp szlig are the zeroes of the polynomial 2x2 - 4x 5 then find the value of aalpha2 szlig2 b 1 alpha 1 szlig c alpha - szlig2 d
vertical asymptotein our graph as the value of x approaches x 0 the graph begin gets extremely large on both sides of the line given by x 0 this
remember that a graph will have a y-intercept at the point 0 f 0 though in this case we have to ignore x 0 and thus this graph will never cross
find out the symmetry of find out the symmetry of
weve some rather simply tests for each of the distinct types of symmetry1 a graph will have symmetry around the x-axis if we get an equal equation
the last set of transformations which were going to be looking at in this section isnt shifts but in spite of they are called reflections amp there
if alphabeta are the zeros of a quadratic polynomial such that alpha beta 24 alpha - beta 8 find a quadratic polynomial having alpha and beta as
we now can also combine the two shifts we only got done looking at into single problem if we know the graph of f x the graph of g x f x c
horizontal shiftsthese are quite simple as well though there is one bit where we have to be carefulgiven the graph of f x the graph of g x f x
if the squared difference of the zeros of the quadratic polynomial x2 px 45 is equal to 144 find the value of
in this section we will see how knowledge of some rather simple graphs can help us graph some more complexes graphs collectively the methods we
on dividing the polynomial 4x4 - 5x3 - 39x2 - 46x - 2 by the polynomial gx the quotient is x2 - 3x - 5 and the remainder is -5x 8find the polynomial
if alpha beta are the zeros of the polynomial x2 8x 6 frame a quadratic polynomial whose zeros are a 1alpha and 1beta b 1 betaalpha 1
miscellaneous functionsthe importance of this section is to introduce you with some other functions that dont really need the work to graph that the
there are also two lines on each of the graph these lines are called asymptotes and as the graphs illustrates as we make x large in both the ve and
note that the right side has to be a 1 to be in standard form the point h k is called the center of the ellipseto graph the ellipse all that we
convert following into the convert following into the
if alphabeta are the zeros of the polynomial 2x2 - 4x 5 find the value of a alpha2 beta2 b alpha - beta2ans p x 2 x2 - 4 x
as a last topic in this section we have to briefly talk about how to take a parabola in the general form amp convert it into the following
as noted earlier most parabolas are not given in that form so we have to take a look at how to graph a parabola which is in the general
lets go through first form of the parabola f x a x - h 2 kthere