If the graph of #f(x) = ax^3+bx^2+cx+d# passes through the points #(0,10)#, #(1,7)#, #(3,-11)# and #(4,-24)# then what are the values of #a, b, c, d# ?
2 Answers
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Explanation:
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Explanation:
Note that given points
#((x-x_2)(x-x_3)(x-x_4))/((x_1-x_2)(x_1-x_3)(x_1-x_4))y_1 + ((x-x_1)(x-x_3)(x-x_4))/((x_2-x_1)(x_2-x_3)(x_2-x_4))y_2+((x-x_1)(x-x_2)(x-x_4))/((x_3-x_1)(x_3-x_2)(x_3-x_4))y_3+((x-x_1)(x-x_2)(x-x_3))/((x_4-x_1)(x_4-x_2)(x_4-x_3))y_4#
That is a little tedious to calculate, so let's use a different approach for the given example.
Suppose the cubic also passes through
Hence we can analyse the
Write out the initial sequence:
#color(blue)(10), 7, p, -11, -14#
Write out the sequence of differences between successive terms:
#color(blue)(-3), p-7, -p-11, -3#
Hmmm. Notice that the first and last differences are both
#color(blue)(-3), -9, -9, -3#
Then the sequence of differences of those differences is:
#color(blue)(-6), 0, 6#
The sequence of differences of those differences is:
#color(blue)(6), 6#
Having arrived at a constant sequence (and incidentally verified our choice of
#f(x) = color(blue)(10)/(0!)+color(blue)(-3)/(1!)x+color(blue)(-6)/(2!)x(x-1)+color(blue)(6)/(3!)x(x-1)(x-2)#
#color(white)(f(x)) = 10-3x-3x^2+3x+x^3-3x^2+2x#
#color(white)(f(x)) = x^3-6x^2+2x+10#
So