How can you tell if a function is linear, quadratic, or exponential when given four points (1,67) (2,80) (3,97) (4,118)?

1 Answer
Aug 1, 2015

Answer:

Since we're given the values for #x=1,2,3,4# which are evenly spaced, look at differences of #y# values, then differences of differences to find that we have a quadratic:

#f(x) = 2x^2+7x+58#

Explanation:

Since the #y# values are given for #x=1, 2, 3, 4# let us write the #y# values as a sequence, then form the difference of those terms, then the difference of the differences...

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Notice that it takes #2# steps to get to a constant sequence, so this is a quadratic.

Next, let us construct the #y# value for #x=0# by adding in a column on the left...

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We do this by writing in another #4# on the last row, then calculating #9# as #13 - 4#, then calculating #58# as #67 - 9#.

Now we have added this row for #x=0# we can use its values to find the equation of the quadratic...

#f(x) = color(red)(58) + color(red)(9) * (x)/(1!) + color(red)(4) * (x(x-1))/(2!)=58+7x+2x^2#

#=2x^2+7x+58#