need to rewrite the table if you know how write the equation for the table

Step-by-step explanation:

A function is linear if x-values are evenly spaced (all have the same difference) and y-values are evenly spaced (all have the same difference).

a) x-values have a difference of 6 -3 = 9 -6 = 12 - 9 = 3. y-values have a difference of 30 -36 = 24 -30 = 18 -24 = -6. Both these differences are constant, so the function is linear.

The ratio of y-differences to x-differences is -6/3 = -2, so that is the slope of the line. We can use the point-slope form to discover an equation in slope-intercept form.

Point-slope form of the equation for a line with slope m through point (h, k) can be written as ...

  y = m(x -h) +k

Here, we have m = -2, and the first point is (h, k) = (3, 36). Then our line's equation can be written as ...

  y = -2(x -3) +36

  y = -2x +42 . . . . . . . eliminating parentheses

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b) The x-values in this table have a constant difference of ...

  3 -1 = 5 -3 = 7 -5 = 2

The y-values have differences of 9 -2 = 7, 20 -9 = 11, 35 -20 = 15. These are not constant, so the relation is non-linear. These first differences have differences of ...

  11 -7 = 15 -11 = 4 . . . . . second differences are constant

When second differences are constant, the relation can be described by a second degree polynomial. We can write some equations to discover what that polynomial is.

Generic form:

  ax² +bx +c = y

Filling in three of the given points, we have three equations in a, b, c:

  a·1² + b·1 +c = 2

  a·3² +b·3 +c = 9

  a·5² +b·5 +c = 20

Subtracting the first equation from the other two eliminates c and gives two equations in a and b:

  a·(9 -1) +b(3 -1) = 9 -2 . . . . . . . . . 8a +2b = 7

  a·(25 -1) + b·(5 -1) = 20 -2 . . . . .  24a +4b = 18

Subtracting twice the first of these equations from the second, we can eliminate b:

  a(24 -2·8) = 18 -2·7

  8a = 4

  a = 1/2 . . . . . divide by 8

Substituting this into the first of the equations in a and b, we get:

  8·(1/2) +2b = 7

  2b = 3 . . . . . subtract 4

  b = 3/2 . . . . divide by 2

Substituting for a and b in the first of our original equations, we find ...

  (1/2)·1 + (3/2)·(1) +c = 2

  2 +c = 2 . . . simplify

  c = 0 . . . . . . subtract 2

So, the table in part b can be described by the quadratic equation ...

  y = (1/2)x² + (3/2)x

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step-by-step explanation: