Minimum Wage: The table shows minimum wage for three different years

Year 1940 1968 1997

Wage ($) 0.25 1.60 5.15

(a) Make a scatterplot of the data in the viewing rectangle

[1930, 2010, 10] by [0, 6, 1]

(b) Find a quadratic function given by f(x) = a(x - h )^2 + k that models the data.

When x = 1940, y = 0.25

0.25 = a(1940 - h)^2 + k

0.25 = a(3763600+h^2 - 3880h) + k----------------------(i)

When x = 1968, y = 1.6

1.6 = a(1968 - h)^2 + k

1.6 = a(3873024 + h^2 - 3936h) + k-----------------------(ii)

When x = 1997, y = 5.15

5.15 = a(1997 - h)^2 + k

5.15 = a(3988009 + h^2 - 3994 h) + k-------------------(iii)

Subtract (i) from (ii), we will get

1.35 =  a(109424 - 56h) -------------(iv)

Subtract (i) from (iii), we will get

4.9 = a(224409 - 114h) -------------(v)

Divide (v) by (iv)

4.9/1.35  = (224409 - 114h)/ (109424 - 56h)

Solve it for h

4.9/1.35 = (224409 - 114h)/ (109424 - 56h)

4.9(109424 - 56h) = 1.35(224409 - 114h)

536177.6 - 274.4h = 302952.15 - 153.9h

- 274.4h + 153.9h = 302952.15 - 536177.6

-120.5h = -233225.45

h = -233225.45/-120.5

h = 1935.48

from (iv)

1.35 = a(109424 - 56*1935.48)

1.35 = a*1037.12

a = 1.35/1037.12

a = 0.0013

From (i)

Now substitute values of h and a in (i0

0.25 = 0.0013(3763600+1935.48^2 - 3880*1935.48) + k

0.25 = 0.0013(20.4304) + k

0.25 = 0.02655952 + k

k = 0.25 - 0.02655952

k = 0.22344

Substitute values of a, h and k into f(x) = a(x - h )^2 + k

The function will be

y = 0.0013(x - 1935.48)^2 + 0.22344

(c) Estimate the minimum wage in 1976 and compare it to the actual value of $2.30.

Put x = 1976

y = 0.0013(1976 - 1935.48)^2 + 0.22344

y = 2.358

2.358 - 2.3 = 0.058

Difference is 0.058.

(d) Estimate when the minimum wage was $1.00.

Put y = 1

1 = 0.0013(x - 1935.48)^2 + 0.22344

0.77656 = 0.0013(x - 1935.48)^2

x = 1911.04, 1959.92

Minimum wage was $1 during the years 1911 and 1959.

(e) If current trends continue, predict the minimum wage in 2009. Compare it to the projected value of $7.25.

Put x = 2009

y = 0.0013(2009 - 1935.48)^2 + 0.22344

y = 7.2502

Difference = 7.2502-7.25 = 0.0002

Difference is 0.0002.