Exercise Module Performance Task Page 494

To find the percentage of total tax, use the following formula. p=tax, t/income, i*100

Piecewise-Defined Function: 0.1x if 0< x≤8925 0.15x-446.25 if 8925< x ≤ 36 250 0.25x-4071.25 if 36 250 < x ≤ 87 850 0.28x-6706.75 if 87 850< x ≤ 100 000 Graph:

Tax for $50 000: 16.9 %
Tax for $100 000: 21.3 %

Practice makes perfect

The table below defines the amount of income tax a single U.S. taxpayer must pay to the federal government for income earned in 2013.

If taxable income is over	but not over	the tax is
$0	$8925	10 % of the amount over $0
$8926	$36 250	$892.50 plus 15 % of the amount over $8925
$36 251	$87 850	$4991.25 plus 25 % of the amount over $36 250
$87 851	$183 250	$17 891.25 plus 28 % of the amount over $87 850
$183 251	$398 350	$44 603.25 plus 33 % of the amount over $183 250
$398 351	$400 000	$115 586.25 plus 35 % of the amount over $398 350
$400 001	no limit	$116 163.75 plus 39.6 % of the amount over $400 000

Piecewise-Defined Function

Now, we will write a piecewise function that gives the income tax y on taxable income of x≤ $100 000. Using the table, we can write the function as the following. 0.1x if 0< x≤8925 892.50+0.15(x-8925) if 8925< x ≤ 36 250 4991.25+0.25(x-36 250) if 36 250 < x ≤ 87 850 17 891.25+0.28(x-87 850) if 87 850< x ≤ 100 000

Let's distribute the percentages to have a better look. 0.1x if 0< x≤8925 0.15x-446.25 if 8925< x ≤ 36 250 0.25x-4071.25 if 36 250 < x ≤ 87 850 0.28x-6706.75 if 87 850< x ≤ 100 000

Graph

Now, lets graph the piecewise function by determining the starting points and ending points of each ray. In order to do that, we will substitute the limits of each ray into its function.

Function	Substitution of Lower Limit	Starting Point	Substitution of Upper Limit	Ending point
0.1x	0.1* 0=0	(0,0)	0.1* 8925=892.5	(8925,892.5)
0.15x-446.25	0.15* 8925-446.25=892.65	(8925,892.5)	0.15* 36 250-446.25=4991.25	(36 250,4991.25)
0.25x-4071.25	0.25* 36 250-4071.25=4991.25	(36 250,4991.25)	0.25* 87 850-4071.25=17 891.25	(87 850,17 891.25)
0.28x-6706.75	0.28* 87 850-6706.75=17 891.25	(87 850,17 891.25)	0.28* 100 000-6706.75=21 293.25	(100 000,21 293.25)

Now, we will graph the function by plotting the points. Notice that, we will indicate the tax and income in $1000 for simplicity.

Percentage of Taxes

We can find the percent of total taxable income that a person pay in income tax as the following. p=tax, t/income, i*100 In order to apply the formula, we need to calculate the tax that each person pays. Using the table and piecewise-defined function that we wrote, we can calculate the tax as the following.

Income	Interval	Corresponding Function	Substitution	Tax
$50 000	36 250 < x ≤ 87 850	y=0.25x-4071.25	y=0.25* 50 000-4071.25	$8428.75
$100 000	87 850< x ≤ 183 250	y=0.28x-6706.75	y=0.28* 100 000-6706.75	$21 293.25