Question

m5 - performance task - 8.f.a.2/3 - functions
mr. and mrs. sanchez plan for a college education when their baby is born. they have already saved $500 and want to invest that money in a savings account at a local bank until their child is 18 years old. one account that the bank offers provides 4% interest to parents for college tuition.

part a
the bank offers two accounts. one account has a 4% simple interest rate. simple interest i can be found with i = prt, using the principal p, the decimal rate r, and the time t in years.
the equation y = 500 + 500(0.04)x represents investing $500 in this account, where y is the total amount in the account and x is the number of years invested.
does the equation represent a function? is it linear? explain your reasoning.

part b
the bank also has an account offering 4% monthly compound interest, which means that the interest will be applied to both the original principal and any interest accumulated each month. the table and graph show how the total amount in the account y will change over x years.

| compound interest |

x (years)	y (total $)
0	500
1	520.37
3	563.64
6	635.37
9	716.24
12	807.39
18	1,025.99

savings accounts
graph with x-axis time (years) from 0 to 18, y-axis amount in account ($) from 500 to 1050, showing a curve labeled compound interest

do the table and graph represent a function? is it linear? explain your reasoning.

Explanation:

Part A

Step1: Simplify the given equation

$y = 500 + 500(0.04)x = 500 + 20x$

Step2: Check function definition

Each input $x$ has exactly one output $y$.

Step3: Check linear form

Matches $y=mx+b$ (linear form).

Step1: Check function definition

Each $x$ (years) has one unique $y$ (amount).

Step2: Check linearity

Check rate of change:
Between $x=0$ and $x=1$: $\frac{520.37-500}{1-0}=20.37$
Between $x=1$ and $x=3$: $\frac{563.64-520.37}{3-1}=21.635$
Rates are not constant; graph is curved.

Answer:

The equation is a function because every input value of $x$ (years) corresponds to exactly one output value of $y$ (account amount). It is linear because it can be rewritten in the standard linear form $y=20x+500$, which follows the structure $y=mx+b$ where $m$ (slope) and $b$ (y-intercept) are constants.

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Part B