Question

nc math 1—released items
questions 4 through 6 are gridded response items that require you to write your answers in the boxes provided on your answer sheet. write only one number or symbol in each box and fill in the circle in each column that matches what you have printed. fill in only one circle in each column.

1. a line, ( y = mx + b ), passes through the point ( (1, 6) ) and is parallel to ( y = 4x + 5 ). what is the value of ( b )?
2. two functions are shown below.

( f(x) = \frac{1}{2} cdot 2^x )
( g(x) = 5x + 2 )
what is the largest integer value of ( x ) such that ( f(x) leq g(x) )?

1. a company models its net income, in thousands of dollars, with the function ( f(x) = 3x^2 - 54x - 144 ), where ( x ) is the number of units of its product sold. how many units of its product does the company need to sell in order for the net income to equal $0?

nc math 1—released items

1. joanne has a total of 50 coins in her purse.

• the coins are either nickels or quarters.
• the total value of the coins is $7.10.

which system of equations can be used to determine the number of nickels, ( n ), and quarters, ( q ), that joanne has in her purse?
a ( \begin{cases} n + q = 50 \\ 0.05n + 0.25q = 7.10 end{cases} )
b ( \begin{cases} n + q = 7.10 \\ 50n + 50q = 7.10 end{cases} )
c ( \begin{cases} 0.05n + 0.25q = 50 \\ n + q = 7.10 end{cases} )
d ( \begin{cases} 0.05n + 0.25q = 7.10 \\ 50n + 50q = 7.10 end{cases} )

1. the function ( f(x) = -0.25x + 5 ) models the height of a candle ( x ) seconds after it is lit. what is the meaning of the ( y )-intercept of the function?

a the initial height of the candle
b the final height of the candle
c the rate at which the candle is burning
d the amount of time it will take the candle to burn

Explanation:

Question 4

Step 1: Determine the slope of the parallel line

Parallel lines have the same slope. The given line is \( y = 4x + 5 \), so its slope \( m \) is 4. Thus, the line \( y = mx + b \) also has \( m = 4 \), so the equation becomes \( y = 4x + b \).

Step 2: Substitute the point (1, 6) into the equation to find \( b \)

Substitute \( x = 1 \) and \( y = 6 \) into \( y = 4x + b \):
\( 6 = 4(1) + b \)
\( 6 = 4 + b \)
Subtract 4 from both sides: \( b = 6 - 4 = 2 \).

• Let \( n \) = number of nickels, \( q \) = number of quarters.
• Total coins: \( n + q = 50 \) (since there are 50 coins total).
• Total value: Nickels are $0.05 each, quarters $0.25 each, so \( 0.05n + 0.25q = 7.10 \).
• This matches option A.

The y - intercept of a function \( f(x)=mx + b \) (or in this case \( f(x)=-0.25x + 5 \)) occurs when \( x = 0 \). When \( x = 0 \) (time = 0 seconds, when the candle is just lit), \( f(0)=5 \). This represents the initial height of the candle. Option B is incorrect (final height would be when the candle burns out, not the y - intercept), option C is the slope (rate of burning, - 0.25), option D is when \( f(x)=0 \) (time to burn out). So the y - intercept is the initial height.

Answer:

2

Question 7