Question

semester 1 review
two organizations are collecting donations for a cause. organization a begins with one donation, and the number of donations triples each month. the table shows the number of donations for organization b.

organization b

time (hours), t	number of donations, y
1	10
2	15
3	20
4	25
5	30
6	35

a) what type of function represents organization a? _____________
b) what type of function represents organization b? _____________

what is the equation of the graph below?
graph of a line with axes labeled, passing through points, with options: a) $y = -4x + 3$; b) $y = -3x + 4$; c) $y = 3x - 4$; d) $y = 4x - 3$

francisco sets up a snow cone stand. he spends $50 on supplies. he charges $3 for each snow cone. if $p$ represents francisco’s profit in dollars of selling $n$ snow cones, which of the following equations represents this situation?
a) $p(n) = 3n - 50$
b) $p(n) = 3n + 50$
c) $p(n) = 50n - 3$
d) $p(n) = 50n + 3$

Explanation:

Part 23a

Step1: Analyze Organization A's growth

Organization A starts with 1 donation and triples each month. So the number of donations at month \( t \) is \( y = 3^t \) (since it's a multiplicative growth by a constant factor each time).

Step2: Identify the function type

A function where the output is multiplied by a constant factor (here 3) for each unit increase in the input (month) is an exponential function.

Step1: Analyze Organization B's data

Looking at the table for Organization B, when \( t \) increases by 1 (from 0 to 1, 1 to 2, etc.), \( y \) increases by 5 (5 to 10, 10 to 15, etc.). The rate of change (slope) is constant (\( \frac{\Delta y}{\Delta t}=\frac{5}{1} = 5 \)).

Step2: Identify the function type

A function with a constant rate of change (slope) is a linear function. The general form is \( y=mt + b \), here \( m = 5 \) and \( b = 5 \) (when \( t = 0 \), \( y = 5 \)).

Step1: Find the slope (\( m \))

The graph passes through points, let's take two points. From the graph, when \( x = 0 \), \( y=- 4 \) (y - intercept, \( b=-4 \)) and when \( x = 1 \), \( y=-1 \) (since from (0, - 4) moving 1 unit right and 3 units up). So slope \( m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{-1-(-4)}{1 - 0}=\frac{3}{1}=3 \).

Step2: Write the equation

Using the slope - intercept form \( y=mx + b \), with \( m = 3 \) and \( b=-4 \), the equation is \( y = 3x-4 \), which is option c.

Answer:

Exponential Function

Part 23b