Question

function A:
(0,-2), (4,-1), and (8,0)

function B:
(1, 0), (4, 9), and (8, 17)

function C:
y = x + 20

1. function with the greatest rate of change _____

2. function with the smallest y-intercept _______

3. which function would benefit you the most if it represented your money earned per hour _______

Function with the highest amount of inital money

Answer 1

Let's analyze the functions based on the information provided:

### Given Functions:
1. **Function A**: Points (0, -2), (4, -1), and (8, 0)
- Slope calculation (rate of change between points):
- Between (0, -2) and (4, -1):
\[
\text{slope} = \frac{-1 - (-2)}{4 - 0} = \frac{1}{4}
\]
- Between (4, -1) and (8, 0):
\[
\text{slope} = \frac{0 - (-1)}{8 - 4} = \frac{1}{4}
\]
- Rate of change: Constant slope of \( \frac{1}{4} \).

2. **Function B**: Points (1, 0), (4, 9), and (8, 17)
- Slope calculation:
- Between (1, 0) and (4, 9):
\[
\text{slope} = \frac{9 - 0}{4 - 1} = \frac{9}{3} = 3
\]
- Between (4, 9) and (8, 17):
\[
\text{slope} = \frac{17 - 9}{8 - 4} = \frac{8}{4} = 2
\]
- Rate of change: Variable, but maximum rate is \( 3 \).

3. **Function C**: \( y = x + 20 \)
- This is a linear function with a constant slope of \( 1 \).

### Answers:
1. **Function with the greatest rate of change**:
- **Function B** (maximum slope of \( 3 \)).

2. **Function with the smallest y-intercept**:
- From Function A, the y-intercept is \( -2 \).
- From Function B, substituting \( x = 0 \) (or looking at (1,0)): \( y = 0 \).
- For Function C, the y-intercept is \( 20 \).
- Therefore, **Function A** has the smallest y-intercept.

3. **Which function would benefit you the most if it represented your money earned per hour**:
- The best option would be the function with a consistently positive increase over time, which is **Function B** as it has the highest rate of change (slope) of \( 3 \), indicating higher earnings as x increases.

4. **Function with the highest initial money**:
- Function A starts at \( -2 \) (at x = 0).
- Function B starts at \( 0 \) (at x = 1).
- Function C starts at \( 20 \) (when x = 0).
- Hence, **Function C** has the highest initial money at \( 20 \).

### Summary of Answers:
1. Function with the greatest rate of change: **Function B**
2. Function with the smallest y-intercept: **Function A**
3. Function that would benefit the most in money earned per hour: **Function B**
4. Function with the highest amount of initial money: **Function C**