Question

Compare the two functions:
Function 1: x -1 0 1 2 4 5
y -1 3 7 11 19 23

Function 2: x 0 1 2 3 4
y 2 3 5 9 17

Which function has the larger y-intercept? Response area

Which function has a constant rate of change? Response area

Which function has the greater rate of change in the interval [0,2]? Response area

Which function has the greater rate of change in the interval [2,4]?

Answer 1

Let's analyze both functions step by step to answer all the provided questions.

### Function 1:
- Data points are ( -1, -1 ), ( 0, 3 ), ( 1, 7 ), ( 2, 11 ), ( 4, 19 ), ( 5, 23 ).

### Function 2:
- Data points are ( 0, 2 ), ( 1, 3 ), ( 2, 5 ), ( 3, 9 ), ( 4, 17 ).

### 1. Which function has the larger y-intercept?
The y-intercept is the value of y when x = 0.

- **Function 1**: When x = 0, y = 3.
- **Function 2**: When x = 0, y = 2.

**Conclusion**: Function 1 has the larger y-intercept (3 vs. 2).

### 2. Which function has a constant rate of change?
A function has a constant rate of change if the increase in y values is consistent for equal increases in x values.

- **Function 1**:
- From x = -1 to 0: y changes from -1 to 3 (change = 4).
- From x = 0 to 1: y changes from 3 to 7 (change = 4).
- From x = 1 to 2: y changes from 7 to 11 (change = 4).
- From x = 2 to 4: y changes from 11 to 19 (change = 8).
- From x = 4 to 5: y changes from 19 to 23 (change = 4).

The change is not constant since the rise from 2 to 4 is different (8).

- **Function 2**:
- From x = 0 to 1: y changes from 2 to 3 (change = 1).
- From x = 1 to 2: y changes from 3 to 5 (change = 2).
- From x = 2 to 3: y changes from 5 to 9 (change = 4).
- From x = 3 to 4: y changes from 9 to 17 (change = 8).

The change is not constant either.

**Conclusion**: Neither function has a constant rate of change.

### 3. Which function has the greater rate of change in the interval [0,2]?
We calculate the rate of change as the change in y over the change in x.

- **Function 1** from x = 0 to 2:
- y from 3 to 11, change in y = 11 - 3 = 8.
- Change in x = 2 - 0 = 2.
- Rate of change = 8 / 2 = 4.

- **Function 2** from x = 0 to 2:
- y from 2 to 5, change in y = 5 - 2 = 3.
- Change in x = 2 - 0 = 2.
- Rate of change = 3 / 2 = 1.5.

**Conclusion**: Function 1 has the greater rate of change in the interval [0, 2] (4 vs. 1.5).

### 4. Which function has the greater rate of change in the interval [2,4]?
We again calculate the rate of change on the given intervals.

- **Function 1** from x = 2 to 4:
- y from 11 to 19, change in y = 19 - 11 = 8.
- Change in x = 4 - 2 = 2.
- Rate of change = 8 / 2 = 4.

- **Function 2** from x = 2 to 4:
- y from 5 to 17, change in y = 17 - 5 = 12.
- Change in x = 4 - 2 = 2.
- Rate of change = 12 / 2 = 6.

**Conclusion**: Function 2 has a greater rate of change in the interval [2, 4] (6 vs. 4).

### Summary of conclusions:
1. **Larger y-intercept**: Function 1
2. **Constant rate of change**: Neither
3. **Greater rate of change [0,2]**: Function 1
4. **Greater rate of change [2,4]**: Function 2