Question

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

(4, 17)
(3, 9)
(2, 5)
(1, 3)
( 0, 2)

What function has the larger y intercept
Which function has a constant rate of change
Which function has the greater rate of change in the interval ( 0, 2)
Which function has the greater rate of change in the interval ( 2, 4)

Function one
Function two
They have the same y Intercept
They have the same rate of change in this interval

Answer 1

To compare the two functions represented by the given points, we will first identify each function.

**Function One:**
This function is represented by the points:
- (-1, -1)
- (0, 3)
- (1, 7)
- (2, 11)
- (4, 19)
- (5, 23)

**Function Two:**
This function is represented by the points:
- (4, 17)
- (3, 9)
- (2, 5)
- (1, 3)
- (0, 2)

### Step 1: Identify y-intercepts
- For **Function One**, the y-intercept is the value of y when x = 0, which is 3.
- For **Function Two**, the y-intercept (when x = 0) is 2.

**Conclusion:** Function One has the larger y-intercept.

### Step 2: Determine which function has a constant rate of change
- **Function One:**
- To find the rate of change, we can calculate the differences in y values over the differences in x values for two intervals:
- From (0, 3) to (1, 7): \((7 - 3) / (1 - 0) = 4\)
- From (1, 7) to (2, 11): \((11 - 7) / (2 - 1) = 4\)
- From (2, 11) to (4, 19): \((19 - 11) / (4 - 2) = 4\)
- From (4, 19) to (5, 23): \((23 - 19) / (5 - 4) = 4\)

- The rate of change for Function One is constant at 4.

- **Function Two:**
- Using the same method:
- From (0, 2) to (1, 3): \((3 - 2) / (1 - 0) = 1\)
- From (1, 3) to (2, 5): \((5 - 3) / (2 - 1) = 2\)
- From (2, 5) to (3, 9): \((9 - 5) / (3 - 2) = 4\)
- From (3, 9) to (4, 17): \((17 - 9) / (4 - 3) = 8\)

- The rate of change for Function Two is not constant.

**Conclusion:** Function One has a constant rate of change.

### Step 3: Calculate the rate of change in the interval (0, 2)
- For **Function One** in the interval (0, 2):
- Change in y: \(11 - 3 = 8\)
- Change in x: \(2 - 0 = 2\)
- Rate of change: \(8 / 2 = 4\)

- For **Function Two** in the interval (0, 2):
- Change in y: \(5 - 2 = 3\)
- Change in x: \(2 - 0 = 2\)
- Rate of change: \(3 / 2 = 1.5\)

**Conclusion:** Function One has the greater rate of change in the interval (0, 2).

### Step 4: Calculate the rate of change in the interval (2, 4)
- For **Function One** in the interval (2, 4):
- Change in y: \(19 - 11 = 8\)
- Change in x: \(4 - 2 = 2\)
- Rate of change: \(8 / 2 = 4\)

- For **Function Two** in the interval (2, 4):
- Change in y: \(17 - 5 = 12\)
- Change in x: \(4 - 2 = 2\)
- Rate of change: \(12 / 2 = 6\)

**Conclusion:** Function Two has the greater rate of change in the interval (2, 4).

### Final Summary
1. **Larger y-intercept:** Function One
2. **Constant rate of change:** Function One
3. **Greater rate of change in (0, 2):** Function One
4. **Greater rate of change in (2, 4):** Function Two