Question

Compare the two functions:

Function 1: y= 2x



Function 2:
(-0.5,-2.25) (0,-2) (1,0)(2,4)
(4 points)
Which function has the larger y-intercept?


Which function has a constant rate of change?


Compare the rate of change over the interval of [0, 1]. Which function has the greater rate of change?

Compare the rate of change over the interval of [1, 2]. Which function has the greater rate of change?

Answer 1

Let's analyze the two functions:

**Function 1: \(y = 2x\)**
This is a linear function with a slope (rate of change) of 2, and a y-intercept of (0, 0).

**Function 2:**
It is defined by the points:
- (-0.5, -2.25)
- (0, -2)
- (1, 0)
- (2, 4)

To determine the characteristics of Function 2, we'll first need to find its rate of change between the provided points.

### 1. Y-Intercept Comparison:
- **Function 1:** The y-intercept is 0 (from \(y = 2x\), when \(x=0\), \(y=0\)).
- **Function 2:** The point (0, -2) indicates that the y-intercept is -2.

**Conclusion:** Function 1 has a larger y-intercept (0) compared to Function 2 (-2).

### 2. Constant Rate of Change Comparison:
- **Function 1:** The rate of change (slope) is constant at 2.
- **Function 2:** We can calculate the slope between the different pairs of points to see if it’s constant.
- From (-0.5, -2.25) to (0, -2):
\[
\text{slope} = \frac{-2 - (-2.25)}{0 - (-0.5)} = \frac{0.25}{0.5} = 0.5
\]
- From (0, -2) to (1, 0):
\[
\text{slope} = \frac{0 - (-2)}{1 - 0} = \frac{2}{1} = 2
\]
- From (1, 0) to (2, 4):
\[
\text{slope} = \frac{4 - 0}{2 - 1} = \frac{4}{1} = 4
\]

Since the slopes between the points of Function 2 are not constant, **Function 1** has a constant rate of change.

### 3. Rate of Change Over the Interval [0, 1]:
- **Function 1:** From \(x = 0\) to \(x = 1\):
\[
\text{slope} = 2 \text{ (constant)}
\]
- **Function 2:** From (0, -2) to (1, 0):
\[
\text{slope} = 2
\]

**Conclusion:** Both functions have the same rate of change of 2 over the interval [0, 1].

### 4. Rate of Change Over the Interval [1, 2]:
- **Function 1:** From \(x = 1\) to \(x = 2\):
\[
\text{slope} = 2 \text{ (constant)}
\]
- **Function 2:** From (1, 0) to (2, 4):
\[
\text{slope} = 4
\]

**Conclusion:** Function 2 has a greater rate of change (4) compared to Function 1 (2) over the interval [1, 2].

### Final Summary:
- Function 1 has the larger y-intercept (0 vs. -2).
- Function 1 has a constant rate of change (2), while Function 2 does not.
- Over the interval [0, 1], both functions have the same rate of change (2).
- Over the interval [1, 2], Function 2 has a greater rate of change (4 vs. 2).