Question

Compare the two functions:

Function 1: y= 2x

Function 2:
graph has 4 points (2,4) (1,0) (0,-2) (-0.5,-2.25)
(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 based on the provided information.

**Function 1: \( y = 2x \)**

1. **Y-Intercept**: The y-intercept of a linear function \( y = mx + b \) is the value of \( y \) when \( x = 0 \). For Function 1:
\[
y = 2(0) = 0
\]
So, the y-intercept is 0.

**Function 2**: The points given are (2, 4), (1, 0), (0, -2), and (-0.5, -2.25). We can find the y-intercept and inspect the rate of change from these points.

2. **Y-Intercept**: Checking the points:
- At \( x = 0 \), \( y = -2 \) (from the point (0, -2)).
Thus, the y-intercept of Function 2 is -2.

**Comparison of Y-Intercepts**:
- Function 1: y-intercept = 0
- Function 2: y-intercept = -2
- Therefore, **Function 1 has the larger y-intercept.**

**Constant Rate of Change**:
- Function 1 is clearly linear and has a constant rate of change of 2 (the slope \( m \)).
- Function 2's rate of change can be determined by calculating the slope between points. Since the points are not linear, the rate of change is not constant.

Therefore, **Function 1 has a constant rate of change** while **Function 2 does not**.

**Rate of Change over the interval [0, 1]**:
- For Function 1:
\[
\text{Rate of change} = \frac{y(1) - y(0)}{1 - 0} = \frac{2(1) - 2(0)}{1 - 0} = \frac{2 - 0}{1} = 2
\]

- For Function 2:
- Points used: (0, -2) and (1, 0)
\[
\text{Rate of change} = \frac{y(1) - y(0)}{1 - 0} = \frac{0 - (-2)}{1 - 0} = \frac{0 + 2}{1} = 2
\]

Comparison:
- Function 1: Rate of change = 2
- Function 2: Rate of change = 2
- Therefore, **both functions have the same rate of change over the interval [0, 1].**

**Rate of Change over the interval [1, 2]**:
- For Function 1:
\[
\text{Rate of change} = \frac{y(2) - y(1)}{2 - 1} = \frac{2(2) - 2(1)}{2 - 1} = \frac{4 - 2}{1} = 2
\]

- For Function 2:
- Points used: (1, 0) and (2, 4)
\[
\text{Rate of change} = \frac{y(2) - y(1)}{2 - 1} = \frac{4 - 0}{2 - 1} = \frac{4}{1} = 4
\]

Comparison:
- Function 1: Rate of change = 2
- Function 2: Rate of change = 4
- Therefore, **Function 2 has a greater rate of change over the interval [1, 2].**

### Summary:
- **Larger y-intercept:** Function 1
- **Constant rate of change:** Function 1
- **Greater rate of change over [0, 1]:** Both functions are equal (rate of change = 2).
- **Greater rate of change over [1, 2]:** Function 2 (rate of change = 4).