Question

Two linear functions are represented in different formats.

Function 1:

x y
0 5
2 ​11​
5 ​20​
8 ​29​
Function 2:

Graph of a line passing through the point begin ordered pair 0 comma negative 1 end ordered pair and the point begin ordered pair 2 comma 0 end ordered pair.

Which statements are true?

Select each correct answer.

Responses

Function 1 has a greater y-intercept than function 2.
Function 1 has a greater y -intercept than function 2.

Function 2 has a greater y-intercept than function 1.
Function 2 has a greater y -intercept than function 1.

Function 1 has a greater rate of change than function 2.
Function 1 has a greater rate of change than function 2.

Function 2 has a greater rate of change than function 1.
Function 2 has a greater rate of change than function 1.

Answer 1

To analyze the two functions, we first need to find the y-intercepts and the rates of change (slopes) for both functions.

**For Function 1:**

We can observe the coordinates given:

- At \( x = 0, y = 5 \) (this gives us the y-intercept)
- At \( x = 2, y = 11 \)
- At \( x = 5, y = 20 \)
- At \( x = 8, y = 29 \)

The y-intercept (when \( x = 0 \)) is clearly 5.

Next, to find the rate of change (slope) of Function 1, we can take any two points and calculate it using the formula:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Using the points (0, 5) and (2, 11):
\[
\text{slope} = \frac{11 - 5}{2 - 0} = \frac{6}{2} = 3
\]

**For Function 2:**

The line passes through the points \( (0, -1) \) and \( (2, 0) \).

The y-intercept is \( -1 \) (when \( x = 0 \)).

Now we calculate the slope using these two points:
\[
\text{slope} = \frac{0 - (-1)}{2 - 0} = \frac{1}{2}
\]

**Summary of results:**

- **Function 1:**
- y-intercept: \( 5 \)
- Slope (rate of change): \( 3 \)

- **Function 2:**
- y-intercept: \( -1 \)
- Slope (rate of change): \( \frac{1}{2} \)

Now we can evaluate the statements:

1. **Function 1 has a greater y-intercept than function 2.**
- True. \( 5 > -1 \)

2. **Function 2 has a greater y-intercept than function 1.**
- False. \( -1 < 5 \)

3. **Function 1 has a greater rate of change than function 2.**
- True. \( 3 > \frac{1}{2} \)

4. **Function 2 has a greater rate of change than function 1.**
- False. \( \frac{1}{2} < 3 \)

**Correct Answers:**
- Function 1 has a greater y-intercept than function 2.
- Function 1 has a greater rate of change than function 2.