Question
Compare the two functions.
Function 1:
x -2 -1 0 1 2 3
y -3 2 7 12 17 22
Function 2:
graph with a line passing points (0,2) (1,4) (2,8)
(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
A) Which function has the larger y-intercept? Response area
B) Which function has a constant rate of change? Response area
C) Which function has the greater rate of change in the interval [0, 2]? Response area
D) Which function has the greater rate of change in the interval [1, 3]? Response area
Function 1:
x -2 -1 0 1 2 3
y -3 2 7 12 17 22
Function 2:
graph with a line passing points (0,2) (1,4) (2,8)
(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
A) Which function has the larger y-intercept? Response area
B) Which function has a constant rate of change? Response area
C) Which function has the greater rate of change in the interval [0, 2]? Response area
D) Which function has the greater rate of change in the interval [1, 3]? Response area
Answers
GPT-4o mini
To compare the two functions, we need to analyze the data provided for Function 1 and the properties of Function 2.
**Function 1 Data:**
- For the points given:
- \( x = 0 \) gives \( y = 7 \)
- Therefore, the y-intercept of Function 1 is 7.
**Function 2 Properties:**
- The points on Function 2 are:
- \( (0, 2) \)
- \( (1, 4) \)
- \( (2, 8) \)
- The y-intercept is 2 (the value of y when \( x = 0 \)).
The slope (or rate of change) of Function 2 can be calculated using the formula for slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
- Between points \( (0, 2) \) and \( (1, 4) \):
- \( m = \frac{4 - 2}{1 - 0} = \frac{2}{1} = 2 \)
- Between points \( (1, 4) \) and \( (2, 8) \):
- \( m = \frac{8 - 4}{2 - 1} = \frac{4}{1} = 4 \)
Thus, the function has a varying rate of change.
**Now, let's answer the questions:**
A) **Which function has the larger y-intercept?**
- **Function 1** has a y-intercept of 7, **Function 2** has a y-intercept of 2.
- **Response:** Function 1
B) **Which function has a constant rate of change?**
- Function 1 can be assessed by looking for a consistent increment in y. The differences are:
- \( 2 - (-3) = 5 \)
- \( 7 - 2 = 5 \)
- \( 12 - 7 = 5 \)
- \( 17 - 12 = 5 \)
- \( 22 - 17 = 5 \)
- Function 2 has varying slopes between the points calculated (2 from 0 to 1 and 4 from 1 to 2).
- **Response:** Function 1
C) **Which function has the greater rate of change in the interval [0, 2]?**
- For Function 1 from \(x=0\) to \(x=2\):
- \( m = \frac{17 - 7}{2 - 0} = \frac{10}{2} = 5 \)
- For Function 2 from \(x=0\) to \(x=2\):
- Calculated earlier: the average rate of change is \( m = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3 \).
- **Response:** Function 1
D) **Which function has the greater rate of change in the interval [1, 3]?**
- For Function 1 from \(x=1\) to \(x=3\):
- \( m = \frac{22 - 12}{3 - 1} = \frac{10}{2} = 5 \)
- Function 2 does not extend to \(x=3\), so we can only evaluate Function 1 here.
- **Response:** Function 1
**Final Response Summary:**
- A) Function 1
- B) Function 1
- C) Function 1
- D) Function 1
**Function 1 Data:**
- For the points given:
- \( x = 0 \) gives \( y = 7 \)
- Therefore, the y-intercept of Function 1 is 7.
**Function 2 Properties:**
- The points on Function 2 are:
- \( (0, 2) \)
- \( (1, 4) \)
- \( (2, 8) \)
- The y-intercept is 2 (the value of y when \( x = 0 \)).
The slope (or rate of change) of Function 2 can be calculated using the formula for slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \):
- Between points \( (0, 2) \) and \( (1, 4) \):
- \( m = \frac{4 - 2}{1 - 0} = \frac{2}{1} = 2 \)
- Between points \( (1, 4) \) and \( (2, 8) \):
- \( m = \frac{8 - 4}{2 - 1} = \frac{4}{1} = 4 \)
Thus, the function has a varying rate of change.
**Now, let's answer the questions:**
A) **Which function has the larger y-intercept?**
- **Function 1** has a y-intercept of 7, **Function 2** has a y-intercept of 2.
- **Response:** Function 1
B) **Which function has a constant rate of change?**
- Function 1 can be assessed by looking for a consistent increment in y. The differences are:
- \( 2 - (-3) = 5 \)
- \( 7 - 2 = 5 \)
- \( 12 - 7 = 5 \)
- \( 17 - 12 = 5 \)
- \( 22 - 17 = 5 \)
- Function 2 has varying slopes between the points calculated (2 from 0 to 1 and 4 from 1 to 2).
- **Response:** Function 1
C) **Which function has the greater rate of change in the interval [0, 2]?**
- For Function 1 from \(x=0\) to \(x=2\):
- \( m = \frac{17 - 7}{2 - 0} = \frac{10}{2} = 5 \)
- For Function 2 from \(x=0\) to \(x=2\):
- Calculated earlier: the average rate of change is \( m = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3 \).
- **Response:** Function 1
D) **Which function has the greater rate of change in the interval [1, 3]?**
- For Function 1 from \(x=1\) to \(x=3\):
- \( m = \frac{22 - 12}{3 - 1} = \frac{10}{2} = 5 \)
- Function 2 does not extend to \(x=3\), so we can only evaluate Function 1 here.
- **Response:** Function 1
**Final Response Summary:**
- A) Function 1
- B) Function 1
- C) Function 1
- D) Function 1