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:
-
Function 1 has a greater y-intercept than function 2.
- True. \( 5 > -1 \)
-
Function 2 has a greater y-intercept than function 1.
- False. \( -1 < 5 \)
-
Function 1 has a greater rate of change than function 2.
- True. \( 3 > \frac{1}{2} \)
-
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.