To analyze the two functions, let’s first determine the equations represented by the points provided.
Function B: Points given are (0, 3), (4, -9), (8, -21). We can calculate the slope (rate of change) using the formula:
\[ \text{slope} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's use points (0, 3) and (4, -9):
\[ \text{slope} = \frac{-9 - 3}{4 - 0} = \frac{-12}{4} = -3 \]
Now let’s find the slope between (4, -9) and (8, -21):
\[ \text{slope} = \frac{-21 - (-9)}{8 - 4} = \frac{-21 + 9}{4} = \frac{-12}{4} = -3 \]
So, Function B has a negative slope (-3), indicating it has a negative rate of change.
Now let's look at Function C, which is given as \(y = x + 5\). The slope here is 1 (positive).
1. Which function has a negative rate of change (slope)?
- Function B has a negative rate of change (slope of -3).
2. Which function has the smallest y-intercept?
- The y-intercept of Function B (when \(x = 0\)) is \(y = 3\).
- The y-intercept of Function C (when \(x = 0\)) is \(y = 5\).
- Function B has the smallest y-intercept (3).
3. Which function would you want to represent your money earned per hour?
- You would want a function with a positive slope to represent money earned per hour. Since Function C has a positive slope (1), it is more appropriate for this context.
4. Which function would give you the highest initial amount of money?
- The initial amount of money corresponds to the y-intercept. As calculated, Function B has a y-intercept of 3, and Function C has a y-intercept of 5.
- Therefore, Function C would give you the highest initial amount of money (5).
Summary:
- Function B has a negative rate of change.
- Function B has the smallest y-intercept.
- Function C is better for representing money earned per hour.
- Function C gives the highest initial amount of money.
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