To analyze the given functions, let's first summarize the information about each one.
Function A: Points given: (2, 3), (0, 2), (-2, 1)
To find the slope (rate of change) of Function A, we can use the formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (2, 3) and (0, 2):
\[ m = \frac{2 - 3}{0 - 2} = \frac{-1}{-2} = \frac{1}{2} \]
Using the points (0, 2) and (-2, 1):
\[ m = \frac{1 - 2}{-2 - 0} = \frac{-1}{-2} = \frac{1}{2} \]
The slope remains consistent at \( \frac{1}{2} \).
Function B: We can find the slope by finding the change in \( y \) over the change in \( x \) for the points given.
Taking the points (0, 3) and (4, -9):
\[ m = \frac{-9 - 3}{4 - 0} = \frac{-12}{4} = -3 \]
Now taking the points (4, -9) and (8, -21):
\[ m = \frac{-21 - (-9)}{8 - 4} = \frac{-21 + 9}{4} = \frac{-12}{4} = -3 \]
The slope of Function B is consistently \( -3 \).
Function C: In the function \( y = x + 5 \), the slope is \( 1 \) (the coefficient of \( x \)).
Now, let's summarize the slopes:
- Function A: \( \frac{1}{2} \)
- Function B: \( -3 \)
- Function C: \( 1 \)
Part A: Greatest Rate of Change (Slope)
The greatest slope is found in Function C, which has a slope of \( 1 \).
Part B: Smallest y-intercept
For y-intercepts:
- Function A: y-intercept is \( 2 \) (at \( x = 0 \))
- Function B: y-intercept is \( 3 \) (at \( x = 0 \))
- Function C: y-intercept is \( 5 \) (at \( x = 0 \))
Function A has the smallest y-intercept at \( 2 \).
Part C: Representing Money Earned Per Hour
A function representing money earned should ideally have a positive slope (indicating earning). Here:
- Function A \( \frac{1}{2} \) (positive)
- Function C \( 1 \) (more positive)
- Function B (-3) indicates loss.
Thus, we would want Function C to represent money earned per hour because of its highest positive slope.
Part D: Highest Initial Amount of Money
Looking at the y-intercepts again:
- Function A has a y-intercept of \( 2 \),
- Function B has a y-intercept of \( 3 \),
- Function C has a y-intercept of \( 5 \).
Function C gives the highest initial amount of money.
Following these evaluations, here are the answers:
a. Which function has the greatest rate of change (slope)? Function C
b. Which function has the smallest y-intercept? Function A
c. Which function would you want to represent your money earned per hour? Function C
d. Which function would give you the highest initial amount of money? Function C
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