To analyze the functions provided, let's break each question down step by step:
Function A
Points: (-4, 0), (-2, 1), (0, 2), (2, 3)
To find the slope (rate of change):
Using two points, for example, (0, 2) and (2, 3):
\[
m = \frac{3 - 2}{2 - 0} = \frac{1}{2}
\]
You can calculate the slope between any two points to find this was consistent; the slope is \( \frac{1}{2} \).
Function B
Table:
\[
\begin{array}{|c|c|}
\hline
x & y \
\hline
0 & 3 \
4 & -9 \
8 & -21 \
\hline
\end{array}
\]
Finding the slope between the first two points (0, 3) and (4, -9):
\[
m = \frac{-9 - 3}{4 - 0} = \frac{-12}{4} = -3
\]
Finding the slope between the second two points (4, -9) and (8, -21):
\[
m = \frac{-21 - (-9)}{8 - 4} = \frac{-12}{4} = -3
\]
The slope is consistently \(-3\).
Function C
Equation: \(y = x + 5\)
The slope (rate of change) is:
1 (the coefficient of x).
Comparing Slopes
- Function A: slope = \( \frac{1}{2} \)
- Function B: slope = \(-3\)
- Function C: slope = \(1\)
Greatest Rate of Change:
- Function C has the greatest slope of \(1\).
Finding y-intercepts
- Function A's y-intercept: When \(x = 0\), \(y = 2\).
- Function B's y-intercept: When \(x = 0\), \(y = 3\).
- Function C's y-intercept: When \(x = 0\), \(y = 5\).
Smallest Y-Intercept:
- Function A: \(2\)
- Function B: \(3\)
- Function C: \(5\)
The smallest y-intercept is from Function A.
Money Earned per Hour
Generally, you might prefer a function with a positive slope representing earnings, and you want it to be as high as possible.
- Function A has a positive slope (\(\frac{1}{2}\)).
- Function C has a slope of \(1\).
Function C would represent the best option for money earned per hour since it has a positive slope and does not dip.
Highest Initial Amount of Money
The initial amount of money is represented by the y-intercept.
- Function A = \(2\)
- Function B = \(3\)
- Function C = \(5\)
The highest initial amount of money is from Function C.
Answers
a. Function C (greatest rate of change)
b. Function A (smallest y-intercept)
c. Function C (best for money earned per hour)
d. Function C (highest initial amount of money)
Would you like to clarify further or provide more detail for any of these points?