Question

question #2:
function a
x (hours) | y (earnings)
4 | 40
8 | 80
12 | 120
16 | 160
rate of change:
function b
graph of function b with x-axis as time (hours) and y-axis as amount earned (usd)
rate of change:
which function has a greater rate of change? why?

Explanation:

Step1: Find Rate of Change for Function A

The rate of change (slope) for a linear function from a table is calculated as $\frac{\Delta y}{\Delta x}$. For Function A, take two points, e.g., $(4, 40)$ and $(8, 80)$. $\Delta y = 80 - 40 = 40$, $\Delta x = 8 - 4 = 4$. So rate of change is $\frac{40}{4} = 10$.

Step2: Find Rate of Change for Function B

For a graph, rate of change is $\frac{\text{Change in } y}{\text{Change in } x}$. From Function B's graph, take points like $(2, 10)$ (since at $x = 2$ hours, $y = 10$ dollars) and $(4, 20)$. $\Delta y = 20 - 10 = 10$, $\Delta x = 4 - 2 = 2$. Wait, no, let's check another pair. Wait, when $x = 0$, $y = 0$; when $x = 2$, $y = 10$? Wait, no, looking at the graph, the first point after origin: at $x = 2$ (hours), $y = 10$? Wait, no, maybe better to take $(2, 10)$ and $(4, 20)$? Wait, no, let's recalculate. Wait, the graph: when $x = 2$ (time in hours), the amount earned (y) is 10? Wait, no, the y-axis is "Amount Earned (USD)". Wait, maybe the grid: each square is 5? Wait, no, the first point: at $x = 2$, y is 10? Wait, no, let's take two clear points. Let's take $(2, 10)$ and $(4, 20)$? Wait, no, the slope formula is $\frac{y_2 - y_1}{x_2 - x_1}$. Let's take $(0, 0)$ and $(2, 10)$: $\frac{10 - 0}{2 - 0} = 5$? Wait, no, that can't be. Wait, maybe I misread the graph. Wait, the Function A has rate 10. Wait, maybe the graph for Function B: let's check the points. Wait, when $x = 2$ (hours), the y (earnings) is 10? No, wait, the grid: the y-axis has marks at 5, 10, 15, 20, 25, 30, 35, 40. The x-axis at 2, 4, 6, 8. Wait, when $x = 2$, the point is at y = 10? Wait, no, maybe the rate is $\frac{10 - 0}{2 - 0} = 5$? Wait, no, that's wrong. Wait, no, let's take $(2, 10)$ and $(4, 20)$: $\frac{20 - 10}{4 - 2} = 5$. Wait, but that's 5? Wait, no, maybe the graph is different. Wait, no, maybe I made a mistake. Wait, the Function A's rate is 10 (since 4 hours earn 40, so 10 per hour). Now, Function B: let's take $x = 2$, $y = 10$; $x = 4$, $y = 20$? No, that's 5 per hour? Wait, no, that can't be. Wait, maybe the graph's y-axis is scaled differently. Wait, no, the problem says "Rate of Change" for Function B. Wait, maybe I misread the graph. Wait, let's check again. Wait, the first point: at $x = 2$ (hours), the amount earned is 10? So from (0,0) to (2,10): slope is 5. But Function A has slope 10. So Function A has a greater rate of change because 10 > 5. Wait, but maybe I messed up Function B's rate. Wait, let's take another pair: (4, 20) and (6, 30). $\frac{30 - 20}{6 - 4} = 5$. So rate of change for Function B is 5.

Step3: Compare Rates

Function A's rate: 10, Function B's rate: 5. So Function A has a greater rate of change because 10 > 5.

Answer:

Function A has a greater rate of change. The rate of change (slope) for Function A is 10 (dollars per hour) and for Function B is 5 (dollars per hour), so 10 > 5.