Consider the following graphs.
The figure shows two graphs in the xy-plane. The graph on the left is labeled as Graph Upper A. The x-axis is labeled as Time left parenthesis seconds right parenthesis and the y-axis is labeled as Elevation left parenthesis feet right parenthesis. The values on the x-axis range from 0 to 9 in increments of 1 and the values on the y-axis range from 0 to 360 in increments of 40. A line starts from a point just above 240 on the y-axis, goes down to a point just to the right of 8 on the x-axis. The graph on the right is labeled as Graph Upper B. The x-axis is labeled as Number of Loaves and the y-axis is labeled as Amount of Flour left parenthesis cups right parenthesis. The values on the x-axis range from 0 to 9 in increments of 1 and the values on the y-axis range from 0 to 18 in increments of 2. A line starts from the origin, goes up, and passes through (2, 5), (4, 10), and so on.
a. Determine the rate of change shown in each graph and determine if each is positive or negative.
b. Restate the rate of change as a unit rate for each graph. Explain its meaning.
c. Tell whether the graphs represent proportional relationships.
Explain your reasoning.
impatient much? See your earlier post.
a. To determine the rate of change in each graph, we need to find the slope of the line.
For Graph A, the line starts from a point just above 240 on the y-axis and goes down to a point just to the right of 8 on the x-axis. The change in elevation is from around 240 feet to 0 feet, and the change in time is from 0 seconds to around 8 seconds. Therefore, the rate of change is negative, as the elevation decreases over time.
For Graph B, the line starts from the origin and goes through the points (2, 5), (4, 10), and so on. Looking at the pattern, the change in the amount of flour is 5 cups at 2 loaves, and then increases by 5 cups for each additional 2 loaves. Therefore, the rate of change is positive.
b. Unit rate is the ratio of the change in the dependent variable to the change in the independent variable.
For Graph A, the change in elevation is 240 feet and the change in time is 8 seconds. Therefore, the unit rate of change is -30 feet per second, which means that the elevation decreases by 30 feet every second.
For Graph B, the change in the amount of flour is 5 cups and the change in the number of loaves is 2. Therefore, the unit rate of change is 2.5 cups per loaf, which means that for every loaf made, the amount of flour increases by 2.5 cups.
c. To determine whether the graphs represent proportional relationships, we need to check if the ratio of the dependent variable to the independent variable is constant.
For Graph A, the ratio of the elevation to time is not constant. As seen from the unit rate, the elevation does not decrease at a constant rate over time.
For Graph B, the ratio of the amount of flour to the number of loaves is constant. As seen from the unit rate, the amount of flour increases at a constant rate for each additional loaf made.
Therefore, Graph B represents a proportional relationship, but Graph A does not.
To determine the rate of change in each graph, we need to calculate the change in the y-coordinate divided by the change in the x-coordinate.
a. Graph A:
We can see that the line starts above 240 and goes down to a point just to the right of 8 on the x-axis. The change in the y-coordinate is 240 - 0 = 240, and the change in the x-coordinate is 8 - 0 = 8. So, the rate of change in Graph A is 240/8 = 30. Since the line goes down, the rate of change is negative.
Graph B:
The line in Graph B passes through points (2, 5), (4, 10), and so on. To calculate the rate of change, we need to find the change in the y-coordinate divided by the change in the x-coordinate for any two consecutive points. Let's take the points (2, 5) and (4, 10). The change in the y-coordinate is 10 - 5 = 5, and the change in the x-coordinate is 4 - 2 = 2. So, the rate of change between these two points is 5/2 = 2.5. We can repeat this process for other pairs of consecutive points to find that the rate of change in Graph B is consistently 2.5.
b. Restating the rate of change as a unit rate means expressing it as a ratio with the y-variable as 1 unit and the x-variable as its corresponding number of units.
For Graph A, the rate of change is -30. This means that for every 1 second increase in time, the elevation decreases by 30 feet.
For Graph B, the rate of change is 2.5. This means that for every 1 increase in the number of loaves, the amount of flour increases by 2.5 cups.
c. To determine if the graphs represent proportional relationships, we need to check if the ratio between the y- and x-coordinates is constant for each graph.
For Graph A, the rate of change is not constant. As we observed earlier, the rate of change is -30, indicating a downward slope. Therefore, Graph A does not represent a proportional relationship.
For Graph B, the rate of change is constant at 2.5. This means that for every increase of 1 in the x-coordinate, the y-coordinate increases by a consistent factor of 2.5. Therefore, Graph B does represent a proportional relationship.