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.
man, could you get any wordier? How the axes are divided makes no freakin' difference.
The rate of change for each graph is in units of y/x
If the graph does not go through (0,0) it is not a proportion (y = kx)
For A, recall that the intercept form of a line is
x/a + y/b = 1
if the intercepts are at (a,0) and (0,b). That makes the slope = -b/a
a. The rate of change in Graph A can be determined by finding the slope of the line. From the given information, we can see that the line starts just above 240 and goes down to a point just to the right of 8 on the x-axis. Since the line is descending, the rate of change is negative.
The rate of change in Graph B can be determined by analyzing the points the line passes through. Given that the line passes through (2, 5), (4, 10), and so on, we can observe that for each increase of 2 in the x-axis (Number of Loaves), the y-axis (Amount of Flour) increases by 5. Therefore, the rate of change is positive.
b. The unit rate for Graph A can be calculated by dividing the change in elevation (y-axis) by the change in time (x-axis). Let's assume the starting point is (0, 240) and the ending point is (8, 0). The change in elevation is 240 - 0 = 240 feet, and the change in time is 8 - 0 = 8 seconds. Therefore, the unit rate of change for Graph A is 240/8 = 30 feet per second. This means that the elevation decreases by 30 feet per second.
The unit rate for Graph B can be calculated by dividing the change in the amount of flour (y-axis) by the change in the number of loaves (x-axis). Let's consider the points (2, 5) and (4, 10). The change in the amount of flour is 10 - 5 = 5 cups, and the change in the number of loaves is 4 - 2 = 2. Hence, the unit rate of change for Graph B is 5/2 = 2.5 cups per loaf. This means that for every 2 loaves, the amount of flour increases by 5 cups.
c. In Graph A, the rate of change (30 feet per second) is constant, indicating a proportional relationship between time (x-axis) and elevation (y-axis).
In Graph B, the rate of change (2.5 cups per loaf) is not constant. It increases as the number of loaves increases. Therefore, Graph B does not represent a proportional relationship.
a. To determine the rate of change in each graph, we need to calculate the change in the dependent variable (y) divided by the change in the independent variable (x):
For Graph A:
Rate of change = (Change in Elevation) / (Change in Time)
From the information given, we can see that the line starts at 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. Therefore:
Change in Elevation = 240 - 0 = 240 feet
Change in Time = 8 - 0 = 8 seconds
Rate of change in Graph A = (240 feet) / (8 seconds) = 30 feet per second
For Graph B:
Rate of change = (Change in Amount of Flour) / (Change in Number of Loaves)
Based on the information given, we can see that the line passes through (2, 5), (4, 10), and so on. Therefore:
Change in Amount of Flour = 10 - 5 = 5 cups
Change in Number of Loaves = 4 - 2 = 2
Rate of change in Graph B = (5 cups) / (2 loaves) = 2.5 cups per loaf
b. The unit rate provides the amount of change in the dependent variable per unit change in the independent variable. This means that for every one unit increase in the independent variable, the dependent variable changes by the unit rate.
For Graph A, the rate of change is 30 feet per second. This means that for every one-second increase in time, the elevation decreases by 30 feet.
For Graph B, the rate of change is 2.5 cups per loaf. This means that for every one-loaf increase in the number of loaves, the amount of flour needed increases by 2.5 cups.
c. Two quantities are said to have a proportional relationship if their ratio or unit rate is constant. Let's analyze each graph:
For Graph A, the rate of change is 30 feet per second. Since the ratio of change in elevation per unit change in time is constant (30 feet per second), we can conclude that Graph A represents a proportional relationship.
For Graph B, the rate of change is 2.5 cups per loaf. Since the ratio of change in the amount of flour per unit change in the number of loaves is constant (2.5 cups per loaf), we can conclude that Graph B represents a proportional relationship as well.
a. To determine the rate of change shown in each graph and whether it is positive or negative, we need to analyze the slope of the lines.
For Graph A:
The line starts just above 240 on the y-axis and goes down to a point just to the right of 8 on the x-axis. Let's assume the starting point is (0, y1) and the ending point is (x2, 0). The rate of change (slope) is calculated by (0 - y1)/(x2 - 0).
For Graph B:
The line starts from the origin and passes through (2, 5), (4, 10), and so on. Let's assume the starting point is (0, 0) and the ending point is (x2, y2). The rate of change (slope) is calculated by (y2 - 0)/(x2 - 0).
b. To restate the rate of change as a unit rate, we divide the change in the dependent variable (y-axis) by the change in the independent variable (x-axis). This gives us the amount of change in the dependent variable for every one unit change in the independent variable.
For Graph A:
Divide the change in elevation (y-axis) by the change in time (x-axis) to get the unit rate. Since elevation is measured in feet and time in seconds, the unit rate would be feet per second. For example, if the elevation changes by 120 feet over the course of 5 seconds, the unit rate would be 120/5 = 24 feet per second.
For Graph B:
Divide the change in the amount of flour (y-axis) by the change in the number of loaves (x-axis) to get the unit rate. Since flour is measured in cups and the number of loaves is a count, the unit rate would be cups per loaf. For example, if the amount of flour increases by 8 cups when the number of loaves increases by 4, the unit rate would be 8/4 = 2 cups per loaf.
The unit rate represents how the dependent variable changes for every one unit change in the independent variable.
c. To determine if the graphs represent proportional relationships, we need to examine if the unit rates (from part b) are consistent for different points on the graph.
For Graph A:
If the unit rate (feet per second) remains the same for any other points on the line, then the relationship is proportional. For example, if for every 1-second increase in time, the elevation consistently increases by 24 feet, then it represents a proportional relationship.
For Graph B:
If the unit rate (cups per loaf) remains the same for any other points on the line, then the relationship is proportional. For example, if for every additional loaf, the amount of flour consistently increases by 2 cups, then it represents a proportional relationship.
By analyzing the consistency of the unit rates, we can determine if the graphs represent proportional relationships or not.