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Determining rates of change from graphs and tables; determine slope.
Questions and Answers
1.
Find the rate of change:
People
Cost (dollars)
2
7.90
3
11.85
4
15.80
5
19.75
6
23.70
Explanation The rate of change in this scenario is $3.95 per person. This means that for every additional person, the cost increases by $3.95. The answer is given as $3.95, $3.95, which suggests that the rate of change is consistent and applies to both the number of people and the cost.
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2.
Find the rate of change:
Explanation The rate of change is 2/3 or two-thirds. This means that for every unit increase in the independent variable, there is a corresponding increase of 2/3 in the dependent variable.
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3.
Find the rate of change:
Explanation The rate of change is 4. This means that for every unit increase in the independent variable, there is a corresponding increase of 4 in the dependent variable.
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4.
Find the slope of the line:
Explanation The given answer, -3/1, represents the slope of the line. The slope is a measure of how steep a line is and is determined by the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In this case, the slope is -3/1, which means that for every 1 unit increase in the horizontal direction, the line decreases by 3 units in the vertical direction. This indicates a steep downward slope. The -3 value alone represents the y-intercept, which is the point where the line crosses the y-axis.
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5.
Find the slope of the line that passes thru each pair of points:
(3, 2) and (5, 6)
Explanation The slope of a line passing through two points can be found using the formula: (y2 - y1) / (x2 - x1). In this case, the coordinates of the two points are (3, 2) and (5, 6). Plugging these values into the formula, we get (6 - 2) / (5 - 3) = 4/2 = 2. Therefore, the slope of the line passing through these two points is 2.
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6.
Find the slope of the line that passes thru each pair of points:
(4, 8) and (8, 11)
Explanation The slope of a line passing through two points can be found using the formula: slope = (change in y)/(change in x). In this case, the change in y is 11 - 8 = 3, and the change in x is 8 - 4 = 4. Therefore, the slope is 3/4.
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7.
Find the slope of the line that passes thru each pair of points:
(-2, 1) and (1, -2)
Explanation The slope of a line passing through two points can be found using the formula: slope = (change in y-coordinates)/(change in x-coordinates). In this case, the change in y-coordinates is (-2 - 1) = -3 and the change in x-coordinates is (1 - (-2)) = 3. Therefore, the slope is -3/3, which simplifies to -1.
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8.
Find the slope of the line that passes thru each pair of points:
(-8, 0) and (1, 5)
Explanation The slope of a line passing through two points can be found using the formula (y2 - y1) / (x2 - x1). In this case, the points are (-8, 0) and (1, 5). Plugging the values into the formula, we get (5 - 0) / (1 - (-8)) which simplifies to 5/9. Therefore, the slope of the line passing through these two points is 5/9. The second option, -5/-9, is the same as 5/9 when simplified.
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9.
Find the slope of the line that passes thru each pair of points:
(-4, -5) and (-9, -1)
Explanation To find the slope of a line passing through two points, we use the formula (y2 - y1) / (x2 - x1). In this case, the points are (-4, -5) and (-9, -1). Plugging in the values, we get (-1 - (-5)) / (-9 - (-4)), which simplifies to 4 / -5. Therefore, the slope of the line is 4 / -5.
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10.
State whether the slope is zero or undefined:
Explanation The correct answer is zero. In mathematics, the slope of a line represents the steepness or incline of the line. A slope of zero indicates that the line is horizontal and has no incline. This means that the line is parallel to the x-axis and does not rise or fall as it extends. Therefore, the slope is zero.
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11.
State whether the slope is zero or undefined:
Explanation The slope is undefined when the line is vertical or parallel to the y-axis. In this case, the answer is "undefined" because it indicates that the slope cannot be determined or does not exist. The answer "Undefined" with a capital U may be a typographical error, but it still conveys the same meaning.
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12.
State whether the slope is zero or undefined:
(3, 4) and (-3, 4)
Explanation The slope between two points is determined by the change in the y-coordinates divided by the change in the x-coordinates. In this case, the y-coordinates of both points are the same (4), while the x-coordinates have a difference of 6. Therefore, the change in y is 0, and the change in x is 6. Dividing 0 by 6 gives us a slope of 0. Hence, the slope between the two points is zero.
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13.
State whether the slope is zero or undefined:
(4, 3) and (4, -3)
Explanation The slope between two points is calculated as the change in y-coordinates divided by the change in x-coordinates. In this case, the x-coordinates of the two points are the same (both 4), which means there is no change in the x-coordinate. Therefore, the change in x is 0. When dividing any number by 0, the result is undefined. Hence, the slope between these two points is undefined.
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14.
Find the rate of change:
The cost of group museum tickets is $48 for four people (4, 48) and $78 for ten people (10, 78).
Explanation The rate of change can be found by calculating the difference in cost divided by the difference in number of people. In this case, the cost increased by $30 (78 - 48) when the number of people increased by 6 (10 - 4). Therefore, the rate of change is $30/6, which simplifies to $5.
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15.
Match the graph with the given information:
(3, 5)
slope: -2/1
A.
B.
C.
D.
Correct Answer A.
Explanation The graph that matches the given information is a line that passes through the point (3, 5) and has a slope of -2/1. This means that for every increase of 1 unit in the x-coordinate, the y-coordinate decreases by 2 units. The line will have a negative slope, meaning it will slope downwards from left to right. The point (3, 5) will be on the line, and the line will continue in both directions.
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16.
Which line is the steepest?
Correct Answer C c
17.
During the 6-week period, which plant had the greatest rate of change?
Correct Answer C c
18.
During the 6-week period, which plant had the least rate of change?
Correct Answer A a
19.
Each pair of points lies on a line with the given slope. Find x.
(2, 4) and (x, 8); slope = -2
Correct Answer x=0 x = 0 0
Explanation The given information states that the two points (2, 4) and (x, 8) lie on a line with a slope of -2. To find the value of x, we can use the slope formula:
slope = (change in y) / (change in x)
Since the slope is -2, and the change in y is 8-4=4, we can substitute these values into the formula:
-2 = 4 / (x-2)
To solve for x, we can cross multiply:
-2(x-2) = 4
-2x + 4 = 4
-2x = 0
Dividing both sides by -2, we get:
x = 0
Therefore, the value of x is 0.
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