Section 3.3 – Investigating Properties Of Linear Relations

1) Rate of change can also be referred to as the _______________ of a line.

The rate of change refers to how much a variable is changing over a given interval. In the context of a line, the rate of change is equivalent to the slope of the line. The slope represents the steepness or incline of the line, indicating how much the dependent variable changes for every unit increase in the independent variable. Therefore, the correct answer for this question is "slope".

Explanation

The rate of change refers to how much a variable is changing over a given interval. In the context of a line, the rate of change is equivalent to the slope of the line. The slope represents the steepness or incline of the line, indicating how much the dependent variable changes for every unit increase in the independent variable. Therefore, the correct answer for this question is "slope".

Submit

2)

Which graph is linear?

Top-Left

Top-Right

Bottom-Left

Bottom-Right

The top-right graph is linear because it shows a straight line that represents a constant rate of change between the x and y variables. The other three graphs show curves or non-linear patterns, indicating a varying rate of change.

Explanation

The top-right graph is linear because it shows a straight line that represents a constant rate of change between the x and y variables. The other three graphs show curves or non-linear patterns, indicating a varying rate of change.

3) If the first differences are _______________ then the relationship is linear.

If the first differences are constant, it means that the difference between consecutive terms in the sequence is the same. This suggests that there is a constant rate of change between each term, indicating a linear relationship. In a linear relationship, the change in the dependent variable is directly proportional to the change in the independent variable. Therefore, if the first differences are constant, it implies a linear relationship between the terms of the sequence.

Explanation

If the first differences are constant, it means that the difference between consecutive terms in the sequence is the same. This suggests that there is a constant rate of change between each term, indicating a linear relationship. In a linear relationship, the change in the dependent variable is directly proportional to the change in the independent variable. Therefore, if the first differences are constant, it implies a linear relationship between the terms of the sequence.

Submit

4) Determine the slope of the line that passes through
(15,-8) and (10, 3)

11/5

1

-1

-11/5

To determine the slope of a line passing through two points, we use the formula: slope = (change in y)/(change in x). In this case, the change in y is -8 - 3 = -11, and the change in x is 15 - 10 = 5. Therefore, the slope is -11/5.

Explanation

To determine the slope of a line passing through two points, we use the formula: slope = (change in y)/(change in x). In this case, the change in y is -8 - 3 = -11, and the change in x is 15 - 10 = 5. Therefore, the slope is -11/5.

5)

Determine the rate of change for the relation shown in the table of values.

-2

-1/2

1/2

2

The rate of change for the relation shown in the table of values is -1/2. This means that for every unit increase in the input variable, the output variable decreases by 1/2.

Explanation

The rate of change for the relation shown in the table of values is -1/2. This means that for every unit increase in the input variable, the output variable decreases by 1/2.

6) Calculate the slope of the line that goes through
A(-4, 5) and B(2, -7)

2

-1/2

1/2

-2

None of the above

The slope of a line can be calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates of point A are (-4, 5) and the coordinates of point B are (2, -7). Plugging these values into the formula, we get (-7 - 5) / (2 - (-4)) = -12 / 6 = -2. Therefore, the slope of the line that goes through A(-4, 5) and B(2, -7) is -2.

Explanation

The slope of a line can be calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates of point A are (-4, 5) and the coordinates of point B are (2, -7). Plugging these values into the formula, we get (-7 - 5) / (2 - (-4)) = -12 / 6 = -2. Therefore, the slope of the line that goes through A(-4, 5) and B(2, -7) is -2.

7)

What is the slope from the bottom right of the hill to the top of the hill.

2.5

2/5

-2/5

-2.5

The slope from the bottom right of the hill to the top of the hill is -2.5.

Explanation

The slope from the bottom right of the hill to the top of the hill is -2.5.

8) What is the slope of the relation y = -4/5x - 4

4

-4

4/5

-4/5

The slope of a linear equation is represented by the coefficient of the x variable. In this case, the equation is y = -4/5x - 4, which means that the slope is -4/5. This means that for every increase of 1 in the x variable, the y variable will decrease by 4/5.

Explanation

The slope of a linear equation is represented by the coefficient of the x variable. In this case, the equation is y = -4/5x - 4, which means that the slope is -4/5. This means that for every increase of 1 in the x variable, the y variable will decrease by 4/5.

9) Calculate the slope of the line that goes through (2, -1) and (0, -2)

2

-1/2

1/2

-2

None of the Above

The slope of a line can be calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates given are (2, -1) and (0, -2). Plugging these values into the formula, we get (-2 - (-1)) / (0 - 2) = (-2 + 1) / (-2) = 1 / (-2) = -1/2. Therefore, the correct answer is -1/2.

Explanation

The slope of a line can be calculated using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the coordinates given are (2, -1) and (0, -2). Plugging these values into the formula, we get (-2 - (-1)) / (0 - 2) = (-2 + 1) / (-2) = 1 / (-2) = -1/2. Therefore, the correct answer is -1/2.

10)

Determine the rate of change for the relation shown in the graph.

2

1/2

-1/2

-2

The rate of change for the relation shown in the graph is -1/2. This means that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 1/2.

Explanation

The rate of change for the relation shown in the graph is -1/2. This means that for every increase of 1 in the x-coordinate, the y-coordinate decreases by 1/2.

11) Determine the slope of the line that passes through
(-9,7) and (2, -6)

-13/11

-11/13

11/13

13/11

To determine the slope of the line passing through the points (-9,7) and (2,-6), we can use the slope formula: m = (y2 - y1)/(x2 - x1). Plugging in the values, we get m = (-6 - 7)/(2 - (-9)) = -13/11. Therefore, the slope of the line is -13/11.

Explanation

To determine the slope of the line passing through the points (-9,7) and (2,-6), we can use the slope formula: m = (y2 - y1)/(x2 - x1). Plugging in the values, we get m = (-6 - 7)/(2 - (-9)) = -13/11. Therefore, the slope of the line is -13/11.

12) Determine which ordered pairs are also in the relation where the rise is 3, the run is 2, and
(-4,10) lies on the line.

(-8, -4) and (0, 16)

(4, 22) and (-2,13)

(-4, -10) and (-16, 8)

(8, 22) and (-8, 4)

The correct answer is (4, 22) and (-2,13). To determine if an ordered pair is in the relation where the rise is 3 and the run is 2, we can use the slope formula: rise/run = (y2 - y1)/(x2 - x1). We can plug in the values for the given ordered pairs and check if the equation is true. For the ordered pair (4, 22), the rise is 22 - 10 = 12 and the run is 4 - (-4) = 8. 12/8 = 3/2, which matches the given rise and run. Similarly, for the ordered pair (-2, 13), the rise is 13 - 10 = 3 and the run is -2 - (-4) = 2. 3/2 = 3/2, which also matches the given rise and run. Therefore, (4, 22) and (-2,13) are in the relation.

Explanation

The correct answer is (4, 22) and (-2,13). To determine if an ordered pair is in the relation where the rise is 3 and the run is 2, we can use the slope formula: rise/run = (y2 - y1)/(x2 - x1). We can plug in the values for the given ordered pairs and check if the equation is true. For the ordered pair (4, 22), the rise is 22 - 10 = 12 and the run is 4 - (-4) = 8. 12/8 = 3/2, which matches the given rise and run. Similarly, for the ordered pair (-2, 13), the rise is 13 - 10 = 3 and the run is -2 - (-4) = 2. 3/2 = 3/2, which also matches the given rise and run. Therefore, (4, 22) and (-2,13) are in the relation.

13) Calculate the rise length for the line going through A(3, 2) and B(-5, 0)

8

-8

-2

2

None of the above

To calculate the rise length for the line going through points A(3, 2) and B(-5, 0), we need to find the difference in the y-coordinates of the two points. The y-coordinate of point A is 2 and the y-coordinate of point B is 0. The difference between these two y-coordinates is 2 - 0 = 2. Therefore, the rise length is 2.

Explanation

To calculate the rise length for the line going through points A(3, 2) and B(-5, 0), we need to find the difference in the y-coordinates of the two points. The y-coordinate of point A is 2 and the y-coordinate of point B is 0. The difference between these two y-coordinates is 2 - 0 = 2. Therefore, the rise length is 2.

14) Which ordered pairs are also in the relation where the rise is -2, the run is 3, and (6,2) lies on the line.

(-3, 4) and (3,8)

(-9, -12) and (-6, 2)

(0, 9) and (-2, 12)

(9, 0) and (12, -2)

The given information states that the rise is -2 and the run is 3. This means that for every 3 units moved horizontally, the line moves down by 2 units vertically. Since (6,2) lies on the line, we can use this information to determine the other ordered pairs that are also on the line.

For example, if we move 3 units to the right from (6,2), we get (9,0) which is one of the answer choices. Similarly, if we move 6 units to the right from (6,2), we get (12,-2) which is also one of the answer choices. Therefore, the correct answer is (9,0) and (12,-2).

Explanation

The given information states that the rise is -2 and the run is 3. This means that for every 3 units moved horizontally, the line moves down by 2 units vertically. Since (6,2) lies on the line, we can use this information to determine the other ordered pairs that are also on the line.

For example, if we move 3 units to the right from (6,2), we get (9,0) which is one of the answer choices. Similarly, if we move 6 units to the right from (6,2), we get (12,-2) which is also one of the answer choices. Therefore, the correct answer is (9,0) and (12,-2).

15) Identify the slope in the equation C = - 200 t + 540

C

200

T

540

None of the above

The equation C = -200t + 540 does not include any term that represents the slope. The slope of a linear equation is typically represented by the coefficient of the independent variable (in this case, t). However, in this equation, the coefficient of t is -200, which represents the rate of change of C with respect to t, not the slope. Therefore, the correct answer is None of the above.

Explanation

The equation C = -200t + 540 does not include any term that represents the slope. The slope of a linear equation is typically represented by the coefficient of the independent variable (in this case, t). However, in this equation, the coefficient of t is -200, which represents the rate of change of C with respect to t, not the slope. Therefore, the correct answer is None of the above.