Question

Determine the initial value and the rate of change in the following description. At a local festival, the entrance fee is $12 per person, and ride tickets cost $2 for every ride.(1 point)

The initial value is b=____, and the rate of change is m=____.
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page 26 of 27

Answer 1

[AI-generated answer — Bot GPT 3.5]

The initial value is $12 (entrance fee) and the rate of change is $2 per ride ticket. So, b=12 and m=2.

Answer 2

Determine the initial value and the rate of change in the following description. A 2,000-gallon tank is half full with 1,000 gallons of water. Water is being pumped into the tank at a rate of 15 gallons per minute.(1 point)

The initial value is b=____, and the rate of change is m=____.

Answer 3

[AI-generated answer — Bot GPT 3.5]

The initial value is 1000 gallons (half full) and the rate of change is 15 gallons per minute. So, b=1000 and m=15.

Answer 4

Use the table to answer the question.

x y
1 4
−3 8
Find the rate of change and the initial value of the linear function given as a table of values.

(1 point)
The rate of change is m=____, and the initial value is b=____.
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Answer 5

[AI-generated answer — Bot GPT 3.5]

To find the rate of change, we need to calculate the difference in y-values divided by the difference in x-values:

Rate of change (m) = (change in y) / (change in x)

For this table:
Change in y = 8 - 4 = 4
Change in x = -3 - 1 = -4

Rate of change (m) = 4 / -4 = -1

To find the initial value (y-intercept or b), we can choose any point from the table and use the slope (rate of change) we found.

Let's use the point (1, 4):
y = mx + b
4 = -1(1) + b
4 = -1 + b
b = 4 + 1
b = 5

Therefore, the rate of change is m = -1 and the initial value is b = 5.

Answer 6

Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. An upward slanting line is plotted on the coordinate plane. The upward slanting line with arrows on both ends passes through the following closed points: left parenthesis 0 comma 0 right parenthesis and left parenthesis 2 comma 3 right parenthesis.

Determine the initial value and the rate of change of the linear function given in the graph.

(1 point)
The initial value is ____, and the rate of change is ____.
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page 26 of 27

Answer 7

[AI-generated answer — Bot GPT 3.5]

To determine the initial value and rate of change of the linear function from the graph, we can use the two points (0, 0) and (2, 3).

The rate of change (slope) can be calculated as:
Rate of change (m) = (change in y) / (change in x)
m = (3 - 0) / (2 - 0)
m = 3/2
m = 1.5

The initial value (y-intercept or b) can be found by substituting one of the points and the slope into the slope-intercept form equation y = mx + b.
Let's use the point (0, 0):
0 = 1.5(0) + b
b = 0

Therefore, the initial value is 0, and the rate of change (slope) is 1.5.

Answer 8

Use the image to answer the question.

An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. A downward slanting line is plotted on the coordinate plane. The downward slanting line with arrows on both ends passes through the following closed points: left parenthesis negative 3 comma 2.5 right parenthesis, left parenthesis 0 comma 1 right parenthesis, and left parenthesis 2 comma 0 right parenthesis.

Determine the initial value and the rate of change of the linear function as given in the graph.

(1 point)
The initial value is ____, and the rate of change is ____.
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page 26 of 27

Answer 9

[AI-generated answer — Bot GPT 3.5]

To determine the initial value and rate of change of the linear function from the graph, we can use the points (-3, 2.5), (0, 1), and (2, 0).

The rate of change (slope) can be calculated using any two distinct points:
For example, let's calculate the rate of change using the points (-3, 2.5) and (0, 1):
m = (1 - 2.5) / (0 - (-3))
m = -1.5 / 3
m = -0.5

Therefore, the rate of change (slope) is -0.5.

To find the initial value (y-intercept or b), we can choose any point and substitute the slope into the slope-intercept form equation y = mx + b. Let's use the point (0, 1):
1 = -0.5(0) + b
b = 1

Therefore, the initial value is 1, and the rate of change (slope) is -0.5.