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)

Question

The initial value is b=(response area), and the rate of change is m=(response area).

Answer 1

In the given description, we can analyze the costs involved in attending the local festival.

1. Initial Value (b): This refers to the fixed cost of entering the festival, which is the entrance fee. Therefore, the initial value \( b = 12 \).

2. Rate of Change (m): This refers to how the total cost changes with respect to the number of rides taken. The cost of ride tickets is $2 for each ride, which means for each additional ride, the total cost increases by $2. Therefore, the rate of change \( m = 2 \).

To summarize:

• The initial value is \( b = 12 \).
• The rate of change is \( m = 2 \).

Answer 2

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=
.

Answer 3

To find the rate of change (slope) and initial value (y-intercept) from a table of values, we can follow these steps:

From the given data:

• For \(x = 1\), \(y = 4\)
• For \(x = -3\), \(y = 8\)

1. Calculate the rate of change (m):

The rate of change \(m\) can be calculated using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Let's use the points \((1, 4)\) and \((-3, 8)\):

• \(x_1 = 1\), \(y_1 = 4\)
• \(x_2 = -3\), \(y_2 = 8\)

Now plug in the values: \[ m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1 \]

So, the rate of change is \(m = -1\).

2. Find the initial value (b):

To find the initial value (y-intercept), we can use the slope \(m\) and one of the points to determine the equation of the line in slope-intercept form \(y = mx + b\).

Using the point \((1, 4)\): \[ 4 = -1(1) + b \] \[ 4 = -1 + b \implies b = 4 + 1 = 5 \]

So the initial value is \(b = 5\).

To summarize:

• The rate of change is \(m = -1\).
• The initial value is \(b = 5\).

Answer 4

Determine the rate of change in the following description. In Ms. Katy’s math class, a homework assignment is worth 100 points. If students turn the homework in late, there will be a deduction of 5 points per day. (1 point)
Responses

m=−100
m equals negative 100

m=100
m equals 100

m=−5
m equals negative 5

m=5

Answer 5

The rate of change in the given scenario refers to how the total points for the homework assignment decrease as the number of days it is turned in late increases.

In this case, the homework is worth a total of 100 points, and for each day it is turned in late, there is a deduction of 5 points.

Thus, the rate at which the points change (reduce) for each day late is:

\[ m = -5 \]

This means that for every day the assignment is late, the score decreases by 5 points, which is a negative change.

So the correct response is:

m = -5.

Answer 6

Use the table to answer the question.

x
y

2.6 −6

3.2 3
Determine the rate of change of the linear function given as a table of values.

(1 point)
Responses

m=−15
m equals negative 15

m=−5
m equals negative 5

m=0.067
m equals 0.067

m=15
m equals 15
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Answer 7

To determine the rate of change (\(m\)) from the given table of values, we can use the formula for the slope:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

From the provided data:

• When \(x = 2.6\), \(y = -6\)
• When \(x = 3.2\), \(y = 3\)

Let's define our points:

• \(x_1 = 2.6\), \(y_1 = -6\)
• \(x_2 = 3.2\), \(y_2 = 3\)

Now plug in the values into the slope formula:

\[ m = \frac{3 - (-6)}{3.2 - 2.6} = \frac{3 + 6}{3.2 - 2.6} = \frac{9}{0.6} \]

Next, calculate \(9 \div 0.6\):

\[ \frac{9}{0.6} = 9 \times \frac{10}{6} = 9 \times \frac{5}{3} = 15 \]

So, the rate of change \(m\) is:

\[ m = 15 \]

The correct response is:

m = 15.

Answer 8

Find the initial value of the linear function, given that the rate of change is m=−47
, and (14,3)
is an (x,y)
value of the linear function.(1 point)
Responses

b=−5
b equals negative 5

b=11
b equals 11

b=12.29
b equals 12.29

b=15.71

Answer 9

To find the initial value (y-intercept) \(b\) of a linear function given the rate of change \(m\) and a point \((x, y)\), we can use the slope-intercept form of the linear equation:

\[ y = mx + b \]

We're given:

• The rate of change \(m = -47\)
• The point \((14, 3)\), where \(x = 14\) and \(y = 3\)

We can substitute the known values into the equation to solve for \(b\):

\[ 3 = -47(14) + b \]

Now calculate \(-47(14)\):

\[ -47 \times 14 = -658 \]

So now we have:

\[ 3 = -658 + b \]

To solve for \(b\), add \(658\) to both sides:

\[ b = 3 + 658 \] \[ b = 661 \]

Since this result does not match any of the provided options, let's check if we can derive the formula correctly or if there might have been a misunderstanding.

If we substitute using the point and the slope we calculated before, there should not be any ambiguity. However, analyzing the answers given indicates that they might pertain to different calculations.

If you would like to derive the initial value using the options provided, please confirm the provided options or provide clarification on any possible miscommunication about the inputs.

Answer 10

Find the initial value of the linear function, given that the rate of change is m=−4/7
, and (14,3)
is an (x,y)
value of the linear function.(1 point)
Responses

b=−5
b equals negative 5

b=11
b equals 11

b=12.29
b equals 12.29

b=15.71

Answer 11

To find the initial value \(b\) of the linear function given the rate of change \(m = -\frac{4}{7}\) and the point \((14, 3)\) (where \(x = 14\) and \(y = 3\)), we can use the slope-intercept form of a linear equation:

\[ y = mx + b \]

Now we will substitute the values into the equation to solve for \(b\):

1. Substitute \(y\), \(m\), and \(x\) into the equation: \[ 3 = -\frac{4}{7}(14) + b \]

2. Calculate \(-\frac{4}{7}(14)\): \[ -\frac{4 \times 14}{7} = -\frac{56}{7} = -8 \]

3. Now substitute this back into the equation: \[ 3 = -8 + b \]

4. Solve for \(b\) by adding \(8\) to both sides: \[ b = 3 + 8 \] \[ b = 11 \]

Thus, the initial value \(b\) is:

b = 11.