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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=12.29 b equals 12.29 b=−5 b equals negative 5 b=15.71 b equals 15.71 b=11

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To find the initial value (y-intercept) \( b \) of the linear function given the slope \( m = -47 \) and the point \( (14, 3) \), we can use the point-slope form of the linear equation:

\[ y - y_1 = m(x - x_1) \]

Where \( (x_1, y_1) \) is the point \( (14, 3) \). Plugging in the values, we have:

\[ y - 3 = -47(x - 14) \]

Expanding this equation:

\[ y - 3 = -47x + 658 \]

Adding 3 to both sides to isolate \( y \):

\[ y = -47x + 658 + 3 \] \[ y = -47x + 661 \]

The y-intercept \( b \) is the constant term when the equation is in the form \( y = mx + b \). Therefore,

\[ b = 661 \]

None of the provided options match \( b = 661 \). Please check the options again, as there seems to be some inconsistency. However, based on the given information, the correct calculation is:

b = 661.

Use the table to answer the question. Height Above Canyon Floor Time (minutes) 360 0 280 3 160 10 80 14 Which statement best interprets the rate of change of the linear model shown in the table? (1 point) Responses The elevation of a hiker who is hiking up from canyon floor changes at a rate of −20 feet per minute. The elevation of a hiker who is hiking up from canyon floor changes at a rate of negative 20 feet per minute. The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute. The elevation of a hiker who is hiking down to a canyon floor changes at a rate of 20 feet per minute. The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute. The elevation of a hiker who is hiking up from a canyon floor changes at a rate of 20 feet per minute. The elevation of a hiker who is hiking down to a canyon floor changes at a rate of −20 feet per minute. The elevation of a hiker who is hiking down to a canyon floor changes at a rate of negative 20 feet per minute. Skip to navigation

To determine the rate of change from the table of height above the canyon floor versus time, we can analyze the provided data points:

  • At 0 minutes, the height is 360 feet.
  • At 3 minutes, the height is 280 feet.
  • At 10 minutes, the height is 160 feet.
  • At 14 minutes, the height is 80 feet.

To find the rate of change, we can calculate the change in height and the change in time between two points.

Let's take the first two points to find the rate of change from 0 to 3 minutes:

  • Change in height: \( 360 - 280 = 80 \) feet
  • Change in time: \( 3 - 0 = 3 \) minutes

Rate of change \( = \frac{\text{Change in height}}{\text{Change in time}} = \frac{-80 \text{ feet}}{3 \text{ minutes}} \approx -26.67 \text{ feet per minute} \)

Next, we can analyze the change from 3 minutes to 10 minutes:

  • Change in height: \( 280 - 160 = 120 \) feet
  • Change in time: \( 10 - 3 = 7 \) minutes

Rate of change \( = \frac{-120 \text{ feet}}{7 \text{ minutes}} \approx -17.14 \text{ feet per minute} \)

Finally, let’s calculate the change from 10 minutes to 14 minutes:

  • Change in height: \( 160 - 80 = 80 \) feet
  • Change in time: \( 14 - 10 = 4 \) minutes

Rate of change \( = \frac{-80 \text{ feet}}{4 \text{ minutes}} = -20 \text{ feet per minute} \)

Conclusion:

The consistent change we see is approximately -20 feet per minute when the hiker is descending.

The best interpretation of this rate of change is:

The elevation of a hiker who is hiking down to a canyon floor changes at a rate of −20 feet per minute.

Thus, the correct statement would be one that captures this meaning properly and mentions the negative rate clearly.

Use the table to answer the question. Number of Months Cost ($) 1 74 2 99 3 124 4 149 The table below shows the cost for a gym membership at the local fitness center with an initial start-up fee charged in the first month. Find the slope, or monthly cost. (1 point) Responses $49 $49 $25 $25 $37.25 $37.25 $74 $74 Skip to navigation

To find the slope (monthly cost) from the table, we can use the formula for slope, which is defined as the change in cost divided by the change in the number of months.

We can analyze two points from the table to calculate the slope, but since the cost appears to be increasing linearly, we'll find the slope using the costs for two consecutive months.

Let's examine the data:

  • Month 1: Cost = $74
  • Month 2: Cost = $99
  • Month 3: Cost = $124
  • Month 4: Cost = $149

Calculating the slope between the first and second month:

  • Change in cost: \( 99 - 74 = 25 \)
  • Change in months: \( 2 - 1 = 1 \)

Using these values to calculate the slope:

\[ \text{slope} = \frac{\text{Change in Cost}}{\text{Change in Months}} = \frac{25}{1} = 25 \]

We can also check another consecutive pair to confirm consistency:

Between Month 2 and Month 3:

  • Change in cost: \( 124 - 99 = 25 \)
  • Change in months: \( 3 - 2 = 1 \)

Calculating the slope again:

\[ \text{slope} = \frac{25}{1} = 25 \]

Or between Month 3 and Month 4:

  • Change in cost: \( 149 - 124 = 25 \)
  • Change in months: \( 4 - 3 = 1 \)

Calculating the slope again:

\[ \text{slope} = \frac{25}{1} = 25 \]

Conclusion:

The slope (or monthly cost) is consistently $25.

Therefore, the correct response is:

$25.

A linear graph of parking fees in dollars based on hours parked has the points (2,20) and (6,44) . How would you interpret the slope of the graph as the rate of change in the parking fee for each additional hour of parking?(1 point) Responses The parking fee rises by $8 with each additional hour. The parking fee rises by $8 with each additional hour. The parking fee rises by $10 with each additional hour. The parking fee rises by $10 with each additional hour. The parking fee rises by $6 with each additional hour. The parking fee rises by $6 with each additional hour. The parking fee rises by $7.33 with each additional hour.

To find the slope of the linear graph representing the parking fees, we can use the two points given: \( (2, 20) \) and \( (6, 44) \). The slope (\( m \)) can be calculated using the formula:

\[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points:

  • \( (x_1, y_1) = (2, 20) \)
  • \( (x_2, y_2) = (6, 44) \)

Substituting these values into the slope formula:

\[ m = \frac{44 - 20}{6 - 2} \]

Calculating the changes:

\[ m = \frac{24}{4} = 6 \]

Conclusion:

The slope of the graph is 6, which means that the parking fee rises by $6 with each additional hour of parking.

Thus, the correct response is:

The parking fee rises by $6 with each additional hour.

Which formula can be used to find the slope of a line from two points (x1,y1) and (x2,y2) ?(1 point) Responses y2−y1x2−x1 Start Fraction y subscript 2 baseline minus y subscript 1 baseline over x subscript 2 baseline minus x subscript 1 baseline End Fraction x2−x1y2−y1 Start Fraction x subscript 2 baseline minus x subscript 1 baseline over y subscript 2 baseline minus y subscript 1 baseline End Fraction y2y1−x2x1 Start Fraction y subscript 2 baseline over y subscript 1 baseline End Fraction minus Start Fraction x subscript 2 baseline over x subscript 1 baseline End Fraction y2+y1x2+x1 Start Fraction y subscript 2 baseline plus y subscript 1 baseline over x subscript 2 baseline plus x subscript 1 baseline End Fraction

The correct formula to find the slope of a line from two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

This formula expresses the slope \( m \) as the change in the y-coordinates (vertical change) divided by the change in the x-coordinates (horizontal change).

Conclusion:

The correct response is:

\( \frac{y_2 - y_1}{x_2 - x_1} \)

This correctly represents the slope formula for a line.