A coordinate plane with a straight line with a positive slope. The line starts at (0, 0) and passes through points labeled A at (2, 1) and B at (4, 2) and through (8, 4). There is a vertical dashed line starting at point A extending two units to the right and one unit up to point B.

Question

What is the vertical change from Point A to Point B?

What is the horizontal change from Point A to Point B?

What is the rate of change shown on the graph? Give the answer as a decimal rounded to the nearest tenth, if necessary

Answer 1

To find the vertical change from Point A to Point B, we determine the difference in their y-coordinates:

• Point A is at (2, 1) and Point B is at (4, 2).
• The y-coordinate for Point A is 1, and for Point B, it is 2.

Vertical Change from A to B: \( \text{Vertical Change} = y_B - y_A = 2 - 1 = 1 \)

So, the vertical change from Point A to Point B is 1.

To find the horizontal change from Point A to Point B, we determine the difference in their x-coordinates:

• The x-coordinate for Point A is 2, and for Point B, it is 4.

Horizontal Change from A to B: \( \text{Horizontal Change} = x_B - x_A = 4 - 2 = 2 \)

Thus, the horizontal change from Point A to Point B is 2.

Next, we calculate the rate of change (slope) shown on the graph, which is given by the ratio of the vertical change to the horizontal change:

Rate of Change (Slope): \( \text{Rate of Change} = \frac{\text{Vertical Change}}{\text{Horizontal Change}} = \frac{1}{2} = 0.5 \)

Therefore, the rate of change shown on the graph is 0.5.

Answer 2

The table below shows the linear relationship between the number of people at a picnic and the total cost of the picnic.

A two column table with five rows. The first column, Number of People, has the entries, 6, 9, 12, 15. The second column, Total Cost in dollars, has the entries, 52, 58, 64, 70.

Which statements about the function described by the table are true? Check all that apply.

The independent variable is the number of people.
The initial value (initial fee) for the picnic is $40.
The rate of change is $8.67 per person.
As the number of people increases, the total cost of the picnic increases.
If 4 people attended the picnic, the total cost would be $46.

Answer 3

Let's analyze the function described by the table before determining which statements are true:

The table is as follows:

| Number of People | Total Cost in Dollars | |------------------|-----------------------| | 6 | 52 | | 9 | 58 | | 12 | 64 | | 15 | 70 |

1. The independent variable is the number of people.

   • True: In a function that describes the total cost based on the number of people, the number of people is indeed the independent variable.

2. The initial value (initial fee) for the picnic is $40.

   • To find the initial value (initial fee) of the picnic, we can look for the y-intercept of the corresponding linear function. Based on the given data points:
     • The total cost for 6 people is $52, and for 9 people, it is $58. There is also a consistent increase in cost with an increase in the number of people.
     • We can determine the cost for 0 people by using the slope formula.
     • From the table, the total cost increases by $6 for each of the first combinations.
     • To find the intercept, we can use the slope \( m \) which can be calculated as follows: \[ m = \frac{58 - 52}{9 - 6} = \frac{6}{3} = 2 \quad \text{(cost increase per person)} \]
     • Therefore, to calculate the total cost for 0 people: \[ \text{Cost} = m \times \text{Number of People} + b \]
     • Taking the point (6, 52): \[ 52 = 2(6) + b \quad \Rightarrow \quad b = 52 - 12 = 40 \] Thus, the initial value is indeed $40.
   • True: The initial value (initial fee) for the picnic is $40.

3. The rate of change is $8.67 per person.

   • From our slope calculation, we found that the cost increases by $2 for each additional person. Therefore, the rate of change is $2 per person, not $8.67.
   • False: The rate of change is not $8.67 per person; it is $2 per person.

4. As the number of people increases, the total cost of the picnic increases.

   • True: As we can see in the table, as the number of people increases from 6 to 15, the total cost also consistently increases from $52 to $70.

5. If 4 people attended the picnic, the total cost would be $46.

   • To check this statement, we use the function \( C = 2n + 40 \) where \( n \) is the number of people: \[ C = 2(4) + 40 = 8 + 40 = 48 \] The calculated cost for 4 people is $48, not $46.
   • False: If 4 people attended the picnic, the total cost would be $48, not $46.

Summary of Statements:

• True: The independent variable is the number of people.
• True: The initial value (initial fee) for the picnic is $40.
• False: The rate of change is $8.67 per person.
• True: As the number of people increases, the total cost of the picnic increases.
• False: If 4 people attended the picnic, the total cost would be $46.

The true statements are:

• The independent variable is the number of people.
• The initial value (initial fee) for the picnic is $40.
• As the number of people increases, the total cost of the picnic increases.