Use the graph to answer this question

Question

The plotted points are : (-3, 4), (0, 3), (3, 2) and (6, 1)
Determine the initial blue and rate of change of the linear function as given in the graph round the answer to three decimal places as needed

Answer 1

To determine the initial value and the rate of change of the linear function represented by the points \((-3, 4), (0, 3), (3, 2), (6, 1)\), we can find the slope (rate of change) and the y-intercept (initial value) of the linear function.

Step 1: Calculate the slope (rate of change)

The slope \(m\) of a linear function can be determined using two points \((x_1, y_1)\) and \((x_2, y_2)\) with the formula:

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

We can use any two points; let's use \((0, 3)\) and \((3, 2)\):

\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} = -0.333 \]

Step 2: Determine the y-intercept (initial value)

The y-intercept is the value of \(y\) when \(x = 0\). According to the points given, when \(x = 0\), \(y = 3\). Therefore, the initial value (y-intercept) is:

\[ b = 3 \]

Final Results

• Initial value (y-intercept): \(3\)
• Rate of change (slope): \(-0.333\) (rounded to three decimal places)

Therefore, the initial blue (y-intercept) and rate of change are:

• Initial value: 3
• Rate of change: -0.333

Answer 2

The graph shows an elevation of a deep sea diver in relation to minutes passed. Using any two points on the graph, what is the rate of change in the divers elevation?
Plotted points: (4, -80), (2, -40)

Answer 3

To determine the rate of change in the diver’s elevation using the two given points \((4, -80)\) and \((2, -40)\), we can use the slope formula:

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

Here, we identify our points:

• Point 1: \((x_1, y_1) = (2, -40)\)
• Point 2: \((x_2, y_2) = (4, -80)\)

Now, we can plug in these coordinates into the formula:

\[ m = \frac{-80 - (-40)}{4 - 2} = \frac{-80 + 40}{2} = \frac{-40}{2} = -20 \]

Rate of Change

Thus, the rate of change in the diver's elevation is \(-20\) units of elevation per minute.

This means the diver is descending at a rate of 20 units per minute.

Answer 4

A pan with four eggs weighs 18 ounces. A pan with 8 eggs weighs 24 ounces. What is the weight of the frying pan?

Answer 5

To solve for the weight of the frying pan, we can set up a system of equations based on the information given.

Let:

• \( p \) = weight of the frying pan (in ounces)
• \( e \) = weight of one egg (in ounces)

From the problem, we have two scenarios:

1. A pan with 4 eggs weighs 18 ounces: \[ p + 4e = 18 \quad \text{(Equation 1)} \]

2. A pan with 8 eggs weighs 24 ounces: \[ p + 8e = 24 \quad \text{(Equation 2)} \]

Now, we can solve these equations step by step.

Step 1: Solve for \( p \)

We can eliminate \( p \) by subtracting Equation 1 from Equation 2:

\[ (p + 8e) - (p + 4e) = 24 - 18 \]

Simplifying this gives:

\[ 8e - 4e = 6 \]

\[ 4e = 6 \quad \Rightarrow \quad e = \frac{6}{4} = 1.5 \text{ ounces} \]

Step 2: Substitute \( e \) back to find \( p \)

Now we can substitute the value of \( e \) back into one of the original equations to find \( p \). We'll use Equation 1:

\[ p + 4(1.5) = 18 \]

Calculating:

\[ p + 6 = 18 \]

Subtracting 6 from both sides:

\[ p = 18 - 6 = 12 \]

Conclusion

The weight of the frying pan is 12 ounces.

Answer 6

Manuel got a new job and was given a hinng bonus along with his monthly salary His earnings aller z months are modeled by the function /(z) = 1,500+300 How much did he receive as a hiring bonus?

Answer 7

The function given for Manuel's earnings after \( z \) months is:

\[ f(z) = 1,500 + 300z \]

This function can be broken down into two parts:

• The constant term \( 1,500 \) represents the monthly salary he receives.
• The variable term \( 300z \) indicates the total amount earned from the hiring bonus, where \( 300 \) is the bonus amount per month and \( z \) is the number of months.

Since the problem specifies that he received a hiring bonus, the constant part \( 1,500 \) could reasonably represent the hiring bonus itself, as bonuses are often paid upfront.

Conclusion:

Manuel received a hiring bonus of 1,500 dollars.

Answer 8

The equation y = mx + b defines what type of function? (1 point)

a linear function

a slope function

an initial value function

a nonlinear function

Answer 9

The equation \( y = mx + b \) defines:

a linear function.

This equation represents a straight line on a graph, where \( m \) represents the slope and \( b \) represents the y-intercept (initial value). Therefore, the correct answer is:

a linear function.

Answer 10

Use the table to answer this question
Table: 6 and 15 are under x and 16 and 43 are under y
Write the equation of the linear function that models the relationship shown in the table
Enter your answer in y = mx + b form

Answer 11

To find the equation of the linear function that models the relationship given in the table, we can use the two points derived from the table. The points are:

• Point 1: \((6, 16)\)
• Point 2: \((15, 43)\)

Step 1: Calculate the slope \(m\)

The slope \(m\) can be calculated using the slope formula:

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

Substituting the coordinates from the points:

\[ m = \frac{43 - 16}{15 - 6} = \frac{27}{9} = 3 \]

Step 2: Use the slope and one point to find \(b\) (the y-intercept)

Now that we have the slope \(m = 3\), we can use one of the points to find the y-intercept \(b\). We'll use the point \((6, 16)\).

The equation of the line in slope-intercept form is:

\[ y = mx + b \]

Substituting \(m\) and one of the points into the equation:

\[ 16 = 3(6) + b \]

Calculating:

\[ 16 = 18 + b \]

Solving for \(b\):

\[ b = 16 - 18 = -2 \]

Step 3: Write the equation

Now we can write the equation of the linear function:

\[ y = 3x - 2 \]

Conclusion

The equation that models the relationship shown in the table is:

\[ \boxed{y = 3x - 2} \]