- The linear model is f(s) = 4.5s + 50.

Question

- The range is {y| 50 ≤ y ≤ 158}.
- The graph is continuous. this is not correct try to change on of the options add or remove something isnt right.

A class of 24 students is planning a field trip to a science museum. A nonrefundable deposit of $50 is required for the day-long program, plus a charge of $4.50 per student.

Determine a linear function that models the cost, c, and the number of students, s.

Which statements about the linear function and its graph are correct? Check all that apply.

The linear model is f(s) = 4.5s + 50.
The linear model is f(c) = 54.5c + 24.
The domain is {x| 0 ≤ x ≤ 24}.
The range is {y| 50 ≤ y ≤ 158}.
The graph is continuous.

Answer 1

Let's analyze the situation and the statements provided in the context of the linear function modeling the cost.

Given Information:

• A nonrefundable deposit of \( $50 \).
• A charge of \( $4.50 \) per student.
• There are \( 24 \) students.

Determining the Function:

The cost \( c \) can be modeled as: \[ c(s) = 4.5s + 50 \] where \( s \) is the number of students.

Evaluation of Statements:

1. The linear model is f(s) = 4.5s + 50.
   (Correct) This is a correct representation of the relationship between the number of students \( s \) and the total cost \( c \).

2. The linear model is f(c) = 54.5c + 24.
   (Incorrect) This statement misrepresents the model. The given equation is incorrect in both format and context as it should model \( c \) in terms of \( s \), not the other way around.

3. The domain is {x| 0 ≤ x ≤ 24}.
   (Correct) This is correct because the number of students \( s \) can vary from \( 0 \) to \( 24 \).

4. The range is {y| 50 ≤ y ≤ 158}.
   (Correct) To find the range, we calculate the total cost for the minimum and maximum values of \( s \):

   • Minimum cost (when \( s = 0 \)): \( c(0) = 4.5(0) + 50 = 50 \)
   • Maximum cost (when \( s = 24 \)): \( c(24) = 4.5(24) + 50 = 108 + 50 = 158 \) So, the range is indeed \( {y| 50 ≤ y ≤ 158} \).

5. The graph is continuous.
   (Correct) The graph of a linear equation is continuous.

Conclusion:

Based on the analysis, the following statements are true:

• The linear model is f(s) = 4.5s + 50. (Correct)
• The domain is {x| 0 ≤ x ≤ 24}. (Correct)
• The range is {y| 50 ≤ y ≤ 158}. (Correct)
• The graph is continuous. (Correct)

Recommendation: Change "The linear model is f(c) = 54.5c + 24" to "The linear model is f(s) = 4.5s + 50." or remove it, as it is an incorrect statement in the context.

Answer 2

The linear model is f(s) = 4.5s + 50.
The domain is {x| 0 ≤ x ≤ 24}.
The range is {y| 50 ≤ y ≤ 158}.

Use the slider to change the value of m.

As the value of m changes from 1 to 3, the graph
✔ gets steeper
.

As the value of m changes from –6 to –1, the graph
.

If the value of m is negative, the slope of the line is
.

when 0 its stays at the middle when going negative its counter clockwise and positive clockwise. and it only reaches at -8 or 8

Answer 3

It seems you're exploring the behavior of a linear function with a slope \( m \) that changes, and this affects the characteristics of the graph. Let’s address the points based on your statements.

Statement Evaluations:

1. As the value of m changes from 1 to 3, the graph gets steeper.
   ✔ (Correct) When the slope \( m \) increases (positive), the steepness of the graph increases as well. A larger positive slope means the line rises more sharply.

2. As the value of m changes from –6 to –1, the graph...
   The correct statement would be: The graph becomes less steep (or the slope becomes shallower). This is true because as \( m \) increases from a negative value toward zero, the slope becomes less negative, which means the line rises less sharply or becomes less steep.

3. If the value of m is negative, the slope of the line is...
   The correct completion could be: the slope of the line is negative. This indicates that as \( x \) increases, \( y \) decreases, meaning the line is sloping downwards.

Additional Information Based on Your Description:

• The mention of "when 0 it stays at the middle" suggests that when \( m = 0 \), the line is horizontal, meaning it retains a constant value across different \( x \) values.
• When \( m \) is negative, the line rotates counterclockwise, reaching a maximum downward slope (steepness) when it is at its most negative (for example, \( m = -8 \)).
• When \( m \) is positive, the line rotates clockwise and can reach a steep upward angle (maximum upward steepness when \( m = 8\)).

Summary:

• When \( m \) increases from \( 1 \) to \( 3\): The graph of the line gets steeper.
• When \( m \) increases from \( -6 \) to \( -1\): The graph becomes less steep.
• If \( m \) is negative: The slope of the line is negative, indicating a downward trend.

Feel free to adjust or let me know if there's anything specific you want to dive deeper into regarding this topic!

Answer 4

Consider the graph of the linear function h(x) = –2/3x + 5. Which could you change to move the graph down 3 units?

the value of b to –3
the value of m to –3
the value of b to 2
the value of m to 2

Answer 5

To change the graph of the linear function \( h(x) = -\frac{2}{3}x + 5 \) down by 3 units, we need to adjust the y-intercept \( b \).

The general form of a linear function is: \[ h(x) = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept.

Currently, in your function:

• \( m = -\frac{2}{3} \)
• \( b = 5 \)

Moving the Graph Down by 3 Units

To move the entire graph down by 3 units, you subtract 3 from the y-intercept \( b \). So the new y-intercept will be: \[ b_{\text{new}} = 5 - 3 = 2 \] Thus, the new equation becomes: \[ h(x) = -\frac{2}{3}x + 2 \]

Conclusion

From the options you provided:

• the value of b to –3 (incorrect, would move it down to a total of \( 5 - 3 = 2 \))
• the value of m to –3 (incorrect, changes the slope instead of moving the graph down)
• the value of b to 2 (correct, since it represents the new y-intercept after moving down)
• the value of m to 2 (incorrect, changes the slope instead of moving the graph down)

The correct choice is "the value of b to 2."

Answer 6

Which graph represents the function y = 2/3 x – 2?A coordinate plane with a line passing through (0, negative 3) and (2, 0).A coordinate plane with a line passing through (0, 2) and (2, negative 1).A coordinate plane with a line passing through (0, negative 2) and (3, 0).A coordinate plane with a line passing through (0, 1.5) and about (2.25, 0).

Answer 7

To determine which graph represents the function \( y = \frac{2}{3}x - 2 \), we can analyze some key points of this linear equation:

1. Y-intercept: The y-intercept (\( b \)) occurs when \( x = 0 \): \[ y = \frac{2}{3}(0) - 2 = -2 \] So the graph will pass through the point \( (0, -2) \).

2. X-intercept: The x-intercept occurs when \( y = 0 \): \[ 0 = \frac{2}{3}x - 2 \] Solving for \( x \): \[ \frac{2}{3}x = 2 \quad \Rightarrow \quad x = 2 \cdot \frac{3}{2} = 3 \] Thus, the graph will also pass through the point \( (3, 0) \).

Summary of Key Points

• The graph has a y-intercept at \( (0, -2) \).
• The graph has an x-intercept at \( (3, 0) \).

Analyzing the Options

Now, let’s analyze the given options:

1. A coordinate plane with a line passing through (0, -3) and (2, 0).

   • Incorrect: The y-intercept is at \((0, -3)\), not \((0, -2)\).

2. A coordinate plane with a line passing through (0, 2) and (2, -1).

   • Incorrect: The y-intercept is at \((0, 2)\), not \((0, -2)\).

3. A coordinate plane with a line passing through (0, -2) and (3, 0).

   • Correct: This matches both intercepts we calculated.

4. A coordinate plane with a line passing through (0, 1.5) and about (2.25, 0).

   • Incorrect: The y-intercept is at \((0, 1.5)\), not \((0, -2)\).

Conclusion

The correct graph that represents the function \( y = \frac{2}{3}x - 2 \) is: "A coordinate plane with a line passing through (0, -2) and (3, 0)."

Answer 8

A coordinate plane with a line passing through (negative 3, negative 5), (0, 1), and (2, 5).A function f(x) is graphed.What is the slope of the function?m =What is the y-intercept of the function?b =Which equation represents the graph of the function?

Answer 9

To find the slope \( m \), the y-intercept \( b \), and the equation of the linear function based on the given points, let's analyze the points and apply the relevant formulas.

Given Points:

1. \( (-3, -5) \)
2. \( (0, 1) \) — This point is the y-intercept itself.
3. \( (2, 5) \)

1. Finding the Slope \( m \)

The slope of a line is calculated using any two points on the line with the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (-3, -5) \) and \( (2, 5) \):

• \( (x_1, y_1) = (-3, -5) \)
• \( (x_2, y_2) = (2, 5) \)

Plugging in the values: \[ m = \frac{5 - (-5)}{2 - (-3)} = \frac{5 + 5}{2 + 3} = \frac{10}{5} = 2 \]

2. Finding the y-intercept \( b \)

The y-intercept is the y-value when \( x = 0 \). From the point \( (0, 1) \), we can see directly that: \[ b = 1 \]

3. Writing the Equation of the Line

The slope-intercept form of a linear equation is given by: \[ y = mx + b \] Substituting in the values we found: \[ y = 2x + 1 \]

Summary of Results:

• The slope \( m = 2 \)
• The y-intercept \( b = 1 \)
• The equation representing the graph of the function is: \[ y = 2x + 1 \]

If you need further assistance or details, feel free to ask!

Answer 10

A function f(x) is graphed.

What is the slope of the function?

m =
✔ 2

What is the y-intercept of the function?

b =
✔ 1

Which equation represents the graph of the function?

✔ y = 2x + 1

Caitlin is designing a railing for a set of stairs. The railing will begin at a height of 36 inches and follow the slant of the stairs, which decreases 9 inches for every 12 horizontal inches.A drawing of the side view of a set of 4 stairs. There are lines drawn to show a railing that is at the highest point of the stairs and labeled 36 inches. Which function can represent the height, y, of the railing in inches according to the horizontal distance in inches, x, from the top of the stairs?y = –StartFraction 3 Over 4 EndFractionx + 36y = –3x + 36y = StartFraction 3 Over 4 EndFractionx + 36y = 3x + 36

Answer 11

To determine the function that represents the height \( y \) of the railing according to the horizontal distance \( x \) from the top of the stairs, we need to analyze the rate at which the height decreases.

Analyzing the Problem

1. Starting Height: The railing starts at a height of \( 36 \) inches.
2. Slope: The slope of the railing is given as decreasing \( 9 \) inches for every \( 12 \) horizontal inches. This represents the change in height relative to the change in horizontal distance.

Finding the Slope

The slope \( m \) can be calculated as: \[ m = \frac{\text{change in height}}{\text{change in horizontal distance}} = \frac{-9 \text{ inches}}{12 \text{ inches}} = -\frac{3}{4} \]

Writing the Function

We can use the slope-intercept form of the linear equation, which is given by: \[ y = mx + b \] Where:

• \( m \) is the slope
• \( b \) is the y-intercept (the starting height, which is \( 36 \))

Substituting the values we found:

• \( m = -\frac{3}{4} \)
• \( b = 36 \)

The equation becomes: \[ y = -\frac{3}{4}x + 36 \]

Evaluating the Options

Now let's evaluate the given options:

1. \( y = -\frac{3}{4}x + 36 \) (This matches what we've derived.)
2. \( y = -3x + 36 \) (This has a steeper slope and is incorrect.)
3. \( y = \frac{3}{4}x + 36 \) (This has a positive slope and is also incorrect.)
4. \( y = 3x + 36 \) (This has a positive slope and is incorrect.)

Conclusion

The correct function that represents the height \( y \) of the railing in inches according to the horizontal distance \( x \) from the top of the stairs is: \[ y = -\frac{3}{4}x + 36 \]

If you have any further questions or need additional details, feel free to ask!