Which of the following represents a linear function with the same slope as the table below? *Select all that apply! (2 points) Responses Cynthia has $200 in her savings account and spends $3 per day on coffee. Cynthia has $200 in her savings account and spends $3 per day on coffee. James increases his muscle mass by 3 kg per day. James increases his muscle mass by 3 kg per day. 6x−2y=12. A graph decreasing from y intercept 10 and goes through between 2 and 4 on the x axis

Question

The table it:

x = -4 f(x) = 14

x = -2 f(x) = 8

x = 3 f(x) = -7

x = 5 f(x) = -13

Answer 1

To determine which of the options represents a linear function with the same slope as the given table, we first need to find the slope of the function from the values provided in the table.

Step 1: Calculate the slope from the table

Using the points in the table:

1. Point 1: \((-4, 14)\)
2. Point 2: \((-2, 8)\)
3. Point 3: \((3, -7)\)
4. Point 4: \((5, -13)\)

We can calculate the slope (\(m\)) between two points using the slope formula:

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

Let's calculate the slope between the first two points \((-4, 14)\) and \((-2, 8)\):

\[ m = \frac{8 - 14}{-2 - (-4)} = \frac{-6}{2} = -3 \]

Step 2: Check each option for the same slope

1. Cynthia's scenario: Cynthia has $200 in her savings account and spends $3 per day:

   • This represents a linear function where the slope is \(-3\) (since she is spending money, which decreases her savings).

2. James's scenario: James increases his muscle mass by 3 kg per day:

   • This has a slope of \(3\) (positive), as he is gaining muscle mass.

3. Equation \(6x - 2y = 12\): Let's convert this to slope-intercept form \(y = mx + b\): \[ 6x - 2y = 12 \implies -2y = -6x + 12 \implies y = 3x - 6 \]

   • This has a slope of \(3\) (positive).

4. Graph: A graph with a decreasing trend from a y-intercept of 10 indicates a negative slope. The slope can be determined through the x intercept, which is between 2 and 4. A graph with a negative slope could potentially represent a slope of \(-3\) given that it is decreasing.

Summary of Slope Analysis:

• Cynthia's function: Slope = -3 (matches)
• James's function: Slope = 3 (does not match)
• \(6x - 2y = 12\): Slope = 3 (does not match)
• The graph (assuming it has a slope of -3 as it is decreasing): Slope = -3 (matches).

Final Answers:

The options that represent a linear function with the same slope of -3 are:

• Cynthia's function
• The graph (if it indeed has a slope of -3).

If the graph’s slope is not explicitly given, further analysis of points could be required to confirm. If it is indeed decreasing and follows the given trend, it can be included as a match.

In conclusion, the correct selections based on the slope are:

• Cynthia's function
• The graph (conditionally, based on visual indication of slope)