Question

Determine the rate of change and the initial value of the linear function given here as a table of values. (1 point) Responses m=−7, b=−3 m equals negative 7 comma b equals negative 3 m=−0.5, b=10.5 m equals negative 0.5 comma b equals 10.5 m=−3, b=−7 m equals negative 3 comma b equals negative 7 m=−0.5, b=5.5

Answer 1

To determine the rate of change (slope) and the initial value (y-intercept) of the linear function from a table of values, you need two points from the table.

The rate of change (slope \( m \)) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the formula:

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

The initial value \( b \) is the value of the function when \( x = 0 \).

Since you did not provide specific points from the table, I can guide you in the right direction if you have a set of numbers or if you identify any two points from the table.

However, looking at the provided options:

• \( m = -7, b = -3 \)
• \( m = -0.5, b = 10.5 \)
• \( m = -3, b = -7 \)
• \( m = -0.5, b = 5.5 \)

You can identify the correct pair by taking potential points from a hypothetical set. If you give me points or values from your table, I can assist you further in calculating anyway you wish.

Answer 2

hypothetically which is it?

Answer 3

Let's analyze the provided options for the slope \( m \) and y-intercept \( b \) to determine which one could be plausible for a hypothetical linear function.

Analyzing the Options

1. Option 1: \( m = -7, b = -3 \)

   • This means for every unit increase in \( x \), \( y \) decreases by 7, and the line crosses the y-axis at -3.

2. Option 2: \( m = -0.5, b = 10.5 \)

   • For every unit increase in \( x \), \( y \) decreases by 0.5, and the line crosses the y-axis at 10.5.

3. Option 3: \( m = -3, b = -7 \)

   • For every unit increase in \( x \), \( y \) decreases by 3, and the line crosses the y-axis at -7.

4. Option 4: \( m = -0.5, b = 5.5 \)

   • For every unit increase in \( x \), \( y \) decreases by 0.5, and the line crosses the y-axis at 5.5.

Determining the Best Option

To determine the most plausible values hypothetically, we generally think about the slope being a smaller negative value (like -0.5 or -3) and consider the y-intercept's behavior:

• If the function is steadily decreasing with a small decline (like Option 2 or Option 4), the slope -0.5 is relatively gentle.
• However, if the function sharply decreases (as in Option 1 or Option 3), we'd expect larger negative slope values like -7 or -3.

Hypothetical Selection

If we assume a moderate decrease and a reasonable y-intercept in the range of typical inputs for a linear function, Option 2: \( m = -0.5, b = 10.5 \), would be a common linear model since a gentle slope is quite frequent in many real-world contexts.

Still, without specific values or context provided, any of these could represent a valid linear relationship based on the data at hand. Therefore, you would ideally select the values based on your particular data if you had them.

If you had data points or further context, please share them for a more tailored analysis!