Compare Linear Function 1 with Linear Function 2. Which one has the greater rate of change?

Question

Linear function 1: y = 15x + 8

Linear function 2:

x      y
5      26
13    34

GPT-4o mini · Answer

To compare the rate of change of Linear Function 1 with Linear Function 2, we first need to identify the slope (rate of change) for each function.

Linear Function 1:

The equation is given as: \[ y = 15x + 8 \] In this equation, the coefficient of \( x \) (which is 15) represents the slope. Therefore, the rate of change for Linear Function 1 is 15.

Linear Function 2:

For Linear Function 2, we can determine the slope using the two points provided: (5, 26) and (13, 34).

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the points into the formula:

Let \((x_1, y_1) = (5, 26)\)
Let \((x_2, y_2) = (13, 34)\)

Now calculate the slope: \[ m = \frac{34 - 26}{13 - 5} = \frac{8}{8} = 1 \]

Summary of Rates of Change:

Linear Function 1 has a rate of change (slope) of 15.
Linear Function 2 has a rate of change (slope) of 1.

Conclusion:

Linear Function 1 has a greater rate of change than Linear Function 2, as 15 is greater than 1.

Answer

you sure

GPT-4o mini · Answer

Yes, let's go over the calculations again step by step to ensure accuracy.

Linear Function 1:

The equation is given as: \[ y = 15x + 8 \]

In this equation:

The slope (rate of change) is the coefficient of \( x \), which is 15.

Linear Function 2:

The points provided are:

Point 1: (5, 26)
Point 2: (13, 34)

To find the slope (rate of change), we'll use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the values:

\( y_1 = 26 \), \( y_2 = 34 \)
\( x_1 = 5 \), \( x_2 = 13 \)

Now plug in the values: \[ m = \frac{34 - 26}{13 - 5} = \frac{8}{8} = 1 \]

Summary:

Linear Function 1 has a slope of 15.
Linear Function 2 has a slope of 1.

Conclusion:

Linear Function 1 does indeed have a greater rate of change (slope of 15) compared to Linear Function 2 (slope of 1). I confirm that the comparison is accurate. Thank you for your patience!