Question

The table shows some of the ordered pairs for linear function M.
x y
-2 6
0 3
2 0
4 -3
6 -6


The equation for linear function N is y= -3/2x + 2


.

Which of the following statements is true?
The rate of change for function M is less than the rate of change for function N.
The rate of change for function M is equal to the rate of change for function N.
The initial value of function M is less than the initial value of function N.
The initial value of function M is equal to the initial value of function N.

Answer 1

To compare the two functions, we will first determine the rate of change (slope) and the initial value (y-intercept) of both functions.

### For Function M:
We will find the slope (rate of change) by using two points from the table. Let's take the points (0, 3) and (2, 0).

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

Using points (0, 3) and (2, 0):
- \( y_1 = 3, x_1 = 0 \)
- \( y_2 = 0, x_2 = 2 \)

\[
m = \frac{0 - 3}{2 - 0} = \frac{-3}{2} = -1.5
\]

Thus, the rate of change for function M is -1.5.

### For Function N:
The equation is given as:
\[
y = -\frac{3}{2}x + 2
\]
From this equation, we can see that the slope (rate of change) is \( -\frac{3}{2} \) or -1.5.

### Comparing Rates of Change:
- Rate of change for function M: -1.5
- Rate of change for function N: -1.5

### Conclusion on Rates of Change:
Since both slopes are equal, we can say:
- The rate of change for function M is equal to the rate of change for function N.

### Finding Initial Values:
The initial value (y-intercept) is found by looking at where \( x = 0 \).

- For function M, when \( x = 0 \), \( y = 3 \).
- For function N, when \( x = 0 \):
\[
y = -\frac{3}{2}(0) + 2 = 2
\]

### Conclusion on Initial Values:
- The initial value of function M is 3.
- The initial value of function N is 2.

Since \( 3 > 2 \), we can say:
- The initial value of function M is greater than the initial value of function N.

### Summary of the True Statement:
The only true statement among the options is:
**The rate of change for function M is equal to the rate of change for function N.**