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=-1/2x+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 determine which statements about linear functions M and N are true, let's first calculate the rate of change (slope) and the initial value (y-intercept) for function M.

### Function M:
We can calculate the rate of change (slope) using two points from the table. We can use the points \( (0, 3) \) and \( (2, 0) \).

1. **Calculate the slope (rate of change) of M**:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{2 - 0} = \frac{-3}{2} = -\frac{3}{2}
\]

2. **Initial value (y-intercept)**:
The initial value is the value of \( y \) when \( x = 0 \). From the table, we can see that:
\[
\text{Initial value of M} = 3
\]

### Function N:
The equation of function N is given as \( y = -\frac{1}{2}x + 2 \).
1. **Slope of N**:
The coefficient of \( x \) is the slope:
\[
\text{slope of N} = -\frac{1}{2}
\]

2. **Initial value of N**:
The y-intercept (initial value) is the constant term:
\[
\text{Initial value of N} = 2
\]

### Comparing the Functions:
1. **Rate of Change**:
- Slope of M: \( -\frac{3}{2} \)
- Slope of N: \( -\frac{1}{2} \)

Since \( -\frac{3}{2} < -\frac{1}{2} \), the rate of change for function M is less than the rate of change for function N.

2. **Initial Values**:
- Initial value of M: \( 3 \)
- Initial value of N: \( 2 \)

Since \( 3 > 2 \), the initial value of function M is greater than the initial value of function N.

### Conclusion:
The true statement is:
**The rate of change for function M is less than the rate of change for function N.**