Question

Determine the initial value and the rate of change of the linear function as given in the graph.
-1,-4/3
-4,-10/3
(1 po
Responses

The initial value is −2
, and the rate of change is 2/3
.
The initial value is negative 2 , and the rate of change is Start Fraction 2 over 3 End Fraction .

The initial value is 16
, and the rate of change is 3/2
.
The initial value is Start Fraction 1 over 6 End Fraction , and the rate of change is Start Fraction 3 over 2 End Fraction .

The initial value is 2/3
, and the rate of change is −2/3
.
The initial value is Start Fraction 2 over 3 End Fraction , and the rate of change is negative Start Fraction 2 over 3 End Fraction .

The initial value is −2/3
, and the rate of change is 2/3
.

Answer 1

To determine the initial value and the rate of change (slope) of the linear function based on the points provided, we need to follow these steps:

1. **Calculate the rate of change (slope)** between the points (-1, -4/3) and (-4, -10/3).

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

Substituting the coordinates:
- \(x_1 = -1\), \(y_1 = -4/3\)
- \(x_2 = -4\), \(y_2 = -10/3\)

We calculate:
\[
m = \frac{-10/3 - (-4/3)}{-4 - (-1)} = \frac{-10/3 + 4/3}{-4 + 1}
\]
\[
= \frac{-6/3}{-3} = \frac{-2}{-3} = \frac{2}{3}
\]

So, the rate of change (slope) is \( \frac{2}{3} \).

2. **Determine the initial value** (y-intercept) of the linear function. The y-intercept can be found using the formula of a linear equation \(y = mx + b\), where \(b\) is the y-intercept.

We can use any of the points to find \(b\). Let's use the point (-1, -4/3):
\[
y = mx + b \implies -\frac{4}{3} = \frac{2}{3}(-1) + b
\]
\[
-\frac{4}{3} = -\frac{2}{3} + b \implies b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]

Thus, the initial value (y-intercept) is \( -\frac{2}{3} \).

**Final Result:**
- The initial value is \( -\frac{2}{3} \).
- The rate of change is \( \frac{2}{3} \).

Therefore, the correct response is:

**The initial value is −2/3, and the rate of change is 2/3.**