Question

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Slope and y-intercepts Quick Check
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Question
Use the image to answer the question.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 5 to 5 in increments of 1. An upward slanting line is plotted on the coordinate plane. The upward slanting line with arrows at both ends passes through the following closed points: left parenthesis negative 4 comma start fraction negative 10 over 3 end fraction right parenthesis, and left parenthesis negative 1 comma start fraction negative 4 over 3 end fraction right parenthesis.

Determine the initial value and the rate of change of the linear function as given in the graph.

(1 point)
Responses

The initial value is 16
, and the rate of change is 32
.
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 23
, and the rate of change is −23
.
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
, and the rate of change is 23
.
The initial value is negative 2 , and the rate of change is Start Fraction 2 over 3 End Fraction .

The initial value is −23
, and the rate of change is 23
.

Answer 1

To determine the initial value (y-intercept) and the rate of change (slope) of the linear function from the points given on the coordinate plane, let's first find the slope from the points:

The points given are:
1. \( (-4, -\frac{10}{3}) \)
2. \( (-1, -\frac{4}{3}) \)

The formula for slope (m) is:

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

Substituting the points into the formula:

\[
m = \frac{-\frac{4}{3} - (-\frac{10}{3})}{-1 - (-4)} = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4}
\]

\[
m = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]

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

Next, we need to determine the y-intercept (initial value). We can use the slope-intercept form of a line \( y = mx + b \), where \( b \) is the y-intercept.

Using one of the points, say \( (-1, -\frac{4}{3}) \):

\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]

\[
-\frac{4}{3} = -\frac{2}{3} + b
\]

Adding \( \frac{2}{3} \) to both sides:

\[
b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]

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

Putting this all together:
- The initial value is \( -\frac{2}{3} \).
- The rate of change is \( \frac{2}{3} \).

Now, by evaluating the response options:

- The initial value is \( -\frac{2}{3} \), and the rate of change is \( \frac{2}{3} \).

Therefore, the correct response is:

**The initial value is \( -\frac{2}{3} \), and the rate of change is \( \frac{2}{3} \).**