Question
page 27 of 27 Slope and y-intercepts Quick Check 4 of 54 of 5 Items 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 −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 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 −23 , and the rate of change is 23 .
Answers
GPT-4o mini
To determine the initial value (y-intercept) and the rate of change (slope) of the linear function represented by the line on the coordinate plane, we can analyze the two points given: \((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\).
### Step 1: Calculate the Rate of Change (Slope)
The formula for calculating the slope (m) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using our points, we have:
- \((x_1, y_1) = (-4, -\frac{10}{3})\)
- \((x_2, y_2) = (-1, -\frac{4}{3})\)
Substituting these values into the slope formula:
\[
m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)}
\]
This simplifies to:
\[
m = \frac{-\frac{4}{3} + \frac{10}{3}}{3} = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]
### Step 2: Determine the Initial Value (Y-Intercept)
Next, we can find the y-intercept (b) of the line, which is the value of y when x = 0. We can use the slope-intercept form of the equation of a line, \(y = mx + b\).
We take one of the points and rearrange to find b. Using the point \((-1, -\frac{4}{3})\):
\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]
\[
-\frac{4}{3} = -\frac{2}{3} + b
\]
\[
b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]
### Conclusion
The initial value (y-intercept) is -\(\frac{2}{3}\), and the rate of change (slope) is \(\frac{2}{3}\).
**The correct answer is:**
The initial value is \(-\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).
### Step 1: Calculate the Rate of Change (Slope)
The formula for calculating the slope (m) of a line given two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using our points, we have:
- \((x_1, y_1) = (-4, -\frac{10}{3})\)
- \((x_2, y_2) = (-1, -\frac{4}{3})\)
Substituting these values into the slope formula:
\[
m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)}
\]
This simplifies to:
\[
m = \frac{-\frac{4}{3} + \frac{10}{3}}{3} = \frac{\frac{6}{3}}{3} = \frac{2}{3}
\]
### Step 2: Determine the Initial Value (Y-Intercept)
Next, we can find the y-intercept (b) of the line, which is the value of y when x = 0. We can use the slope-intercept form of the equation of a line, \(y = mx + b\).
We take one of the points and rearrange to find b. Using the point \((-1, -\frac{4}{3})\):
\[
-\frac{4}{3} = \frac{2}{3}(-1) + b
\]
\[
-\frac{4}{3} = -\frac{2}{3} + b
\]
\[
b = -\frac{4}{3} + \frac{2}{3} = -\frac{2}{3}
\]
### Conclusion
The initial value (y-intercept) is -\(\frac{2}{3}\), and the rate of change (slope) is \(\frac{2}{3}\).
**The correct answer is:**
The initial value is \(-\frac{2}{3}\), and the rate of change is \(\frac{2}{3}\).