Determine the initial value and the rate of change of the linear function as given in the graph.
-4,-10/3 -1,-4/3
A. The initial value is 1/6 and the rate of change is 3/2
B. The initial value is - 2/3, and the rate of change is 2/3
C. The initial value is -2, and the rate of change is 2/3
D. The initial value is 2/3 , and the rate of change is -2/3
To determine the rate of change, we can use the formula:
Rate of change = (change in y) / (change in x)
Given the points (-4, -10/3) and (-1, -4/3), the change in y is -4/3 - (-10/3) = -4/3 + 10/3 = 6/3 = 2.
The change in x is -1 - (-4) = -1 + 4 = 3.
Therefore, the rate of change = 2/3.
To determine the initial value, we can use the point-slope form of a linear function:
y - y1 = m(x - x1), where m is the rate of change and (x1, y1) is a point on the line.
Using the point (-4, -10/3), we have:
y - (-10/3) = (2/3)(x - (-4))
Simplifying this equation gives:
y + 10/3 = (2/3)(x + 4)
Multiplying both sides by 3 gives:
3y + 10 = 2x + 8
Simplifying further:
3y = 2x - 2
Dividing both sides by 3 gives:
y = (2/3)x - 2/3
Therefore, the initial value is -2/3.
Thus, the correct answer is:
B. The initial value is -2/3, and the rate of change is 2/3.
what is the inital value
The initial value refers to the y-intercept of the linear function, which is the value of y when x = 0.
From the equation y = (2/3)x - 2/3, we can see that when x = 0, y = -2/3.
Therefore, the initial value is -2/3.
To determine the initial value and the rate of change of a linear function, we need to find the equation of the line that represents the given data points (-4, -10/3) and (-1, -4/3).
First, let's find the rate of change (also known as the slope) of the line. The rate of change is found by calculating the change in the y-values divided by the change in the x-values.
Rate of change = (change in y) / (change in x)
Using the given points, we can calculate the change in y and the change in x as follows:
Change in y = (y2 - y1) = (-4/3 - (-10/3)) = (-4/3 + 10/3) = 6/3 = 2
Change in x = (x2 - x1) = (-1 - (-4)) = (-1 + 4) = 3
Therefore, the rate of change (slope) is 2/3.
Now, to find the initial value (y-intercept) of the line, we can choose either of the given points and substitute it into the equation y = mx + b, where m is the slope and b is the y-intercept.
Let's use the point (-4, -10/3) to find the initial value:
-10/3 = (2/3)(-4) + b
-10/3 = -8/3 + b
b = -10/3 + 8/3
b = -2/3
Therefore, the initial value (y-intercept) of the line is -2/3.
Based on our calculations, the initial value is -2/3 (option B) and the rate of change is 2/3. Therefore, the correct answer is option B: The initial value is -2/3, and the rate of change is 2/3.