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 −23, and the rate of change is 23.
The initial value is negative Start Fraction 2 over 3 End Fraction , and the rate of change is 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 .
Skip to navigation
Since the line passes through the points (-4, -10/3) and (-1, -4/3), we can calculate the rate of change by finding the difference in the y-values and dividing by the difference in the x-values:
Rate of change = (y2 - y1) / (x2 - x1) = (-4/3 - (-10/3)) / (-1 - (-4)) = (-4/3 + 10/3) / (3) = 6/3 / 3 = 2/3.
Therefore, the rate of change is 2/3.
To find the initial value, we can substitute one of the given points into the equation of a line: y = mx + b, where m is the rate of change and b is the initial value.
Using the point (-4, -10/3), we can substitute x = -4 and y = -10/3 into the equation and solve for b:
-10/3 = (2/3)(-4) + b
-10/3 = -8/3 + b
-10/3 + 8/3 = b
-2/3 = b
Therefore, the initial value is -2/3.
So, the correct answer is:
The initial value is -2/3, and the rate of change is 2/3.