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 −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.
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 23, and the rate of change is −23.
From the given information, we can find the equation of the line using the slope-intercept form: y = mx + b, where m is the rate of change and b is the initial value.
The slope, or rate of change, can be found using the two points (-4, -10/3) and (-1, -4/3):
m = (change in y)/(change in x)
m = (-4/3 - (-10/3))/(-1 - (-4))
m = (-4/3 + 10/3)/(3)
m = 6/3 = 2
Now, we can plug in one of the points to find the initial value, b. Let's use (-4, -10/3):
-10/3 = (2)(-4) + b
-10/3 = -8 + b
b = -10/3 + 8
b = 14/3
Therefore, the initial value is 14/3, and the rate of change is 2.
The correct answer is: The initial value is Start Fraction 1 over 6 End Fraction, and the rate of change is Start Fraction 3 over 2 End Fraction.
ARE YOU SURE
I apologize for any confusion caused. Let's recheck the calculations:
The slope, or rate of change, can be found using the two points (-4, -10/3) and (-1, -4/3):
m = (change in y)/(change in x)
m = (-4/3 - (-10/3))/(-1 - (-4))
m = (-4/3 + 10/3)/(3)
m = 6/3 = 2
Now, we can plug in one of the points to find the initial value, b. Let's use (-4, -10/3):
-10/3 = (2)(-4) + b
-10/3 = -8 + b
b = -10/3 + 8
b = 14/3
Therefore, the initial value is 14/3, and the rate of change is 2.
The correct answer is: The initial value is 14/3, and the rate of change is 2.
To determine the initial value and rate of change of the linear function depicted in the graph, we need to use the given information about the points the line passes through.
First, let's identify two points on the line: ( -4, -10/3 ) and ( -1, -4/3 ).
The initial value of the linear function is the y-intercept, which is the value of y when x = 0. In this case, we can observe that when x = 0, y is approximately -2, based on the graph.
Next, we calculate the rate of change, also known as the slope, by determining the change in y divided by the change in x between the two given points.
The change in y is the difference between the y-coordinates of the two points: (-4/3) - (-10/3) = 6/3 = 2.
The change in x is the difference between the x-coordinates of the two points: -1 - (-4) = 3.
So, the rate of change is 2/3.
Therefore, the correct answer is: The initial value is -2, and the rate of change is 2/3.