compare linear function 1 with linear function 2. which one has the greater rate of change?
linear function 1: y = x + 8
linear function 2: (2,6) (4,10)
To find the rate of change for a linear function, we can look at the slope of the line.
For linear function 1, y = x + 8, the slope is 1. This means that for every increase of 1 in the x-value, the y-value increases by 1. So the rate of change is 1.
For linear function 2, we can find the slope by using the formula: slope = (change in y) / (change in x). Given the points (2,6) and (4,10), we can calculate the changes: change in y = 10 - 6 = 4, and change in x = 4 - 2 = 2. Therefore, the slope is 4/2 = 2. This means that for every increase of 1 in the x-value, the y-value increases by 2. So the rate of change is 2.
Comparing the two linear functions, linear function 2 has a greater rate of change because its slope is 2, while linear function 1 has a rate of change of 1.
graph: (-3,4) (0,3) (3,2) (6,1)
determine the initial value and the rate of change of the linear function as given in the graph, round the answer to three decimal places as needed
To determine the initial value and the rate of change from the graph, we can find the equation of the line represented by the given points using the slope-intercept form: y = mx + b, where m is the rate of change (slope) and b is the initial value (y-intercept).
Given the points (-3,4), (0,3), (3,2), and (6,1), we can find the slope by using the formula: slope = (change in y) / (change in x).
1. Between points (-3,4) and (0,3):
change in y = 3 - 4 = -1
change in x = 0 - (-3) = 3
slope = (-1) / 3 = -0.33333 (rounded to three decimal places)
2. Between points (0,3) and (3,2):
change in y = 2 - 3 = -1
change in x = 3 - 0 = 3
slope = (-1) / 3 = -0.33333 (rounded to three decimal places)
3. Between points (3,2) and (6,1):
change in y = 1 - 2 = -1
change in x = 6 - 3 = 3
slope = (-1) / 3 = -0.33333 (rounded to three decimal places)
The slope is the same for all three pair of points. This means that the rate of change is constant throughout the line.
To find the initial value (y-intercept), we can select any point on the line and substitute the values into the slope-intercept form: y = mx + b.
Using the point (-3,4):
4 = -0.33333 * (-3) + b
4 = 0.99999 + b
b = 3.00001 (rounded to three decimal places)
Therefore, the initial value (y-intercept) is approximately 3.000.
The linear function represented by the given graph is y = -0.333x + 3.000 (rounded to three decimal places).
graph: (-3,4) (0,3) (3,2) (6,1)
determine the initial value and the rate of change of the linear function as given in the graph, round the answer to three decimal places as needed
To determine the initial value and rate of change of the linear function from the given graph, we need to find the equation of the line that passes through these points.
The rate of change (slope) can be calculated using the formula: m = (change in y) / (change in x).
1. Between points (-3,4) and (0,3):
change in y = 3 - 4 = -1
change in x = 0 - (-3) = 3
slope = (-1) / 3 = -0.333 (rounded to three decimal places)
2. Between points (0,3) and (3,2):
change in y = 2 - 3 = -1
change in x = 3 - 0 = 3
slope = (-1) / 3 = -0.333 (rounded to three decimal places)
3. Between points (3,2) and (6,1):
change in y = 1 - 2 = -1
change in x = 6 - 3 = 3
slope = (-1) / 3 = -0.333 (rounded to three decimal places)
The slope is the same for all three pairs of points. Therefore, the rate of change is constant throughout the line.
Next, to find the initial value (y-intercept), we can use one of the known points (-3,4) and substitute it into the slope-intercept form: y = mx + b.
Using (-3,4):
4 = -0.333 * (-3) + b
4 = 0.999 + b
b ≈ 3.001 (rounded to three decimal places)
So, the initial value (y-intercept) is approximately 3.001.
Therefore, the linear function represented by the given graph can be written as:
y = -0.333x + 3.001 (rounded to three decimal places).
graph: (-3,4) (0,3) (3,2) (6,1)
determine the initial value and the rate of change of the linear function as given in the graph
To determine the initial value and rate of change of the linear function from the given graph, we'll use the formula y = mx + b, where m represents the rate of change (slope) and b represents the initial value (y-intercept).
1. We can find the rate of change (slope) by using any two points on the graph. Let's use the first and last points, (-3,4) and (6,1).
Rate of change = (change in y) / (change in x)
Rate of change = (1 - 4) / (6 - (-3))
Rate of change = -3 / 9
Rate of change = -1/3
2. To find the initial value (y-intercept), we can substitute one of the given points into the equation and solve for b. Let's use the point (0,3).
3 = (-1/3)(0) + b
3 = 0 + b
b = 3
Therefore, the linear function represented by the given graph can be written as:
y = (-1/3)x + 3
The initial value (y-intercept) is 3 and the rate of change is -1/3.