Which function represents the relationship shown in the table?
|x| 1 | 2 | 3 | 4 |
|y| 3 | 1 |-1 |-3 |
The answer is a equation
x1 = 1 , y1 = 3
x2 = 2 , y2 = 1
x3 = 3 , y3 = - 1
x4 = 4 , y4 = - 3
m = rate of change = change y / change x
m = ( y2 - y1 ) / ( x2 - x1 ) = ( 1 - 3 ) / ( 2 - 1 ) = - 2 / 1 = - 2
m = ( y3 - y2 ) / ( x3 - x2 ) = ( - 1 - 1 ) / ( 3 - 2 ) = - 2 / 1 = - 2
m = ( y4 - y3 ) / ( x4 - x3 ) = [ - 3 - ( - 1 ) ] / ( 4 - 3 ) = ( - 3 + 1 ) / 1 = - 2 / 1 = - 2
A linear function y = m x + b has a constant rate of change.
In this case:
x1 = 1 , y1 = 3
y = m x + b
3 = ( - 2 ) ∙ 1 + b
3 = - 2 + b
Add 2 to both sides
5 = b
b = 5
Your linear function y = m x + b is:
y = - 2 x + 5
You can check that:
x1 = 1 , y = - 2 ∙ 1 + 5 = - 2 + 5 = 3 , y1 = 3
x2 = 2 , y = - 2 ∙ 2 + 5 = - 4 + 5 = 1 , y2 = 1
x3 = 3 , y = - 2 ∙ 3 + 5 = - 6 + 5 = - 1 , y3 = - 1
x4 = 4 , y = - 2 ∙ 4 + 5 = - 8 + 5 = - 3 , y4 = - 3
So the answer is:
y = - 2 x + 5
To find the equation that represents the relationship shown in the table, we need to identify the pattern or rule that relates the values of x and y.
Looking at the table, we can observe that as x increases by 1, y decreases by 2. This indicates a linear relationship between the variables, where there is a constant decrease of 2 in y for every increase of 1 in x.
To express this relationship as an equation, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.
In this case, we know that the slope (m) is -2, since y decreases by 2 for every increase of 1 in x. Now, we need to determine the y-intercept (b).
To find the y-intercept, we can choose any point from the table. Let's use the point (1, 3), where x = 1 and y = 3.
Plugging these values into the slope-intercept form equation, we have:
3 = -2(1) + b
Simplifying:
3 = -2 + b
b = 3 + 2
b = 5
Therefore, the equation that represents the relationship shown in the table is y = -2x + 5.